STEM 隨筆︰鬼月談化學︰☲ 麗 《光明》‧階

有人

天下『一切』事情,都不過是個『分類』。

有人』就『有事』,天下何得『无事』?天下能得『無人』乎?

有人

酸葡萄『沒有』想『』,甜檸檬『』卻想『不要』;

都是一種『心理』。

有人

有所謂『』與『』? 分別著『』或『』,

』其所『』,『』其所『』。

有人

蜜蜂』為何不見了?

只因『』和『』!!

─── 《『蜜蜂』為何不見了?

 

即便講︰識物辨名始於分門別類。重要的還是門類之界定性徵。若從完備無餘來看 A \cdot z + B 變換形式,僅止於平面上的縮放、旋轉 、平移爾,實未得仿射變換之全也。

──── 《GOPIGO 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《變換‧B 》

 

『自然』不說話,『原理』如天書!天書本『無字』,辨名識物『科學』生?

莫問『軟體』為何沒『手冊』??

 

引君『好奇』自查明!!

Eyring equation

The Eyring equation (occasionally also known as Eyring–Polanyi equation) is an equation used in chemical kinetics to describe the variance of the rate of a chemical reaction with temperature. It was developed almost simultaneously in 1935 by Henry Eyring,Meredith Gwynne Evans and Michael Polanyi. This equation follows from the transition state theory (a.k.a. activated-complex theory) and (if one assumes constant enthalpy of activation and constant entropy of activation) is similar to the empirical Arrhenius equation, although the Arrhenius equation is empirical, and the Eyring equation has a statistical mechanical justification.

General form

The general form of the Eyring–Polanyi equation somewhat resembles the Arrhenius equation:

\displaystyle \ k={\frac {\kappa k_{\mathrm {B} }T}{h}}\mathrm {e} ^{-{\frac {\Delta G^{\ddagger }}{RT}}}

where ΔG is the Gibbs energy of activation, κ is the transmission coefficient, kB is Boltzmann’s constant, and h is Planck’s constant. The transmission coefficient is often assumed to be equal to one as it reflects what fraction of the flux through the transition state proceeds to the product without recrossing the transition state, so a transmission coefficient equal to one means that the fundamental no-recrossing assumption of transition state theory holds perfectly.

It can be rewritten as:

\displaystyle k={\frac {\kappa k_{\mathrm {B} }T}{h}}\mathrm {e} ^{\frac {\Delta S^{\ddagger }}{R}}\mathrm {e} ^{-{\frac {\Delta H^{\ddagger }}{RT}}}

One can put this equation in the following form:

\displaystyle \ln {\frac {k}{T}}={\frac {-\Delta H^{\ddagger }}{R}}\cdot {\frac {1}{T}}+\ln {\frac {\kappa k_{\mathrm {B} }}{h}}+{\frac {\Delta S^{\ddagger }}{R}}

where:

If one assumes constant enthalpy of activation, constant entropy of activation, and constant transmission coefficient, this equation can be used as follows: A certain chemical reaction is performed at different temperatures and the reaction rate is determined. The plot of  \displaystyle \ \ln(k/T) versus  \displaystyle \ 1/T gives a straight line with slope  \displaystyle \ -\Delta H^{\ddagger }/R from which the enthalpy of activation can be derived and with intercept  \displaystyle \ \ln(\kappa k_{\mathrm {B} }/h)+\Delta S^{\ddagger }/R from which the entropy of activation is derived.

Accuracy

Transition state theory requires a value of the transmission coefficient, called  \displaystyle \ \kappa in that theory. This value is often taken to be unity (i.e., the species passing through the transition state  \displaystyle \ AB^{\ddagger } always proceed directly to products  \displaystyle \ AB and never revert to reactants  \displaystyle \ A and  \displaystyle \ B ). To avoid specifying a value of  \displaystyle \ \kappa , the rate constant can be compared to the value of the rate constant at some fixed reference temperature (i.e.,  \displaystyle \ k(T)/k(T_{Ref}) ) which eliminates the  \displaystyle \ \kappa factor in the resulting expression if one assumes that the transmission coefficient is independent of temperature.

 

倘心已發願知『級數』︰

Rate equation

The rate law or rate equation for a chemical reaction is an equation that links the reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reaction orders).[1] For many reactions the rate is given by a power law such as

\displaystyle r\;=\;k[\mathrm {A} ]^{x}[\mathrm {B} ]^{y}

where [A] and [B] express the concentration of the species A and B (usually in moles per liter (molarity, M)). The exponents x and y are the partial orders of reaction for A and B and the overall reaction order is the sum of the exponents. These are often positive integers, but they may also be zero, fractional, or negative. The constant k is the reaction rate constant or rate coefficient of the reaction and has units of 1/time. Its value may depend on conditions such as temperature, ionic strength, surface area of an adsorbent, or light irradiation.

Elementary (single-step) reactions have reaction orders equal to the stoichiometric coefficients for each reactant. The overall reaction order, i.e. the sum of stoichiometric coefficients of reactants, is always equal to the molecularity of the elementary reaction. Complex (multi-step) reactions may or may not have reaction orders equal to their stoichiometric coefficients.

The rate equation of a reaction with an assumed multi-step mechanism can often be derived theoretically using quasi-steady state assumptions from the underlying elementary reactions, and compared with the experimental rate equation as a test of the assumed mechanism. The equation may involve a fractional order, and may depend on the concentration of an intermediate species.

A reaction can also have an undefined reaction order with respect to a reactant if the rate is not simply proportional to some power of the concentration of that reactant; for example, one cannot talk about reaction order in the rate equation for a bimolecular reaction between adsorbed molecules:

\displaystyle r=k{\frac {K_{1}K_{2}C_{A}C_{B}}{(1+K_{1}C_{A}+K_{2}C_{B})^{2}}}.

Determination of reaction order

The order of a reaction cannot be deduced from the chemical equation of the reaction. It must be determined by experiment.

Method of initial rates

The order of a reaction for each reactant can be estimated[2][3] from the variation in initial rate with the concentration of that reactant, using the natural logarithm of the typical rate equation

\displaystyle \ln r=\ln k+x\ln[A]+y\ln[B]+...

For example, the initial rate can be measured in a series of experiments at different initial concentrations of reactant A with all other concentrations [B], [C], … kept constant, so that

\displaystyle \ln r=x\ln[A]+{\textrm {constant}}

The slope of a graph of \displaystyle \ln r as a function of \displaystyle \ln[A] then corresponds to the order x with respect to reactant A.

However, this method is not always reliable because

  1. measurement of the initial rate requires accurate determination of small changes in concentration in short times (compared to the reaction half-life) and is sensitive to errors, and
  2. the rate equation will not be completely determined if the rate also depends on substances not present at the beginning of the reaction, such as intermediates or products.

Integral method

The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the concentrations measured over a longer time (several half-lives) with the integrated form of the rate equation.

For example, the integrated rate law for a first-order reaction is

\displaystyle \ \ln {[A]}=-kt+\ln {[A]_{0}} ,

where [A] is the concentration at time t and [A]0 is the initial concentration at zero time. The first-order rate law is confirmed if \displaystyle \ln {[A]} is in fact a linear function of time. In this case the rate constant \displaystyle k is equal to the slope with sign reversed.[4][5]

Method of flooding

The partial order with respect to a given reactant can be evaluated by the method of flooding (or of isolation) of Ostwald. In this method, the concentration of one reactant is measured with all other reactants in large excess so that their concentration remains essentially constant. For a reaction a·A + b·B → c·C with rate law: \displaystyle r=k\cdot [{\rm {A}}]^{\alpha }\cdot [{\rm {B}}]^{\beta } , the partial order α with respect to A is determined using a large excess of B. In this case

\displaystyle r=k'\cdot [{\rm {A}}]^{\alpha } with \displaystyle k'=k\cdot [{\rm {B}}]^{\beta },

and α may be determined by the integral method. The order β with respect to B under the same conditions (with B in excess) is determined by a series of similar experiments with a range of initial concentration [B]0 so that the variation of k’ can be measured.[6]

Zero order

For zero-order reactions, the reaction rate is independent of the concentration of a reactant, so that changing its concentration has no effect on the speed of the reaction. Thus, the concentration changes linearly with time. This may occur when there is a bottleneck which limits the number of reactant molecules that can react at the same time, for example if the reaction requires contact with an enzyme or a catalytic surface.[7]

Many enzyme-catalyzed reactions are zero order, provided that the reactant concentration is much greater than the enzyme concentration which controls the rate, so that the enzyme is saturated. For example, the biological oxidation of ethanol to acetaldehyde by the enzyme liver alcohol dehydrogenase (LADH) is zero order in ethanol.[8]

Similarly reactions with heterogeneous catalysis can be zero order if the catalytic surface is saturated. For example, the decomposition of phosphine (PH3) on a hot tungsten surface at high pressure is zero order in phosphine which decomposes at a constant rate.[7]

In homogeneous catalysis zero order behavior can come about from reversible inhibition. For example, ring-opening metathesis polymerization using third-generation Grubbs catalyst exhibits zero order behavior in catalyst due to the reversible inhibition that is occur between the pyridine and the ruthenium center.[9]

First order

A first order reaction depends on the concentration of only one reactant (a unimolecular reaction). Other reactants can be present, but each will be zero order. The rate law for such a reaction is

\frac {d[{\ce {A}}]}{dt}}=k[{\ce {A}}],

The half-life is independent of the starting concentration and is given by \displaystyle t_{\frac {1}{2}}={\frac {\ln {(2)}}{k}} .

Examples of such reactions are:

  • \displaystyle {\ce {H2O2 (l) -> {H2O (l)}+ 1/2O2 (g)}}
  • \displaystyle {\ce {SO2Cl2 (l) -> {SO2 (g)}+ Cl2 (g)}}
  • \displaystyle {\ce {2N2O5 (g) -> {4NO2 (g)}+ O2 (g)}}

In organic chemistry, the class of SN1 (nucleophilic substitution unimolecular) reactions consists of first-order reactions. For example, in the reaction of aryldiazonium ions with nucleophiles in aqueous solution ArN2+ + X → ArX + N2, the rate equation is r = k[ArN2+], where Ar indicates an aryl group.[10]

Second order

A reaction is said to be second order when the overall order is two. The rate of a second-order reaction may be proportional to one concentration squared \displaystyle r=k[A]^{2} , or (more commonly) to the product of two concentrations \displaystyle r=k[A][B] . As an example of the first type, the reaction NO2 + CO → NO + CO2 is second-order in the reactant NO2 and zero order in the reactant CO. The observed rate is given by \displaystyle r=k[{{\ce {NO2}}}]^{2} , and is independent of the concentration of CO.[11]

For the rate proportional to a single concentration squared, the time dependence of the concentration is given by

\displaystyle {\ce {{\frac {1}{[A]}}= {\frac {1}{[A]0}}+ kt}}

The time dependence for a rate proportional to two unequal concentrations is

\displaystyle {\ce {{\frac {[A]}{[B]}}={\frac {[A]0}{[B]0}}{\mathit {e}}^{([A]0-[B]0){\mathit {kt}}}}} ;

if the concentrations are equal, they satisfy the previous equation.

The second type includes nucleophillic addition-elimination reactions, such as the alkaline hydrolysis of ethyl acetate:[10]

CH3COOC2H5 + OH → CH3COO + C2H5OH

This reaction is first-order in each reactant and second-order overall: r = k[CH3COOC2H5][OH]

If the same hydrolysis reaction is catalyzed by imidazole, the rate equation becomes[10] r = k[imidazole][CH3COOC2H5]. The rate is first-order in one reactant (ethyl acetate), and also first-order in imidazole which as a catalyst does not appear in the overall chemical equation.

Another well-known class of second-order reactions are the SN2 (bimolecular nucleophilic substitution) reactions, such as the reaction of n-butyl bromide with sodium iodide in acetone:

CH3CH2CH2CH2Br + NaI → CH3CH2CH2CH2I + NaBr↓

This same compound can be made to undergo a bimolecular (E2) elimination reaction, another common type of second-order reaction, if the sodium iodide and acetone are replaced with sodium tert-butoxide as the salt and tert-butanol as the solvent:

CH3CH2CH2CH2Br + NaOt-Bu → CH3CH2CH=CH2 + NaBr + HOt-Bu

Pseudo-first order

If the concentration of a reactant remains constant (because it is a catalyst, or because it is in great excess with respect to the other reactants), its concentration can be included in the rate constant, obtaining a pseudo–first-order (or occasionally pseudo–second-order) rate equation. For a typical second-order reaction with rate equation r = k[A][B], if the concentration of reactant B is constant then r = k[A][B] = k'[A], where the pseudo–first-order rate constant k’ = k[B]. The second-order rate equation has been reduced to a pseudo–first-order rate equation, which makes the treatment to obtain an integrated rate equation much easier.

One way to obtain a pseudo-first order reaction is to use a large excess of one reactant (say, [B]≫[A]) so that, as the reaction progresses, only a small fraction of the reactant in excess (B) is consumed, and its concentration can be considered to stay constant. For example, the hydrolysis of esters by dilute mineral acids follows pseudo-first order kinetics where the concentration of water is present in large excess:

CH3COOCH3 + H2O → CH3COOH + CH3OH

The hydrolysis of sucrose in acid solution is often cited as a first-order reaction with rate r = k[sucrose]. The true rate equation is third-order, r = k[sucrose][H+][H2O]; however, the concentrations of both the catalyst H+ and the solvent H2O are normally constant, so that the reaction is pseudo–first-order.[12]

Summary for reaction orders 0, 1, 2, and n

Elementary reaction steps with order 3 (called ternary reactions) are rare and unlikely to occur. However, overall reactions composed of several elementary steps can, of course, be of any (including non-integer) order.

Where M stands for concentration in molarity (mol · L−1), t for time, and k for the reaction rate constant. The half-life of a first order reaction is often expressed as t1/2 = 0.693/k (as ln2 = 0.693).

Fractional order

In fractional order reactions, the order is a non-integer, which often indicates a chemical chain reaction or other complex reaction mechanism. For example, the pyrolysis of acetaldehyde (CH3CHO) into methane and carbon monoxide proceeds with an order of 1.5 with respect to acetaldehyde: r = k[CH3CHO]3/2.[14] The decomposition of phosgene (COCl2) to carbon monoxide and chlorine has order 1 with respect to phosgene itself and order 0.5 with respect to chlorine: r = k[COCl2] [Cl2]1/2.[15]

The order of a chain reaction can be rationalized using the steady state approximation for the concentration of reactive intermediates such as free radicals. For the pyrolysis of acetaldehyde, the Rice-Herzfeld mechanism is[14][16]

Initiation
CH3CHO → •CH3 + •CHO
Propagation
•CH3 + CH3CHO → CH3CO• + CH4
CH3CO• → •CH3 + CO
Termination
2 •CH3 → C2H6

where • denotes a free radical. To simplify the theory, the reactions of the •CHO to form a second •CH3 are ignored.

In the steady state, the rates of formation and destruction of methyl radicals are equal, so that

\displaystyle {\frac {d[{{\ce {.CH3}}}]}{dt}}=k_{i}[{{\ce {CH3CHO}}}]-k_{t}[{{\ce {.CH3}}}]^{2}=0 ,

so that the concentration of methyl radical satisfies

\displaystyle {\ce {[.CH3]\quad \propto \quad [CH3CHO]^{1/2}}} .

The reaction rate equals the rate of the propagation steps which form the main reaction products CH4 and CO:

\displaystyle v={\frac {d[{\ce {CH4}}]}{dt}}=k_{p}{\ce {[.CH3][CH3CHO]}}\quad \propto \quad {\ce {[CH3CHO]^{3/2}}}

in agreement with the experimental order of 3/2.[14][16]

Mixed order

More complex rate laws have been described as being mixed order if they approximate to the laws for more than one order at different concentrations of the chemical species involved. For example, a rate law of the form \displaystyle r=k_{1}[A]+k_{2}[A]^{2} represents concurrent first order and second order reactions (or more often concurrent pseudo-first order and second order) reactions, and can be described as mixed first and second order.[17] For sufficiently large values of [A] such a reaction will approximate second order kinetics, but for smaller [A] the kinetics will approximate first order (or pseudo-first order). As the reaction progresses, the reaction can change from second order to first order as reactant is consumed.

Another type of mixed-order rate law has a denominator of two or more terms, often because the identity of the rate-determining step depends on the values of the concentrations. An example is the oxidation of an alcohol to a ketone by hexacyanoferrate (III) ion [Fe(CN)63−] with ruthenate (VI) ion (RuO42−) as catalyst.[18] For this reaction, the rate of disappearance of hexacyanoferrate (III) is \displaystyle r={\frac {{\ce {[Fe(CN)6]^2-}}}{k_{\alpha }+k_{\beta }{\ce {[Fe(CN)6]^2-}}}}

This is zero-order with respect to hexacyanoferrate (III) at the onset of the reaction (when its concentration is high and the ruthenium catalyst is quickly regenerated), but changes to first-order when its concentration decreases and the regeneration of catalyst becomes rate-determining.

Notable mechanisms with mixed-order rate laws with two-term denominators include:

  • Michaelis-Menten kinetics for enzyme-catalysis: first-order in substrate (second-order overall) at low substrate concentrations, zero order in substrate (first-order overall) at higher substrate concentrations; and
  • the Lindemann mechanism for unimolecular reactions: second-order at low pressures, first-order at high pressures.

Negative order

A reaction rate can have a negative partial order with respect to a substance. For example, the conversion of ozone (O3) to oxygen follows the rate equation \displaystyle r=k{\frac {{\ce {[O_3]^2}}}{{\ce {[O_2]}}}} in an excess of oxygen. This corresponds to second order in ozone and order (-1) with respect to oxygen.[19]

When a partial order is negative, the overall order is usually considered as undefined. In the above example for instance, the reaction is not described as first order even though the sum of the partial orders is 2 + (−1) = 1, because the rate equation is more complex than that of a simple first-order reaction.

 

原始碼『解』,不登階!!??

 

 

 

 

 

 

 

STEM 隨筆︰鬼月談化學︰☲ 麗 《光明》‧史

中國傳統服飾-先秦天子冕服

中國結

玉珮

先秦天子冕服有許多象徵意義上衣、象以日夜,故為之青黑下裳,徵之富饒,用中透十二章紋述說一年,上下均分說著夏冬,其數實乃是周易的乾坤。古代上衣下裳的形制,因能方便耕作勞動,也就變成大眾傳統服飾了。

古之讀書人常有配戴玉珮』的習慣,這是因為──

禮記‧聘義

子貢問於孔子曰︰敢問君子貴玉賤媒者,何也?為與?孔子曰︰非為媒之多,故賤之也;玉之寡,故貴之也。夫昔者君子比德於玉焉,溫潤而澤、也,縝密以栗、也,廉而不劌、也,垂之如隊、也,叩之其聲清越以長其終詘然 、也,瑕不掩瑜、瑜不掩瑕、也,孚尹旁達、也,氣如白虹、也,精神見於山川、也,圭璋特達、也。天下莫不貴者、道也云︰言念君子.溫其如玉,故君子貴之也──。

……

周易經傳在數千年的歷史中,不知被『結構』和『解構』過多少次 ,這是為什麼呢?或許是川流不息生命之河,自有它亙古奧秘,也是永恆追求!!誰又能知有沒有『因明』的一天呢? ?如果『詮釋學』使山海經中『夸父追日』的精神,不再那麼飄渺迷茫,也許訴說著人類問著『為什麼』的童年;那中國歷史上最早的一部天文曆算著作《周髀算經》則是『好奇後』的老成,似乎已經揭示了日月星辰的運行規律、四季更替、氣候變化甚至包涵南北有極晝夜相推的道理。可是誰又能一直持有年少時那顆天真好奇之心呢?

假使一個人果能站在前人學問的基石上,又天真好奇孜孜不倦,那就會如孔子在《論語‧子罕》:

後生可畏焉知來者之不如今也。 四十、五十而無 ── ㄨㄣˊ陽關道 ──焉,斯亦不足畏也已。

,裡所說的一樣。甚至要能如下面所引的『一則故事』那樣

歐陽修,一向治學嚴謹,直至晚年,不減當初。他常將自己平生所寫的文章,清理出來進行修改,每字每句反覆推敲,甚是認真。為此,他整天辛苦勞累,有時直忙到深夜。夫人見他年歲已高,還如此盡心費神,恐其操勞過度,影響健康,十分擔心,目前制止。她關切地對丈夫說:『官人,何必如此用功,不惜貴體安康,為這些文字吃這樣多的苦頭,官人已年邁致仕(退休),難道還怕先生責難生氣嗎?』歐陽修回答說:『不怕先生生氣,只怕後生生譏』,『後生可畏耶!』

活到老學到老

── 英特乃者何耶?右尊讀之,是乃特英也,

特英者理念所鑄之祈立人願達人!! ──

─── 《後生可畏!?

 

『概念』實難憑空而生!『創見』偶在夢中出現?

當人們在『夢中』喃喃自語,人們真的說了些什麼嗎?若是講有人『記下』了『夢中』所憶︰

pi@raspberrypi ~ python3 Python 3.2.3 (default, Mar  1 2013, 11:53:50)  [GCC 4.6.3] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>>  >>>  >>> from pyDatalog import pyDatalog >>> pyDatalog.create_terms('T萬物, 有力, 長生') >>> +有力('聖人', '能力') >>> +有力('大盜', '能力') >>> 長生(T萬物, '難死') <= 有力(T萬物, '能力') 長生(T萬物,'難死') <= 有力(T萬物,'能力') >>> print(長生(T萬物,'難死')) T萬物 --- 聖人  大盜  >>> print(有力(T萬物,'能力')) T萬物 --- 大盜  聖人  >>> >>> pyDatalog.create_terms('有死') >>> pyDatalog.create_terms('定律') >>> 定律(T萬物, '會死') <= 有死(T萬物, '人') 定律(T萬物,'會死') <= 有死(T萬物,'人') >>> +有死('聖人', '人') >>> +有死('大盜', '人') >>> print(有死(T萬物,'人')) T萬物 --- 聖人  大盜  >>> print(定律(T萬物,'會死')) T萬物 --- 大盜  聖人  >>> >>> print((有死(T萬物, '人')) & ~(定律(T萬物, '會死')))  [] >>> >>> print((T萬物 == '聖人') & ~(定律(T萬物, '會死'))) [] >>> print((T萬物 == '大盜') & ~(定律(T萬物, '會死'))) [] >>> print((定律('聖人', '會死')) & ~(定律('大盜', '會死')))  [] >>></pre>    <span style="color: #008080;">那麼是否應該『設想』他可能『證明』了『聖人不死,大盜不止』的呢?然後在考察『邏輯』之後,『認同』他真的……的耶??</span>  <span style="color: #008080;">也許與『清醒』的人談論『真、假』以及『是、非』都未必可信,那又將怎麽說那『夢中』之事?真的有誰知此『夢』會不會是一個『夢中之夢』!還不曉那人哪時方將『醒來』的哩!!雖說你不能『證明』上帝『存在』,那你能說祂『不存在』的嗎?反之你不能『證明』上帝『不存在』,那你就能說祂『存在』的耶??此所以</span><span style="color: #ff9900;">《<a href="http://cls.hs.yzu.edu.tw/hlm/read/text/text.asp">紅樓夢</a>》講︰世間事終難定。</span><span style="color: #008080;">因為『<a style="color: #008080;" href="http://www.freesandal.org/?p=1969">時間矛盾</a>』早已經在那裡了 ,這又與『知或不知』有什麼關係的呢?於是『邏輯』與『宇宙』的關係到底是什麼??!!也許真正困惑的是『人世間』正在追求之『價值』的方向的吧!!??</span>  <span style="color: #008080;">有的人以疏落的觀點看待『語言』,認為『凡是可以詮釋的現象,都是大自然之言說』。如是一沙一石皆有所說,更別講鳥語花香,以至於『動物語言』的哩!這樣的人是否更容易了解『程式語言』的呢?也有人以嚴格之想法處理『語言』,認為『只有人類的語言才能稱得上言語』。因此海豚雖可溝通,卻不會講話,動物吼叫聲除了警示意謂,了無它意,若說到花草的榮枯根本毫無意義的勒!這樣的人是否更容易了解『程式語言』的嗎?那麼什麼是『語言』的呢?什麼又是『程式語言』的哩?假使給個『定義』是否就能將之釐定清楚,大家都講同家話的耶!考之於歷史,此事希望渺茫,難保不正因這種『多樣性』開拓了視野,加深了認識的嗎??也許還是多些『兩極對話』的好!!</span>  <span style="color: #808080;">─── 摘自《<a style="color: #808080;" href="http://www.freesandal.org/?p=37117">勇闖新世界︰ 《 PYDATALOG 》 導引《五》</a>》</span>     <span style="color: #666699;">所以孔子喜夢周公乎??一代大哲<a style="color: #666699;" href="http://www.freesandal.org/?p=37117">熊十力</a>斷言</span>  <span style="color: #ff99cc;">真積力,久則入。</span>  <span style="color: #666699;">也!!</span>  <span style="color: #666699;">如斯者,</span>  <span style="color: #666699;">可以讀『史』︰</span> <h1 id="firstHeading" class="firstHeading" lang="en"><span style="color: #008080;"><a style="color: #008080;" href="https://en.wikipedia.org/wiki/Law_of_mass_action">Law of mass action</a></span></h1> <span style="color: #008080;">In chemistry, the <b>law of mass action</b> is the proposition that the <a style="color: #008080;" title="Reaction rate" href="https://en.wikipedia.org/wiki/Reaction_rate">rate</a> of a <a style="color: #008080;" title="Chemical reaction" href="https://en.wikipedia.org/wiki/Chemical_reaction">chemical reaction</a> is directly proportional to the product of the <a class="mw-redirect" style="color: #008080;" title="Activity (chemistry)" href="https://en.wikipedia.org/wiki/Activity_(chemistry)">activities</a> or <a style="color: #008080;" title="Concentration" href="https://en.wikipedia.org/wiki/Concentration">concentrations</a> of the <a style="color: #008080;" title="Reagent" href="https://en.wikipedia.org/wiki/Reagent">reactants</a>.<sup id="cite_ref-ÉrdiTóth1989_1-0" class="reference"><a style="color: #008080;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-ÉrdiTóth1989-1">[1]</a></sup> It explains and predicts behaviors of <a class="mw-redirect" style="color: #008080;" title="Solutions" href="https://en.wikipedia.org/wiki/Solutions">solutions</a> in <a style="color: #008080;" title="Dynamic equilibrium" href="https://en.wikipedia.org/wiki/Dynamic_equilibrium">dynamic equilibrium</a>. Specifically, it implies that for a chemical reaction mixture that is in equilibrium, the ratio between the concentration of reactants and <a style="color: #008080;" title="Product (chemistry)" href="https://en.wikipedia.org/wiki/Product_(chemistry)">products</a> is constant.<sup id="cite_ref-uwaterloo_cact_2-0" class="reference"><a style="color: #008080;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-uwaterloo_cact-2">[2]</a></sup></span>  <span style="color: #008080;">Two aspects are involved in the initial formulation of the law: 1) the equilibrium aspect, concerning the composition of a <a style="color: #008080;" title="Chemical reaction" href="https://en.wikipedia.org/wiki/Chemical_reaction">reaction</a> mixture at <a style="color: #008080;" title="Chemical equilibrium" href="https://en.wikipedia.org/wiki/Chemical_equilibrium">equilibrium</a> and 2) the <a style="color: #008080;" title="Chemical kinetics" href="https://en.wikipedia.org/wiki/Chemical_kinetics">kinetic</a> aspect concerning the <a style="color: #008080;" title="Rate equation" href="https://en.wikipedia.org/wiki/Rate_equation">rate equations</a> for <a style="color: #008080;" title="Elementary reaction" href="https://en.wikipedia.org/wiki/Elementary_reaction">elementary reactions</a>. Both aspects stem from the research performed by <a style="color: #008080;" title="Cato Maximilian Guldberg" href="https://en.wikipedia.org/wiki/Cato_Maximilian_Guldberg">Cato M. Guldberg</a> and <a style="color: #008080;" title="Peter Waage" href="https://en.wikipedia.org/wiki/Peter_Waage">Peter Waage</a> between 1864 and 1879 in which equilibrium constants were derived by using kinetic data and the rate equation which they had proposed. Guldberg and Waage also recognized that chemical equilibrium is a dynamic process in which <a style="color: #008080;" title="Reaction rate" href="https://en.wikipedia.org/wiki/Reaction_rate">rates of reaction</a> for the forward and backward reactions must be equal at <a style="color: #008080;" title="Chemical equilibrium" href="https://en.wikipedia.org/wiki/Chemical_equilibrium">chemical equilibrium</a>. In order to derive the expression of the equilibrium constant appealing to kinetics, the expression of the rate equation must be used. The expression of the rate equations was rediscovered later independently by <a style="color: #008080;" title="Jacobus Henricus van 't Hoff" href="https://en.wikipedia.org/wiki/Jacobus_Henricus_van_%27t_Hoff">Jacobus Henricus van 't Hoff</a>.</span>  <span style="color: #008080;">The law is a statement about equilibrium and gives an expression for the <a style="color: #008080;" title="Equilibrium constant" href="https://en.wikipedia.org/wiki/Equilibrium_constant">equilibrium constant</a>, a quantity characterizing <a style="color: #008080;" title="Chemical equilibrium" href="https://en.wikipedia.org/wiki/Chemical_equilibrium">chemical equilibrium</a>. In modern chemistry this is derived using <a style="color: #008080;" title="Equilibrium thermodynamics" href="https://en.wikipedia.org/wiki/Equilibrium_thermodynamics">equilibrium thermodynamics</a>.</span> <h2><span id="History" class="mw-headline" style="color: #339966;">History</span></h2> <span style="color: #339966;">Two chemists generally expressed the composition of a mixture in terms of numerical values relating the amount of the product to describe the equilibrium state. <a style="color: #339966;" title="Cato Maximilian Guldberg" href="https://en.wikipedia.org/wiki/Cato_Maximilian_Guldberg">Cato Maximilian Guldberg</a> and <a style="color: #339966;" title="Peter Waage" href="https://en.wikipedia.org/wiki/Peter_Waage">Peter Waage</a>, building on <a class="mw-redirect" style="color: #339966;" title="Berthollet" href="https://en.wikipedia.org/wiki/Berthollet">Claude Louis Berthollet</a>'s ideas<sup id="cite_ref-3" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-3">[3]</a></sup><sup id="cite_ref-Levere_4-0" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-Levere-4">[4]</a></sup> about<a style="color: #339966;" title="Chemical equilibrium" href="https://en.wikipedia.org/wiki/Chemical_equilibrium">reversible chemical reactions</a>, proposed the law of mass action in 1864.<sup id="cite_ref-GW1_5-0" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW1-5">[5]</a></sup><sup id="cite_ref-GW2_6-0" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW2-6">[6]</a></sup><sup id="cite_ref-GW3_7-0" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW3-7">[7]</a></sup> These papers, in Danish, went largely unnoticed, as did the later publication (in French) of 1867 which contained a modified law and the experimental data on which that law was based.<sup id="cite_ref-8" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-8">[8]</a></sup><sup id="cite_ref-GW4_9-0" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW4-9">[9]</a></sup></span>  <span style="color: #339966;">In 1877 <a style="color: #339966;" title="Jacobus Henricus van 't Hoff" href="https://en.wikipedia.org/wiki/Jacobus_Henricus_van_%27t_Hoff">van 't Hoff</a> independently came to similar conclusions,<sup id="cite_ref-10" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-10">[10]</a></sup> but was unaware of the earlier work, which prompted Guldberg and Waage to give a fuller and further developed account of their work, in German, in 1879.<sup id="cite_ref-GW5_11-0" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW5-11">[11]</a></sup> Van 't Hoff then accepted their priority.</span> <h3><span id="1864" class="mw-headline" style="color: #339966;">1864</span></h3> <h4><span style="color: #339966;"><span id="The_equilibrium_state_.28composition.29"></span><span id="The_equilibrium_state_(composition)" class="mw-headline">The equilibrium state (composition)</span></span></h4> <span style="color: #339966;">In their first paper,<sup id="cite_ref-GW1_5-1" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW1-5">[5]</a></sup> Guldberg and Waage suggested that in a reaction such as</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle {\ce {{A}+ B <=> {A'}+ B'}}</span></span></dd> </dl> <span style="color: #339966;">the "chemical affinity" or "reaction force" between A and B did not just depend on the chemical nature of the reactants, as had previously been supposed, but also depended on the amount of each reactant in a reaction mixture. Thus the Law of Mass Action was first stated as follows:</span> <dl>  	<dd><span style="color: #339966;">When two reactants, A and B, react together at a given temperature in a "substitution reaction," the affinity, or chemical force between them, is proportional to the active masses, [A] and [B], each raised to a particular power</span> <dl>  	<dd><span style="color: #339966;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle {\text{affinity}}=\alpha [{{\ce {A}}}]^{a}[{{\ce {B}}}]^{b} </span></span>.</span></dd> </dl> </dd> </dl> <span style="color: #339966;">In this context a substitution reaction was one such as\displaystyle {\ce {{alcohol}+ acid <=> {ester}+ water}} . Active mass was defined in the 1879 paper as "the amount of substance in the sphere of action".<sup id="cite_ref-12" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-12">[12]</a></sup> For species in solution active mass is equal to concentration. For solids, active mass is taken as a constant.\displaystyle \alpha , a and b were regarded as empirical constants, to be determined by experiment.</span>  <span style="color: #339966;">At <a style="color: #339966;" title="Chemical equilibrium" href="https://en.wikipedia.org/wiki/Chemical_equilibrium">equilibrium</a>, the chemical force driving the forward reaction must be equal to the chemical force driving the reverse reaction. Writing the initial active masses of A,B, A' and B' as p, q, p' and q' and the dissociated active mass at equilibrium as\displaystyle \xi  , this equality is represented by</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \alpha (p-\xi )^{a}(q-\xi )^{b}=\alpha '(p'+\xi )^{a'}(q'+\xi )^{b'}</span></span></dd> </dl> <span style="color: #339966;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \xi  </span></span>represents the amount of reagents A and B that has been converted into A' and B'. Calculations based on this equation are reported in the second paper.<sup id="cite_ref-GW2_6-1" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW2-6">[6]</a></sup></span> <h4><span id="Dynamic_approach_to_the_equilibrium_state" class="mw-headline" style="color: #339966;">Dynamic approach to the equilibrium state</span></h4> <span style="color: #339966;">The third paper of 1864<sup id="cite_ref-GW3_7-1" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW3-7">[7]</a></sup> was concerned with the kinetics of the same equilibrium system. Writing the dissociated active mass at some point in time as x, the rate of reaction was given as</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \left({\frac {dx}{dt}}\right)_{forward}=k(p-x)^{a}(q-x)^{b}</span></span></dd> </dl> <span style="color: #339966;">Likewise the reverse reaction of A' with B' proceeded at a rate given by</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \left({\frac {dx}{dt}}\right)_{reverse}=k'(p'+x)^{a'}(q'+x)^{b'}</span></span></dd> </dl> <span style="color: #339966;">The overall rate of conversion is the difference between these rates, so at equilibrium (when the composition stops changing) the two rates of reaction must be equal. Hence</span> <dl>  	<dd><span style="color: #339966;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle (p-x)^{a}(q-x)^{b}={\frac {k'}{k}}(p'+x)^{a'}(q'+x)^{b'} </span></span>...</span></dd> </dl> <h3><span id="1867" class="mw-headline" style="color: #339966;">1867</span></h3> <span style="color: #339966;">The rate expressions given in the 1864 paper could not be differentiated, so they were simplified as follows.<sup id="cite_ref-GW4_9-1" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW4-9">[9]</a></sup> The chemical force was assumed to be directly proportional to the product of the active masses of the reactants.</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle {\mbox{affinity}}=\alpha [A][B]</span></span></dd> </dl> <span style="color: #339966;">This is equivalent to setting the exponents a and b of the earlier theory to one. The proportionality constant was called an affinity constant, k. The equilibrium condition for an "ideal" reaction was thus given the simplified form</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle k[A]_{\text{eq}}[B]_{\text{eq}}=k'[A']_{\text{eq}}[B']_{\text{eq}}</span></span></dd> </dl> <span style="color: #339966;">[A]<sub>eq</sub>, [B]<sub>eq</sub> etc. are the active masses at equilibrium. In terms of the initial amounts reagents p,q etc. this becomes</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle (p-\xi )(q-\xi )={\frac {k'}{k}}(p'+\xi )(q'+\xi )</span></span></dd> </dl> <span style="color: #339966;">The ratio of the affinity coefficients, k'/k, can be recognized as an equilibrium constant. Turning to the kinetic aspect, it was suggested that the velocity of reaction, v, is proportional to the sum of chemical affinities (forces). In its simplest form this results in the expression</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle v=\psi (k(p-x)(q-x)-k'(p'+x)(q'+x))</span></span></dd> </dl> <span style="color: #339966;">where\displaystyle \psi  is the proportionality constant. Actually, Guldberg and Waage used a more complicated expression which allowed for interaction between A and A', etc. By making certain simplifying approximations to those more complicated expressions, the rate equation could be integrated and hence the equilibrium quantity\displaystyle \xi  could be calculated. The extensive calculations in the 1867 paper gave support to the simplified concept, namely,</span> <dl>  	<dd><span style="color: #339966;">The rate of a reaction is proportional to the product of the active masses of the reagents involved.</span></dd> </dl> <span style="color: #339966;">This is an alternative statement of the Law of Mass Action.</span> <h3><span id="1879" class="mw-headline" style="color: #339966;">1879</span></h3> <span style="color: #339966;">In the 1879 paper<sup id="cite_ref-GW5_11-1" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-GW5-11">[11]</a></sup> the assumption that reaction rate was proportional to the product of concentrations was justified microscopically in terms of <a style="color: #339966;" title="Collision theory" href="https://en.wikipedia.org/wiki/Collision_theory">collision theory</a>, as had been developed for gas reactions. It was also proposed that the original theory of the equilibrium condition could be generalised to apply to any arbitrary chemical equilibrium.</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle {\text{affinity}}=k[{{\ce {A}}}]^{\alpha }[{{\ce {B}}}]^{\beta }\dots</span></span></dd> </dl> <span style="color: #339966;">The exponents α, β etc. are explicitly identified for the first time as the <a style="color: #339966;" title="Stoichiometry" href="https://en.wikipedia.org/wiki/Stoichiometry">stoichiometric coefficients</a> for the reaction.</span> <h2><span id="Modern_statement_of_the_law" class="mw-headline" style="color: #339966;">Modern statement of the law</span></h2> <span style="color: #339966;">The affinity constants, k<sub>+</sub> and k<sub>−</sub>, of the 1879 paper can now be recognised as <a class="mw-redirect" style="color: #339966;" title="Rate constant" href="https://en.wikipedia.org/wiki/Rate_constant">rate constants</a>. The equilibrium constant, K, was derived by setting the rates of forward and backward reactions to be equal. This also meant that the chemical affinities for the forward and backward reactions are equal. The resultant expression</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle K={\frac {[S]^{\sigma }[T]^{\tau }\dots }{[A]^{\alpha }[B]^{\beta }\dots }}</span></span></dd> </dl> <span style="color: #339966;">is correct<sup id="cite_ref-uwaterloo_cact_2-1" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-uwaterloo_cact-2">[2]</a></sup> even from the modern perspective, apart from the use of concentrations instead of activities (the concept of chemical activity was developed by <a style="color: #339966;" title="Josiah Willard Gibbs" href="https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs">Josiah Willard Gibbs</a>, in the 1870s, but was not <a style="color: #339966;" title="Josiah Willard Gibbs" href="https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs#Scientific_recognition">widely known</a> in Europe until the 1890s). The derivation from the reaction rate expressions is no longer considered to be valid. Nevertheless, Guldberg and Waage were on the right track when they suggested that the driving force for both forward and backward reactions is equal when the mixture is at equilibrium. The term they used for this force was chemical affinity. Today the expression for the equilibrium constant is derived by setting the <a style="color: #339966;" title="Chemical potential" href="https://en.wikipedia.org/wiki/Chemical_potential">chemical potential</a> of forward and backward reactions to be equal. The generalisation of the Law of Mass Action, in terms of affinity, to equilibria of arbitrary stoichiometry was a bold and correct conjecture.</span>  <span style="color: #339966;">The hypothesis that reaction rate is proportional to reactant concentrations is, strictly speaking, only true for <a style="color: #339966;" title="Elementary reaction" href="https://en.wikipedia.org/wiki/Elementary_reaction">elementary reactions</a> (reactions with a single mechanistic step), but the empirical rate expression</span> <dl>  	<dd><span class="mwe-math-element" style="color: #339966;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle r_{f}=k_{f}[A][B]</span></span></dd> </dl> <span style="color: #339966;">is also applicable to <a style="color: #339966;" title="Rate equation" href="https://en.wikipedia.org/wiki/Rate_equation">second order</a> reactions that may not be concerted reactions. Guldberg and Waage were fortunate in that reactions such as ester formation and hydrolysis, on which they originally based their theory, do indeed follow this rate expression.</span>  <span style="color: #339966;">In general many reactions occur with the formation of reactive intermediates, and/or through parallel reaction pathways. However, all reactions can be represented as a series of elementary reactions and, if the mechanism is known in detail, the rate equation for each individual step is given by the\displaystyle r_{f} expression so that the overall rate equation can be derived from the individual steps. When this is done the equilibrium constant is obtained correctly from the rate equations for forward and backward reaction rates.</span>  <span style="color: #339966;">In biochemistry, there has been significant interest in the appropriate mathematical model for chemical reactions occurring in the intracellular medium. This is in contrast to the initial work done on chemical kinetics, which was in simplified systems where reactants were in a relatively dilute, pH-buffered, aqueous solution. In more complex environments, where bound particles may be prevented from disassociation by their surroundings, or diffusion is slow or anomalous, the model of mass action does not always describe the behavior of the reaction kinetics accurately. Several attempts have been made to modify the mass action model, but consensus has yet to be reached. Popular modifications replace the rate constants with functions of time and concentration. As an alternative to these mathematical constructs, one school of thought is that the mass action model can be valid in intracellular environments under certain conditions, but with different rates than would be found in a dilute, simple environment<sup class="noprint Inline-Template Template-Fact">[<i><a style="color: #339966;" title="Wikipedia:Citation needed" href="https://en.wikipedia.org/wiki/Wikipedia:Citation_needed"><span title="This claim needs references to reliable sources. (July 2007)">citation needed</span></a></i>]</sup>.</span>  <span style="color: #339966;">The fact that Guldberg and Waage developed their concepts in steps from 1864 to 1867 and 1879 has resulted in much confusion in the literature as to which equation the Law of Mass Action refers. It has been a source of some textbook errors.<sup id="cite_ref-13" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-13">[13]</a></sup> Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula, <sup id="cite_ref-14" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-14">[14]</a></sup> <sup id="cite_ref-15" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-15">[15]</a></sup> <sup id="cite_ref-16" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-16">[16]</a></sup> <sup id="cite_ref-17" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-17">[17]</a></sup> <sup id="cite_ref-18" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-18">[18]</a></sup> <sup id="cite_ref-19" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-19">[19]</a></sup> <sup id="cite_ref-20" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-20">[20]</a></sup> <sup id="cite_ref-21" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-21">[21]</a></sup> <sup id="cite_ref-22" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-22">[22]</a></sup> <sup id="cite_ref-23" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-23">[23]</a></sup> and at other times to the (usually incorrect)\displaystyle r_{f} rate formula. <sup id="cite_ref-24" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-24">[24]</a></sup> <sup id="cite_ref-25" class="reference"><a style="color: #339966;" href="https://en.wikipedia.org/wiki/Law_of_mass_action#cite_note-25">[25]</a></sup></span>     <span style="color: #666699;">可以觀『鹽』自得︰</span>  <span style="font-size: 18pt;"><strong><a style="color: #ff9900;" href="https://zh.wikipedia.org/zh-tw/%E6%BA%B6%E8%A7%A3%E5%B9%B3%E8%A1%A1">溶解平衡</a></strong></span>  <span style="color: #ff9900;"><b>溶解平衡</b>是一種關於<a style="color: #ff9900;" title="化合物" href="https://zh.wikipedia.org/wiki/%E5%8C%96%E5%90%88%E7%89%A9">化合物</a><a style="color: #ff9900;" title="溶解" href="https://zh.wikipedia.org/wiki/%E6%BA%B6%E8%A7%A3">溶解</a>的<a style="color: #ff9900;" title="化學平衡" href="https://zh.wikipedia.org/wiki/%E5%8C%96%E5%AD%A6%E5%B9%B3%E8%A1%A1">化學平衡</a>。溶解平衡能作用於化合物的應用,並且可以用於預測特定情況下化合物的<a style="color: #ff9900;" title="溶解性" href="https://zh.wikipedia.org/wiki/%E6%BA%B6%E8%A7%A3%E6%80%A7">溶解度</a>。</span>  <span style="color: #ff9900;">溶解的固體可以是<a class="mw-redirect" style="color: #ff9900;" title="共價化合物" href="https://zh.wikipedia.org/wiki/%E5%85%B1%E4%BB%B7%E5%8C%96%E5%90%88%E7%89%A9">共價化合物</a>(<a style="color: #ff9900;" title="有機化合物" href="https://zh.wikipedia.org/wiki/%E6%9C%89%E6%9C%BA%E5%8C%96%E5%90%88%E7%89%A9">有機化合物</a>:<a style="color: #ff9900;" title="糖類" href="https://zh.wikipedia.org/wiki/%E7%B3%96%E7%B1%BB">糖</a>和<a style="color: #ff9900;" title="無機化合物" href="https://zh.wikipedia.org/wiki/%E6%97%A0%E6%9C%BA%E5%8C%96%E5%90%88%E7%89%A9">無機化合物</a>:<a style="color: #ff9900;" title="氯化氫" href="https://zh.wikipedia.org/wiki/%E6%B0%AF%E5%8C%96%E6%B0%A2">氯化氫</a>)或<a style="color: #ff9900;" title="離子化合物" href="https://zh.wikipedia.org/wiki/%E7%A6%BB%E5%AD%90%E5%8C%96%E5%90%88%E7%89%A9">離子化合物</a>(如<a style="color: #ff9900;" title="食鹽" href="https://zh.wikipedia.org/wiki/%E9%A3%9F%E7%9B%90">食鹽</a>,即<a style="color: #ff9900;" title="氯化鈉" href="https://zh.wikipedia.org/wiki/%E6%B0%AF%E5%8C%96%E9%92%A0">氯化鈉</a>),它們溶解時的主要區別是離子化合物會在溶於<a style="color: #ff9900;" title="水" href="https://zh.wikipedia.org/wiki/%E6%B0%B4">水</a>時<a style="color: #ff9900;" title="電離" href="https://zh.wikipedia.org/wiki/%E7%94%B5%E7%A6%BB">電離</a>為<a style="color: #ff9900;" title="離子" href="https://zh.wikipedia.org/wiki/%E7%A6%BB%E5%AD%90">離子</a>(部分共價化合物亦可,如<a class="mw-redirect" style="color: #ff9900;" title="醋酸" href="https://zh.wikipedia.org/wiki/%E9%86%8B%E9%85%B8">醋酸</a>、<a style="color: #ff9900;" title="氯化氫" href="https://zh.wikipedia.org/wiki/%E6%B0%AF%E5%8C%96%E6%B0%A2">氯化氫</a>、<a style="color: #ff9900;" title="硝酸" href="https://zh.wikipedia.org/wiki/%E7%A1%9D%E9%85%B8">硝酸</a>、<a class="mw-redirect" style="color: #ff9900;" title="醋酸鉛" href="https://zh.wikipedia.org/wiki/%E9%86%8B%E9%85%B8%E9%93%85">醋酸鉛</a>等)。水是最常用的<a style="color: #ff9900;" title="溶劑" href="https://zh.wikipedia.org/wiki/%E6%BA%B6%E5%89%82">溶劑</a>,但同樣的原則適用於任何溶劑。</span>  <span style="color: #ff9900;">在<a style="color: #ff9900;" title="環境科學" href="https://zh.wikipedia.org/wiki/%E7%8E%AF%E5%A2%83%E7%A7%91%E5%AD%A6">環境科學</a>中,溶解在水中的全部固體物質(無論是否達到<a class="mw-redirect" style="color: #ff9900;" title="飽和溶液" href="https://zh.wikipedia.org/wiki/%E9%A5%B1%E5%92%8C%E6%BA%B6%E6%B6%B2">飽和</a>)的<a style="color: #ff9900;" title="濃度" href="https://zh.wikipedia.org/wiki/%E6%B5%93%E5%BA%A6">濃度</a>被稱為<a style="color: #ff9900;" title="總溶解固體" href="https://zh.wikipedia.org/wiki/%E6%80%BB%E6%BA%B6%E8%A7%A3%E5%9B%BA%E4%BD%93">總溶解固體</a>(<a class="mw-redirect" style="color: #ff9900;" title="TDS" href="https://zh.wikipedia.org/wiki/TDS">TDS</a>)。</span>  <img class="alignnone size-full wp-image-89729" src="http://www.freesandal.org/wp-content/uploads/220px-SaltInWaterSolutionLiquid.jpg" alt="" width="220" height="417" />  <span style="color: #cc99ff;">食鹽(氯化鈉)易溶於水</span> <h2><span id="非离子化合物" class="mw-headline" style="color: #808080;">非離子化合物</span></h2> <span style="color: #808080;">有機固體的溶解平衡是其固態部分與溶解部分之間的平衡:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \mathrm {{C}_{12}{H}_{22}{O}_{11}(s)} \rightleftharpoons \mathrm {{C}_{12}{H}_{22}{O}_{11}(aq)}</span></span></dd> </dl> <span style="color: #808080;">而<a class="new" style="color: #808080;" title="平衡表達式(頁面不存在)" href="https://zh.wikipedia.org/w/index.php?title=%E5%B9%B3%E8%A1%A1%E8%A1%A8%E8%BE%BE%E5%BC%8F&action=edit&redlink=1">平衡表達式</a>可以如下所寫(這適用於任何此類反應):</span> <dl>  	<dd><span style="color: #808080;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle K={\frac {\left\{\mathrm {{C}_{12}{H}_{22}{O}_{11}} (aq)\right\}}{\left\{\mathrm {{C}_{12}{H}_{22}{O}_{11}} (s)\right\}}}</span></span><sup id="cite_ref-Atkins_1-0" class="reference"><a style="color: #808080;" href="https://zh.wikipedia.org/zh-tw/%E6%BA%B6%E8%A7%A3%E5%B9%B3%E8%A1%A1#cite_note-Atkins-1">[1]</a></sup></span></dd> </dl> <span style="color: #808080;">K 是<a style="color: #808080;" title="平衡常數" href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E8%A1%A1%E5%B8%B8%E6%95%B0">平衡常數</a>,花括號代表相應物質的<a class="mw-redirect" style="color: #808080;" title="活度" href="https://zh.wikipedia.org/wiki/%E6%B4%BB%E5%BA%A6">活度</a>,而根據定義,固體物質的活度是1。如果<a class="new" style="color: #808080;" title="離子氛(頁面不存在)" href="https://zh.wikipedia.org/w/index.php?title=%E7%A6%BB%E5%AD%90%E6%B0%9B&action=edit&redlink=1">離子氛</a>之間的作用可以忽略(一種常見的情形是溶液的濃度極低時),則活度也可用<a style="color: #808080;" title="濃度" href="https://zh.wikipedia.org/wiki/%E6%B5%93%E5%BA%A6">濃度</a>代替:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle K_{s}=\left[\mathrm {{C}_{12}{H}_{22}{O}_{11}} (aq)\right]</span></span></dd> </dl> <span style="color: #808080;">方括號代表<a class="mw-redirect" style="color: #808080;" title="摩爾濃度" href="https://zh.wikipedia.org/wiki/%E6%91%A9%E5%B0%94%E6%B5%93%E5%BA%A6">摩爾濃度</a>(通常用M表示)。</span>  <span style="color: #808080;">這個表達式是說在達到溶解平衡時,水中含有的已溶解的糖的濃度等於K。在 25℃時,當標準濃度為1mol/L時,蔗糖的K=1.971,這是在 25℃時能溶解的蔗糖的最大量,這時的溶液被稱為「飽和的」 。如果當前溶液濃度低於<a class="new" style="color: #808080;" title="飽和濃度(頁面不存在)" href="https://zh.wikipedia.org/w/index.php?title=%E9%A5%B1%E5%92%8C%E6%B5%93%E5%BA%A6&action=edit&redlink=1">飽和濃度</a>,固體會繼續溶解直到兩者相等或所有固體均已經溶解;如果當前溶液濃度高於飽和濃度,這時的溶液是「過飽和的」,溶液中的蔗糖將會以固體形式析出,直到兩者相等。這個過程可能是緩慢的,但是平衡常數描述的是體系平衡時的狀態,不是體系達到平衡的速度。</span> <h2><span style="color: #808080;"><span id=".E7.A6.BB.E5.AD.90.E5.8C.96.E5.90.88.E7.89.A9"></span><span id="离子化合物" class="mw-headline">離子化合物</span></span></h2> <span style="color: #808080;"><a style="color: #808080;" title="離子化合物" href="https://zh.wikipedia.org/wiki/%E7%A6%BB%E5%AD%90%E5%8C%96%E5%90%88%E7%89%A9">離子化合物</a>在<a style="color: #808080;" title="溶解" href="https://zh.wikipedia.org/wiki/%E6%BA%B6%E8%A7%A3">溶解</a>時通常會發生<a style="color: #808080;" title="電離" href="https://zh.wikipedia.org/wiki/%E7%94%B5%E7%A6%BB">電離</a>,即在<a style="color: #808080;" title="水" href="https://zh.wikipedia.org/wiki/%E6%B0%B4">水</a>的作用下<a class="mw-redirect" style="color: #808080;" title="解離" href="https://zh.wikipedia.org/wiki/%E8%A7%A3%E9%9B%A2">解離</a>為<a style="color: #808080;" title="離子" href="https://zh.wikipedia.org/wiki/%E7%A6%BB%E5%AD%90">離子</a> 。例如<a style="color: #808080;" title="硫酸鈣" href="https://zh.wikipedia.org/wiki/%E7%A1%AB%E9%85%B8%E9%92%99">硫酸鈣</a>:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \mathrm {CaSO} _{4}(s)\rightleftharpoons {\mbox{Ca}}^{2+}(aq)+{\mbox{SO}}_{4}^{2-}(aq)</span></span></dd> </dl> <span style="color: #808080;">對上例而言,<a class="new" style="color: #808080;" title="平衡表達式(頁面不存在)" href="https://zh.wikipedia.org/w/index.php?title=%E5%B9%B3%E8%A1%A1%E8%A1%A8%E8%BE%BE%E5%BC%8F&action=edit&redlink=1">平衡表達式</a>為:</span> <dl>  	<dd><span style="color: #808080;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle K={\frac {\left\{{\mbox{Ca}}^{2+}(aq)\right\}\left\{{\mbox{SO}}_{4}^{2-}(aq)\right\}}{\left\{{\mbox{CaSO}}_{4}(s)\right\}}}</span></span><sup id="cite_ref-Atkins_1-1" class="reference"><a style="color: #808080;" href="https://zh.wikipedia.org/zh-tw/%E6%BA%B6%E8%A7%A3%E5%B9%B3%E8%A1%A1#cite_note-Atkins-1">[1]</a></sup></span></dd> </dl> <span style="color: #808080;">K 被稱作<a style="color: #808080;" title="平衡常數" href="https://zh.wikipedia.org/wiki/%E5%B9%B3%E8%A1%A1%E5%B8%B8%E6%95%B0">平衡常數</a>,而花括號代表<a class="mw-redirect" style="color: #808080;" title="活度" href="https://zh.wikipedia.org/wiki/%E6%B4%BB%E5%BA%A6">活度</a>。固態物質的活度,根據定義,等於 1。當<a style="color: #808080;" title="溶液" href="https://zh.wikipedia.org/wiki/%E6%BA%B6%E6%B6%B2">溶液</a>的<a style="color: #808080;" title="濃度" href="https://zh.wikipedia.org/wiki/%E6%B5%93%E5%BA%A6">濃度</a>極低,即離子的活度可以看做 1時,這個表達式可以改寫為以下的「<a style="color: #808080;" title="溶度積" href="https://zh.wikipedia.org/wiki/%E6%BA%B6%E5%BA%A6%E7%A7%AF">溶度積</a>」表達式:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle K_{\mathrm {sp} }=\left[{\mbox{Ca}}^{2+}(aq)\right]\left[{\mbox{SO}}_{4}^{2-}(aq)\right].</span></span></dd> </dl> <span style="color: #808080;">這個表達式說明了硫酸鈣的水溶液達平衡時,由硫酸鈣電離出的兩種<a class="new" style="color: #808080;" title="離子濃度(頁面不存在)" href="https://zh.wikipedia.org/w/index.php?title=%E7%A6%BB%E5%AD%90%E6%B5%93%E5%BA%A6&action=edit&redlink=1">離子濃度</a>的乘積等於 K<sub>sp</sub>,即溶度積。硫酸鈣的溶度積為4.93×10<sup>−5</sup>。如果溶液中只含硫酸鈣,即只含由其電離出的 Ca<sup>2+</sup>和 SO<sub>4</sub><sup>2−</sup>,那麼每種離子的濃度為:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle {\sqrt {K_{\mathrm {sp} }}}={\sqrt {4.93\times 10^{-5}}}=7.02\times 10^{-3}=\left[{\mbox{Ca}}^{2+}\right]=\left[{\mbox{SO}}_{4}^{2-}\right].</span></span></dd> </dl> <span style="color: #808080;">當一種溶質電離為<a class="new" style="color: #808080;" title="計量數(頁面不存在)" href="https://zh.wikipedia.org/w/index.php?title=%E8%AE%A1%E9%87%8F%E6%95%B0&action=edit&redlink=1">計量數</a>不相等的幾部分時:</span> <dl>  	<dd><span style="color: #808080;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \mathrm {Ca(OH)_{2}} (s)\rightleftharpoons {\mbox{Ca}}^{2+}(aq)+{\mbox{2OH}}^{-}(aq) </span></span>,</span></dd> </dl> <span style="color: #808080;">K<sub>sp</sub>的確定會稍有複雜。對於如下電離過程:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle \mathrm {A} (s)\rightleftharpoons {\mbox{xB}}^{p+}(aq)+{\mbox{yC}}^{q-}(aq)</span></span></dd> </dl> <span style="color: #808080;">溶度積和溶解度的關係由以下方程確定:</span> <dl>  	<dd><span class="mwe-math-element" style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y">\displaystyle {\sqrt[{n}]{K_{\mathrm {sp} } \over {x^{x}\cdot y^{y}}}}={C \over M_{M}}$

其中:

  • n是電離方程右邊的總計量數(對上例,x+y),無量綱
  • x是所有陽離子的總計量數,無量綱;
  • y是所有陰離子的總計量數,無量綱;
  • Ksp是溶度積,(mol/kg)n
  • C是化合物 A的溶解度(A的質量比溶液的質量),無量綱;
  • MM是化合物 A的摩爾質量,kg/mol。

上述方程假設電離過程發生在純溶劑(無同離子效應發生),亦不存在絡合水解(即溶液中只存在Bp+和Cq-),且濃度小到離子活度可被認為等於1。

 

 

焉不能互動求『解』呦◎

 

 

 

 

 

 

 

 

STEM 隨筆︰鬼月談化學︰☴ 入《平衡》

楞嚴經‧卷三

阿難。如汝所言四大和合。發明世間種種變化。阿難。若彼大性體非和合。則不能與諸大雜和。猶如虛空。不和諸色。若和合者。同於變化。始終相成。生滅相續。生死死生。生生死死。如旋火輪。未有休息。阿難。如水成冰。冰還成水。汝觀地性。麤為大地。細為微塵。至鄰虛塵。析彼極微色邊際相。七分所成。更析鄰虛。即實空性。阿難。若此鄰虛。析成虛空。當知虛空。出生色相。汝今問言。由和合故。出生世間諸變化相 。汝且觀此一鄰虛塵。用幾虛空和合而有。不應鄰虛合成鄰虛。又鄰虛塵。析入空者。用幾色相 。合成虛空。若色合時。合色非空。若空合時。合空非色。色猶可析。空云何合。汝元不知。如來藏中。性色真空。性空真色。清淨本然。周遍法界。隨眾生心。應所知量。循業發現。世間無知。惑為因緣及自然性。皆是識心分別計度。但有言說。都無實義。

若問光子有大小嗎?以波動性論耶?用粒子性說乎?既是時空中之存在,如何沒有大小!既然沒有大小,又怎麼講可以量子生滅呢!解析大地粗細,以至於鄰虛塵,可知性色真空、性空真色 的乎!!卡西米爾議論真空漲落,加總無量光子能量 ,竟得有限量的耶??假使真知光子本來面目,得見如來的吧??!!

─── 《光的世界︰【□○閱讀】顯微鏡《上》

 

雙手推開窗前月!一石驚破井中天?

莫非『日光白』所說事?

Presence of Light

Light provides necessary activation energy to the starting materials, therefore, most of the reactions becomes faster in the presence of light

 

因此娑婆世界『變化多』!

一時巽入『鄰虛塵』︰

Dynamic equilibrium

In chemistry, a dynamic equilibrium exists once a reversible reaction ceases to change its ratio of reactants/products, but substances move between the chemicals at an equal rate, meaning there is no net change. It is a particular example of a system in a steady state. In thermodynamics, a closed system is in thermodynamic equilibrium when reactions occur at such rates that the composition of the mixture does not change with time. Reactions do in fact occur, sometimes vigorously, but to such an extent that changes in composition cannot be observed. Equilibrium constants can be expressed in terms of the rate constants for elementary reactions.

Examples

In a new bottle of soda the concentration of carbon dioxide in the liquid phase has a particular value. If half of the liquid is poured out and the bottle is sealed, carbon dioxide will leave the liquid phase at an ever-decreasing rate and the partial pressure of carbon dioxide in the gas phase will increase until equilibrium is reached. At that point, due to thermal motion, a molecule of CO2 may leave the liquid phase, but within a very short time another molecule of CO2 will pass from the gas to the liquid, and vice versa. At equilibrium the rate of transfer of CO2 from the gas to the liquid phase is equal to the rate from liquid to gas. In this case, the equilibrium concentration of CO2 in the liquid is given by Henry’s law, which states that the solubility of a gas in a liquid is directly proportional to the partial pressure of that gas above the liquid.[1] This relationship is written as

\displaystyle c=kp

where k is a temperature-dependent constant, p is the partial pressure and c is the concentration of the dissolved gas in the liquid Thus the partial pressure of CO2 in the gas has increased until Henry’s law is obeyed. The concentration of carbon dioxide in the liquid has decreased and the drink has lost some of its fizz.

Henry’s law may be derived by setting the chemical potentials of carbon dioxide in the two phases to be equal to each other. Equality of chemical potential defines chemical equilibrium. Other constants for dynamic equilibrium involving phase changes, include partition coefficient and solubility product. Raoult’s law defines the equilibrium vapor pressure of an ideal solution

Dynamic equilibrium can also exist in a single-phase system. A simple example occurs with acid-base equilibrium such as the dissociation of acetic acid, in aqueous solution.

CH3CO2H \rightleftharpoons CH3CO2 + H+

At equilibrium the concentration quotient, K, the acid dissociation constant, is constant (subject to some conditions)

\displaystyle K_{c}=\mathrm {\frac {[CH_{3}CO_{2}^{-}][H^{+}]}{[CH_{3}CO_{2}H]}}

In this case, the forward reaction involves the liberation of some protons from acetic acid molecules and the backward reaction involves the formation of acetic acid molecules when an acetate ion accepts a proton. Equilibrium is attained when the sum of chemical potentials of the species on the left-hand side of the equilibrium expression is equal to the sum of chemical potentials of the species on the right-hand side. At the same time the rates of forward and backward reactions are equal to each other. Equilibria involving the formation of chemical complexes are also dynamic equilibria and concentrations are governed by the stability constants of complexes.

Dynamic equilibria can also occur in the gas phase as, for example, when nitrogen dioxide dimerizes.

\displaystyle \rightleftharpoons Kp=[N2O4][NO2]2{\displaystyle K_{p}=\mathrm {\frac {[N_{2}O_{4}]}{[NO_{2}]^{2}}}

In the gas phase, square brackets indicate partial pressure. Alternatively, the partial pressure of a substance may be written as P(substance).[2]

Relationship between equilibrium and rate constants

In a simple reaction such as the isomerization:

\displaystyle A\rightleftharpoons B

there are two reactions to consider, the forward reaction in which the species A is converted into B and the backward reaction in which B is converted into A. If both reactions are elementary reactions, then the rate of reaction is given by[3]

\displaystyle {\frac {d[A]}{dt}}=-k_{f}[A]_{t}+k_{b}[B]_{t}

where kf is the rate constant for the forward reaction and kb is the rate constant for the backward reaction and the square brackets, [..] denote concentration . If only A is present at the beginning, time t=0, with a concentration [A]0, the sum of the two concentrations, [A]t and [B]t, at time t, will be equal to [A]0.

\displaystyle {\frac {d[A]}{dt}}=-k_{f}[A]_{t}+k_{b}\left([A]_{0}-[A]_{t}\right)

% concentrations of species in isomerization reaction. kf = 2 s−1, kr = 1 s−1

The solution to this differential equation is

\displaystyle [A]_{t}={\frac {k_{b}+k_{f}e^{-\left(k_{f}+k_{b}\right)t}}{k_{f}+k_{b}}}[A]_{0}

and is illustrated at the right. As time tends towards infinity, the concentrations [A]t and [B]t tend towards constant values. Let t approach infinity, that is, t→∞, in the expression above:

\displaystyle [A]_{\infty }={\frac {k_{b}}{k_{f}+k_{b}}}[A]_{0};[B]_{\infty }={\frac {k_{f}}{k_{f}+k_{b}}}[A]_{0}

In practice, concentration changes will not be measurable after \displaystyle t\gtrapprox {\frac {10}{k_{f}+k_{b}}} . Since the concentrations do not change thereafter, they are, by definition, equilibrium concentrations. Now, the equilibrium constant for the reaction is defined as

\displaystyle K={\frac {[B]_{eq}}{[A]_{eq}}}

It follows that the equilibrium constant is numerically equal to the quotient of the rate constants.

\displaystyle K={\frac {{\frac {k_{f}}{k_{f}+k_{b}}}[A]_{0}}{{\frac {k_{b}}{k_{f}+k_{b}}}[A]_{0}}}={\frac {k_{f}}{k_{b}}}

In general they may be more than one forward reaction and more than one backward reaction. Atkins states[4] that, for a general reaction, the overall equilibrium constant is related to the rate constants of the elementary reactions by

\displaystyle K=\left({\frac {k_{f}}{k_{b}}}\right)_{1}\times \left({\frac {k_{f}}{k_{b}}}\right)_{2}\dots

 

遊戲範例『文字』間!!

Copper(II)amine complexes (speciation)

 

忽憶『指令』無從來?★

只得先啃『原始』碼

Source code for chempy.chemistry

# -*- coding: utf-8 -*-
from __future__ import (absolute_import, division, print_function)

from collections import OrderedDict, defaultdict
from functools import reduce
from itertools import chain
from operator import mul, add
import math
import warnings

from .util.arithmeticdict import ArithmeticDict
from .util._expr import Expr
from .util.periodic import mass_from_composition
from .util.parsing import (
    formula_to_composition, to_reaction,
    formula_to_latex, formula_to_unicode, formula_to_html
)

from .units import default_units, is_quantity, unit_of, to_unitless
from ._util import intdiv
from .util.pyutil import deprecated, DeferredImport, ChemPyDeprecationWarning


ReactionSystem = DeferredImport('chempy.reactionsystem', 'ReactionSystem',
                                [deprecated(use_instead='chempy.ReactionSystem')])


[docs]class Substance(object):
    """ Class representing a chemical substance

    Parameters
    ----------
    name : str
    charge : int (optional, default: None)
        Will be stored in composition[0], prefer composition when possible.
    latex_name : str
    unicode_name : str
    html_name : str
    composition : dict or None (default)
        Dictionary (int -> number) e.g. {atomic number: count}, zero has special
        meaning (net charge). Avoid using the key 0 unless you specifically mean
        net charge. The motivation behind this is that it is easier to track a
        net-charge of e.g. 6 for U(VI) than it is to remember that uranium has 92
        electrons and use 86 as the value).
    data : dict
        Free form dictionary. Could be simple such as ``{'mp': 0, 'bp': 100}``
        or considerably more involved, e.g.: ``{'diffusion_coefficient': {\
 'water': lambda T: 2.1*m**2/s/K*(T - 273.15*K)}}``.

    Attributes
    ----------
    mass
        Maps to data['mass'], and when unavailable looks for ``formula.mass``.
    attrs
        A tuple of attribute names for serialization.
    composition : dict or None
        Dictionary mapping fragment key (str) to amount (int).
    data
        Free form dictionary.

    Examples
    --------
    >>> ammonium = Substance('NH4+', 1, 'NH_4^+', composition={7: 1, 1: 4},
    ...     data={'mass': 18.0385, 'pKa': 9.24})
    >>> ammonium.name
    'NH4+'
    >>> ammonium.composition == {0: 1, 1: 4, 7: 1}  # charge represented by key '0'
    True
    >>> ammonium.data['mass']
    18.0385
    >>> ammonium.data['pKa']
    9.24
    >>> ammonium.mass  # mass is a special case (also attribute)
    18.0385
    >>> ammonium.pKa
    Traceback (most recent call last):
        ...
    AttributeError: 'Substance' object has no attribute 'pKa'
    >>> nh4p = Substance.from_formula('NH4+')  # simpler
    >>> nh4p.composition == {7: 1, 1: 4, 0: 1}
    True
    >>> nh4p.latex_name
    'NH_{4}^{+}'

    """

    attrs = (
        'name', 'latex_name', 'unicode_name', 'html_name',
        'composition', 'data'
    )
...

 

再論其餘說

沉澱

沉澱化學上指從溶液中析出固體物質的過程;也指在沉澱過程中析出的固體物質。事實上沉澱多為難溶物(20°C時溶解度<0.01g) 。在化學實驗和生產中廣泛應用沉澱方法進行物質的分離。

水處理中指懸浮物在水中下沉,是懸浮物和水在密度上的差異形成的。

原因

化學中沉澱的產生是由於化學反應而生成溶解度較小的物質,或者由於溶液的濃度大於該溶質的溶解度所引起的;

標識

通常在化學反應方程式中沉澱會被標上「↓」,如:

Ca(OH)2+CO2CaCO3↓+H2O

從液相中產生可分離固相物的過程

!。 

 

 

 

 

 

 

 

STEM 隨筆︰鬼月談化學︰☴ 入《平衡》‧內

U348P5058DT20141115014820
英國漢學家閔福德
歷時十二年之易經英譯

Xiantianbagua

Sun_Wen_Red_Chamber_14

紅樓夢‧大觀園

易經》第二十卦‧風地觀

:盥而不荐,有孚顒若。

彖曰:大觀在上,順而巽,中正以觀天下。觀,盥而不荐,有孚顒若,下觀而化也。 觀天之神道 ,而四時不忒, 聖人以神道設教 ,而天下服矣。

象曰:風行地上,觀﹔先王以省方,觀民設教。

初六:童觀,小人無咎,君子吝 。
象曰:初六童觀,小人道也。

六二:窺觀,利女貞。
象曰:窺觀女貞,亦可丑也。

六三:觀我生,進退。
象曰:觀我生,進退﹔未失道也。

六四:觀國之光,利用賓于王。
象曰:觀國之光,尚賓也。

九五:觀我生,君子無咎。
象曰:觀我生,觀民也。

上九:觀其生,君子無咎。
象曰:觀其生,志未平也。

論語‧學而篇
子曰:不患人之不己知,患不知人也。

論語 ‧為政篇
子曰:視其所以,觀其所由,察其所安;人焉廋哉。人焉廋哉。 

若果】『一葉知秋』,履霜繼至堅冰,所以或有『深謀遠慮』,善採『先天下之憂而憂』之『舉措之旨』。所謂的『大觀在上』是說『自然人生法則』,『順而巽』是講『順應自然律』而且『深入其理』,這就是『』之大者!!

……

天地之《生》是一,《生生》故為多。所謂『數』始於一,然後二,然後三,然後很多很多。恰似古今中外,都有個『函三』『惟‧唯』一的說法︰

西︰ 聖父,聖子,聖靈

東︰ 太易、太初、太始

於是千百億年之後,『眾生』蔚為『大觀』,有人開始用『理性』之『眼』,欲觀宇宙『大霹靂』之『開天闢地』的『溫柔』。尋找元初的

立德,立功,立言

,卻是

不見其『德』,有『道』

難覓其『功』,成『果』

不知其『言』,現『象』

,當真『不識廬山真面目』的耶!

因此現今有人欲察『派生』之『博多』,當如何求『其一』的呢?『物』有『始』『中』『終』,或求『其始』的吧︰

東西』 Object 為本!

─── 《W!O 的派生‧十日談之《八》

 

『養身』可以『養生』耶?故耳黃帝問『風』也!

《黃帝內經‧素問》風論篇第四十二

黃帝問曰:風之傷人也,或為寒熱,或為熱中,或為寒中,或為癘風,或為偏枯,或為風也,其病各異,其名不同。或內至五臟六腑 ,不知其解,願聞其說。

岐伯對曰:風氣藏在皮膚之間,內不得通,外不得洩。風者,善行而數變,腠理開,則灑然寒,閉則熱而悶。其寒也,則衰食飲;其熱也,則消肌肉。故使人怢慄而不能食,名曰寒熱。風氣與陽明入胃,循脈而上至目內眥,其人肥,則風氣不得外洩,則為熱中而目黃;人瘦則外洩而寒,則為寒中而泣出。風氣與太陽俱入,行諸脈俞,散於分肉之間,與衛氣相干,其道不利。故使肌肉憤●而有瘍,衛氣有所凝而不行,故其肉有不仁也。癘者,有榮氣熱腑,其氣不清,故使其鼻柱壞而色敗,皮膚瘍潰。風寒客於脈而不去,名曰癘風,或名曰寒熱。以春甲乙傷於風者為肝風,以夏丙丁傷於風者為心風,以季夏戊己傷於邪者為脾風,以秋庚辛中於邪者為肺風,以冬壬癸中於邪者為腎風。風中五臟六腑之俞,亦為臟腑之風,各入其門戶,所中則為偏風。風氣循風府而上,則為胸風,風入系頭,則為目風,眼寒。飲酒中風,則為漏風。入房汗出中風,則為內風 。新沐中風,則為首風。久風入中,則為腸風,飧洩。外在腠理,則為洩風。故風者,百病之長也,至其變化,乃為他病也,無常方 ,然致有風氣也。

………

所謂

風邪

風邪中醫學上對一類外界環境致病因素的稱呼,為六淫之一。《黃帝內經》中描述為:「風者,善行而數變。」[1]。風邪為陽邪 ,侵襲部位多在表、在上,具有遊走性,病情變化較多且較迅速 。感受風邪的常見症狀有頭痛、寒熱汗出、遍身遊走疼痛和肌膚瘙癢等。風邪常與其他病邪結合一同致病,如風寒風熱風濕風燥等。

者,內外門戶乎??

因是知『平衡』

Chemical equilibrium

In a chemical reaction, chemical equilibrium is the state in which both reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system.[1] Usually, this state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but equal. Thus, there are no net changes in the concentrations of the reactant(s) and product(s). Such a state is known as dynamic equilibrium.[2][3]

仰賴『環境』呦!!

Historical introduction

The concept of chemical equilibrium was developed after Berthollet (1803) found that some chemical reactions are reversible.[4] For any reaction mixture to exist at equilibrium, the rates of the forward and backward (reverse) reactions are equal. In the following chemical equation with arrows pointing both ways to indicate equilibrium,[5] A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

α A + β B ⇌ σ S + τ T

The equilibrium concentration position of a reaction is said to lie “far to the right” if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be “far to the left” if hardly any product is formed from the reactants.

Guldberg and Waage (1865), building on Berthollet’s ideas, proposed the law of mass action:

質量作用定律

質量作用定律英語:Law of mass action)是化學領域的概念。定律包含兩個方向:

  1. 平衡方向,關於平衡時的反應混合物的組成。
  2. 動力學方向,關於基元反應的速率式。

表述

對於一個簡單地基元反應: \displaystyle aX(g)+bY(g)=cM(g)+dN(g) ,記化學反應速率為 \displaystyle v ,則有: \displaystyle v=k[X]^{a}[Y]^{b} 。其中[X]表示反應物X的濃度,k表示反應的速率常數。

作為對動力學的表述:一個基元反應[1]的速率,是參與反應的分子的濃度的乘積的比例。 在近代化學中,這是用統計力學推導出來的。

而作為對平衡的陳述,這個定律給出了平衡常數的關係式,平衡常數是可以描述化學平衡的一個量。在近代化學中,這是由平衡熱力學推導出來的。

 

\displaystyle {\begin{aligned}{\text{forward reaction rate}}&=k_{+}{\ce {A}}^{\alpha }{\ce {B}}^{\beta }\\{\text{backward reaction rate}}&=k_{-}{\ce {S}}^{\sigma }{\ce {T}}^{\tau }\end{aligned}}

where A, B, S and T are active masses and k+ and k are rate constants. Since at equilibrium forward and backward rates are equal:

\displaystyle k_{+}\left\{{\ce {A}}\right\}^{\alpha }\left\{{\ce {B}}\right\}^{\beta }=k_{-}\left\{{\ce {S}}\right\}^{\sigma }\left\{{\ce {T}}\right\}^{\tau }

and the ratio of the rate constants is also a constant, now known as an equilibrium constant.

\displaystyle K_{c}={\frac {k_{+}}{k_{-}}}={\frac {\{{\ce {S}}\}^{\sigma }\{{\ce {T}}\}^{\tau }}{\{{\ce {A}}\}^{\alpha }\{{\ce {B}}\}^{\beta }}}

By convention the products form the numerator. However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed (see, for example, nucleophilic aliphatic substitution by SN1 or reaction of hydrogen and bromine to form hydrogen bromide). Equality of forward and backward reaction rates, however, is a necessary condition for chemical equilibrium, though it is not sufficient to explain why equilibrium occurs.

Despite the failure of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the van ‘t Hoff equation. Adding a catalyst will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.[2][6]

Although the macroscopic equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case of acetic acid dissolved in water and forming acetate and hydronium ions,

CH3CO2H + H2O ⇌ CH3CO2 + H3O+

a proton may hop from one molecule of acetic acid on to a water molecule and then on to an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example of dynamic equilibrium. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.

Le Châtelier’s principle (1884) gives an idea of the behavior of an equilibrium system when changes to its reaction conditions occur. If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change. For example, adding more S from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).

If mineral acid is added to the acetic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:

\displaystyle K={\frac {\{{\ce {CH3CO2-}}\}\{{\ce {H3O+}}\}}{{\ce {\{CH3CO2H\}}}}}

If {H3O+} increases {CH3CO2H} must increase and CH3CO2 must decrease. The H2O is left out, as it is the solvent and its concentration remains high and nearly constant.

A quantitative version is given by the reaction quotient.

J. W. Gibbs suggested in 1873 that equilibrium is attained when the Gibbs free energy of the system is at its minimum value (assuming the reaction is carried out at constant temperature and pressure). What this means is that the derivative of the Gibbs energy with respect to reaction coordinate (a measure of the extent of reaction that has occurred, ranging from zero for all reactants to a maximum for all products) vanishes, signalling a stationary point.

反應坐標

圖示為一個反應坐標,當中位於能量最高點的均為過渡態。紅色線表示了沒有加入催化劑(酶)因此具有較高的活化能,藍色線是加了催化劑後的情況。

反應坐標,或稱作能線圖是一種根據反應途徑加以圖像化的一維坐標,用以表達化學反應的進行過程[1]。利用簡單的幾何圖表,加以表示單個或數個分子實體化學反應中所經歷轉變的途徑。

這類坐標有時也可用鍵長鍵角參數來表示,但最常見的則是用自由能作為主要參數。一些非幾何參數有時會在更複雜的反應中使用,例如是不同分子實體的鍵級

過渡態理論中,反應坐標就是一組以曲線組成的坐標,用以表示某種化學實體由本身的結構,經變化成過渡態,最後演變成某結構的生成物的整個過程。在坐標中上升斜率最大的一個途徑,就是該反應的速率控制步驟

反應進度

反應進度英文Extent of reaction)是用於量化反映化學反應進行程度的化學量,在運算中通常以希臘字母ξ(Xi)代表。一般反應進度的單位為摩爾。一個進行中或已完成的化學反應,無論計算哪一個反應物生成物的反應程度,都會得到相同的結果,因為反應進度代表整個反應的進度,而不是單一物質的反應進度。

基本定義

考慮一化學反應計量式:

A ⇌ B

這個化學反應計量式表示化學物質A在特定條件下等量地與化學物質B相互轉換。假設有無限少量的A物質透過此反應生成B物質,則 A 物質數量的變化量為 dnA=-dξ ,而 B 物質數量的變化量為 dnB=dξ[1] ,則反應程度 ξ 將定義為:[2][3] 更一般地,若化學反應計量式中,第i個化學物質(反應物生成物)的化學計量係數是 \displaystyle \nu _{i} ,其物質的量為 \displaystyle n_{i} ,則

\displaystyle d\xi ={\frac {dn_{i}}{\nu _{i}}}

其中,需要特別注意的是 \displaystyle \nu _{i} 值的正負號。選定了反應的正方向後,當選取的物質為反應物時, \displaystyle \nu _{i} 為負值;若為生成物,則為正值。唯有如此規範,才能使反應程度 ξ 之值恆正。

若非無限少量的反應,而是有限變化量的反應,則可將上式通過對時間積分而改寫為:

\displaystyle \Delta \xi ={\frac {\Delta n_{i}}{\nu _{i}}}

若假設 t=0 時,反應進度 ξ=0,則反應進度ξ可表示為:

\displaystyle \xi ={\frac {\Delta n_{i}}{\nu _{i}}}={\frac {n_{equilibrium}-n_{initial}}{\nu _{i}}}

 

This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between the chemical potentials of reactants and products at the composition of the reaction mixture.[1] This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or Helmholtz energy at constant volume reactions) is the “driving force” for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard Gibbs free energy change for the reaction by the equation

\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K_{\mathrm {eq} }

where R is the universal gas constant and T the temperature.

When the reactants are dissolved in a medium of high ionic strength the quotient of activity coefficients

活度係數

活度係數英語:Activity coefficient),又稱活度因子英語 :Activity factor),是熱力學中的一個係數,反映的是真實溶液中某組分i的行為偏離理想溶液的程度[1],量綱為1。引入活度係數後 ,適用於理想溶液的各種關係可以相應修正為適用於真實溶液。類似的,逸度係數是表示真實氣體混合物中某組分和理想行為的偏離的係數。

定義

在理想溶液中,溶液組分 i 遵循拉午耳定律

\displaystyle x_{i}={\frac {p_{i}}{p_{i}^{\star }}}

其中 \displaystyle x_{i} 是組分 i 在溶液中的莫耳分率\displaystyle p_{i} 和 \displaystyle p_{i}^{\star } 分別是組分 i 的分壓和飽和蒸氣壓。 而組分 i 的化學勢 \displaystyle \mu _{i} 可由下式表達:

\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln x_{i}

這裡的 \displaystyle \mu _{i}^{\ominus } 代表組分 i 在標準狀態下的化學勢。而在真實溶液中,組分 i-組分 i 間的作用力和組分 i-其他組分間的作用力並不相等,導致了組分i並不滿足拉午耳定律,其化學勢也不滿足以上關係,即偏離了理想溶液的行為,為此吉爾伯特·牛頓·路易士引入了活度和活度係數的概念。 定義:

\displaystyle a_{x,i}=\gamma _{x,i}x_{i}

這裡的 \displaystyle a_{x,i} 是組分 i 以莫耳分率所表示的活度\displaystyle \gamma _{x,i} 則是組分i用莫耳分率所表示的活度係數。引入活度和活度係數後,拉午耳定律可以修正為:

\displaystyle a_{x,i}=\gamma _{x,i}x_{i}=\gamma _{x,i}{\frac {p_{i}}{p_{i}^{\star }}}

組分i的化學勢則可以修正為:

\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}

真實溶液的濃度越稀,溶劑的活度係數就越接近1,活度和莫耳分率近乎相等,其行為越接近理想溶液。濃度越高,活度係數越偏離1,真實溶液的行為偏差理想溶液就越大,比如對於濃度較高的電解質溶液,其活度就無法用莫耳分率取代,這一點在電化學土壤化學中十分常見[2]

 

may be taken to be constant. In that case the concentration quotient, Kc,

\displaystyle K_{\ce {c}}={\frac {[{\ce {S}}]^{\sigma }[{\ce {T}}]^{\tau }}{[{\ce {A}}]^{\alpha }[{\ce {B}}]^{\beta }}}

where [A] is the concentration of A, etc., is independent of the analytical concentration of the reactants. For this reason, equilibrium constants for solutions are usually determined in media of high ionic strength. Kc varies with ionic strength, temperature and pressure (or volume). Likewise Kp for gases depends on partial pressure. These constants are easier to measure and encountered in high-school chemistry courses.

Thermodynamics

At constant temperature and pressure, one must consider the Gibbs free energy, G, while at constant temperature and volume, one must consider the Helmholtz free energy: A, for the reaction; and at constant internal energy and volume, one must consider the entropy for the reaction: S.

The constant volume case is important in geochemistry and atmospheric chemistry where pressure variations are significant. Note that, if reactants and products were in standard state (completely pure), then there would be no reversibility and no equilibrium. Indeed, they would necessarily occupy disjoint volumes of space. The mixing of the products and reactants contributes a large entropy (known as entropy of mixing) to states containing equal mixture of products and reactants. The standard Gibbs energy change, together with the Gibbs energy of mixing, determine the equilibrium state.[7][8]

In this article only the constant pressure case is considered. The relation between the Gibbs free energy and the equilibrium constant can be found by considering chemical potentials.[1]

At constant temperature and pressure, the Gibbs free energy, G, for the reaction depends only on the extent of reaction: ξ (Greek letter xi), and can only decrease according to the second law of thermodynamics. It means that the derivative of G with ξ must be negative if the reaction happens; at the equilibrium the derivative being equal to zero.

\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=0~ :     equilibrium

In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative of G with respect to the extent of reaction: ξ, must be zero. It can be shown that in this case, the sum of chemical potentials of the products is equal to the sum of those corresponding to the reactants. Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.

\displaystyle \alpha \mu _{\mathrm {A} }+\beta \mu _{\mathrm {B} }=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }

where μ is in this case a partial molar Gibbs energy, a chemical potential. The chemical potential of a reagent A is a function of the activity, {A} of that reagent.

\displaystyle \mu _{\mathrm {A} }=\mu _{A}^{\ominus }+RT\ln\{\mathrm {A} \}

(where μoA is the standard chemical potential).

The definition of the Gibbs energy equation interacts with the fundamental thermodynamic relation to produce

\displaystyle dG=Vdp-SdT+\sum _{i=1}^{k}\mu _{i}dN_{i} .

Inserting dNi = νi dξ into the above equation gives a Stoichiometric coefficient (\displaystyle \nu _{i}~) and a differential that denotes the reaction occurring once (). At constant pressure and temperature the above equations can be written as

\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\sum _{i=1}^{k}\mu _{i}\nu _{i}=\Delta _{\mathrm {r} }G_{T,p} which is the “Gibbs free energy change for the reaction .

This results in:

\displaystyle \Delta _{\mathrm {r} }G_{T,p}=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }-\alpha \mu _{\mathrm {A} }-\beta \mu _{\mathrm {B} } .

By substituting the chemical potentials:

\displaystyle \Delta _{\mathrm {r} }G_{T,p}=(\sigma \mu _{\mathrm {S} }^{\ominus }+\tau \mu _{\mathrm {T} }^{\ominus })-(\alpha \mu _{\mathrm {A} }^{\ominus }+\beta \mu _{\mathrm {B} }^{\ominus })+(\sigma RT\ln\{\mathrm {S} \}+\tau RT\ln\{\mathrm {T} \})-(\alpha RT\ln\{\mathrm {A} \}+\beta RT\ln\{\mathrm {B} \}) ,

the relationship becomes:

\displaystyle \Delta _{\mathrm {r} }G_{T,p}=\sum _{i=1}^{k}\mu _{i}^{\ominus }\nu _{i}+RT\ln {\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}
\displaystyle \sum _{i=1}^{k}\mu _{i}^{\ominus }\nu _{i}=\Delta _{\mathrm {r} }G^{\ominus } :

which is the standard Gibbs energy change for the reaction that can be calculated using thermodynamical tables. The reaction quotient is defined as:

\displaystyle Q_{\mathrm {r} }={\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}

Therefore,

\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G_{T,p}=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} }

At equilibrium:

\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G_{T,p}=0

leading to:

\displaystyle 0=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln K_{\mathrm {eq} }

and

\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=-RT\ln K_{\mathrm {eq} }

Obtaining the value of the standard Gibbs energy change, allows the calculation of the equilibrium constant.

Addition of reactants or products

For a reactional system at equilibrium: Qr = Keq; ξ = ξeq.

  • If are modified activities of constituents, the value of the reaction quotient changes and becomes different from the equilibrium constant: Qr ≠ Keq
\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} }~
and
\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=-RT\ln K_{eq}~
then
\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=RT\ln \left({\frac {Q_{\mathrm {r} }}{K_{\mathrm {eq} }}}\right)~
  • If activity of a reagent i increases
\displaystyle Q_{\mathrm {r} }={\frac {\prod (a_{j})^{\nu _{j}}}{\prod (a_{i})^{\nu _{i}}}}~ , the reaction quotient decreases.
then
\displaystyle Q_{\mathrm {r} }<K_{\mathrm {eq} }~    and    \displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}<0~
The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).
  • If activity of a product j increases
then
\displaystyle Q_{\mathrm {r} }>K_{\mathrm {eq} }~     and     \displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}>0~
The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).

Note that activities and equilibrium constants are dimensionless numbers.

Treatment of activity

The expression for the equilibrium constant can be rewritten as the product of a concentration quotient, Kc and an activity coefficient quotient, Γ.

\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}\times {\frac {{\gamma _{\mathrm {S} }}^{\sigma }{\gamma _{\mathrm {T} }}^{\tau }...}{{\gamma _{\mathrm {A} }}^{\alpha }{\gamma _{\mathrm {B} }}^{\beta }...}}=K_{\mathrm {c} }\Gamma

[A] is the concentration of reagent A, etc. It is possible in principle to obtain values of the activity coefficients, γ. For solutions, equations such as the Debye–Hückel equation or extensions such as Davies equation[9] Specific ion interaction theory or Pitzer equations[10]may be used.Software (below) However this is not always possible. It is common practice to assume that Γ is a constant, and to use the concentration quotient in place of the thermodynamic equilibrium constant. It is also general practice to use the term equilibrium constant instead of the more accurate concentration quotient. This practice will be followed here.

For reactions in the gas phase partial pressure is used in place of concentration and fugacity coefficient in place of activity coefficient. In the real world, for example, when making ammonia in industry, fugacity coefficients must be taken into account. Fugacity, f, is the product of partial pressure and fugacity coefficient. The chemical potential of a species in the gas phase is given by

\displaystyle \mu =\mu ^{\ominus }+RT\ln \left({\frac {f}{\mathrm {bar} }}\right)=\mu ^{\ominus }+RT\ln \left({\frac {p}{\mathrm {bar} }}\right)+RT\ln \gamma

so the general expression defining an equilibrium constant is valid for both solution and gas phases.

Metastable mixtures

A mixture may appear to have no tendency to change, though it is not at equilibrium. For example, a mixture of SO2 and O2 is metastable as there is a kinetic barrier to formation of the product, SO3.

2 SO2 + O2 ⇌ 2 SO3

The barrier can be overcome when a catalyst is also present in the mixture as in the contact process, but the catalyst does not affect the equilibrium concentrations.

Likewise, the formation of bicarbonate from carbon dioxide and water is very slow under normal conditions

CO2 + 2 H2O ⇌ HCO
3
+ H3O+

but almost instantaneous in the presence of the catalytic enzyme carbonic anhydrase.

Pure substances

When pure substances (liquids or solids) are involved in equilibria their activities do not appear in the equilibrium constant[12] because their numerical values are considered one.

Applying the general formula for an equilibrium constant to the specific case of a dilute solution of acetic acid in water one obtains

CH3CO2H + H2O ⇌ CH3CO2 + H3O+
\displaystyle K_{\mathrm {c} }={\frac {\mathrm {[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]} }{\mathrm {[{CH_{3}CO_{2}H}][{H_{2}O}]} }}

For all but very concentrated solutions, the water can be considered a “pure” liquid, and therefore it has an activity of one. The equilibrium constant expression is therefore usually written as

\displaystyle K={\frac {\mathrm {[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]} }{\mathrm {[{CH_{3}CO_{2}H}]} }}=K_{\mathrm {c} } .

A particular case is the self-ionization of water itself

2 H2O ⇌ H3O+ + OH

Because water is the solvent, and has an activity of one, the self-ionization constant of water is defined as

\displaystyle K_{\mathrm {w} }=\mathrm {[H^{+}][OH^{-}]}

It is perfectly legitimate to write [H+] for the hydronium ion concentration, since the state of solvation of the proton is constant (in dilute solutions) and so does not affect the equilibrium concentrations. Kw varies with variation in ionic strength and/or temperature.

The concentrations of H+ and OH are not independent quantities. Most commonly [OH] is replaced by Kw[H+]−1 in equilibrium constant expressions which would otherwise include hydroxide ion.

Solids also do not appear in the equilibrium constant expression, if they are considered to be pure and thus their activities taken to be one. An example is the Boudouard reaction:[12]

2 CO ⇌ CO2 + C

for which the equation (without solid carbon) is written as:

\displaystyle K_{\mathrm {c} }={\frac {\mathrm {[CO_{2}]} }{\mathrm {[CO]^{2}} }}

Multiple equilibria

Consider the case of a dibasic acid H2A. When dissolved in water, the mixture will contain H2A, HA and A2−. This equilibrium can be split into two steps in each of which one proton is liberated.

\displaystyle {\begin{array}{rl}{\ce {H2A <=> HA^- + H+}}:&K_{1}={\frac {{\ce {[HA-][H+]}}}{{\ce {[H2A]}}}}\\{\ce {HA- <=> A^2- + H+}}:&K_{2}={\frac {{\ce {[A^{2-}][H+]}}}{{\ce {[HA-]}}}}\end{array}}

K1 and K2 are examples of stepwise equilibrium constants. The overall equilibrium constant, βD, is product of the stepwise constants.

\displaystyle {\ce {{H2A}<=>{A^{2-}}+{2H+}}} :     \displaystyle \beta _{{\ce {D}}}={\frac {{\ce {[A^{2-}][H^+]^2}}}{{\ce {[H_2A]}}}}=K_{1}K_{2}

Note that these constants are dissociation constants because the products on the right hand side of the equilibrium expression are dissociation products. In many systems, it is preferable to use association constants.

\displaystyle {\begin{array}{ll}{\ce {A^2- + H+ <=> HA-}}:&\beta _{1}={\frac {{\ce {[HA^-]}}}{{\ce {[A^{2-}][H+]}}}}\\{\ce {A^2- + 2H+ <=> H2A}}:&\beta _{2}={\frac {{\ce {[H2A]}}}{{\ce {[A^{2-}][H+]^2}}}}\end{array}}

β1 and β2 are examples of association constants. Clearly β1 = 1/K2 and β2 = 1/βD; log β1 = pK2 and log β2 = pK2 + pK1[13] For multiple equilibrium systems, also see: theory of Response reactions.

Effect of temperature

The effect of changing temperature on an equilibrium constant is given by the van ‘t Hoff equation

\displaystyle {\frac {d\ln K}{dT}}={\frac {\Delta H_{\mathrm {m} }^{\ominus }}{RT^{2}}}
Thus, for exothermic reactions (ΔH is negative), K decreases with an increase in temperature, but, for endothermic reactions, (ΔH is positive) K increases with an increase temperature. An alternative formulation is
\displaystyle {\frac {d\ln K}{d(T^{-1})}}=-{\frac {\Delta H_{\mathrm {m} }^{\ominus }}{R}}

At first sight this appears to offer a means of obtaining the standard molar enthalpy of the reaction by studying the variation of K with temperature. In practice, however, the method is unreliable because error propagation almost always gives very large errors on the values calculated in this way.

 

 

 

 

 

 

STEM 隨筆︰鬼月談化學︰☳ 動 《碰撞》

據『歷史典故』上說,東晉慧遠大師主持東林寺,立下了規矩『影不出山迹不入谷』;一過虎溪,寺後山虎則吼。一日大詩人陶淵明和道士陸修靜來訪,談的投機,送行時不覺過了虎溪橋,待聞得虎嘯後方恍然大悟,相視大笑而別,後世稱作『虎溪三笑』。其後有清朝唐蝸寄題的廬山東林寺三笑庭名聯:

橋跨虎溪,三教三源流,三人三笑語;
蓮開僧舍,一花一世界,一葉一如來。

今天的人或許較熟悉英國詩人布莱克的『一沙一世界,一花一天堂。』名句。這個名句出自一首長詩《純真的徵兆》的起頭︰

snadworld

天堂鳥-花

220px-Blake_jacobsladder

Auguries of Innocence

To see a world in a grain of sand,
一粒沙裡世界
And a heaven in a wild flower,
一朵花中天堂
Hold infinity in the palm of your hand,
掌尺足無限
And eternity in an hour.
時針能永恆

布莱克生於 1757 年,幼年就個性獨特討厭正統學校的教條氣息,因而拒絕入學,博覽眾書自學成家,由於潛心研讀洛克博克經驗主義哲學著作,於是對這個大千世界有了深刻認識早熟的他為減輕家計重擔和考慮弟妹前途,放棄了畫家夢想,十四歲時就選擇了去雕版印刷作坊當個學徒,二十二歲出師,…
是英國浪漫主義詩人的第一人
雅各的天梯,布莱克的版畫,布莱克『自爬』?

 

博克的名著【壯美優美觀念起源之哲學探究】,布莱克用來觀察飛鳥之姿』── Auguries ──,體驗預示藝術參與,果然恰當!!就像『掌尺』的可成無限,用時針的『循環』以度永恆一樣;也許布莱克的浪漫充滿著理性思辨,其要總在觀察

─── 《一個奇想!!

 

山 ☶ 高不能止飛鳥,

活化能

活化能Activation energy)是一個化學名詞,又被稱為閾能。這一名詞是由阿瑞尼士在 1889 年引入,用來定義一個化學反應的發生所需要克服的能量障礙。活化能可以用於表示一個化學反應發生所需要的最小能量,因此活化能越高,反應越難進行。反應的活化能通常表示為 E,單位是千焦耳摩爾(kJ/mol)。

活化能基本上是表示勢壘(有時稱為能壘)的高度。

圖中的火花是在用鐵塊敲擊燧石提供活化能以點燃本生燈時所產生 。在火花消失後,藍色火焰還可以持續,這是因為火焰燃燒所釋放的能量足以維持其自身。

 

水 ☵ 流緩緩石可穿!

靜謐無聲誰人羨?

乎耳雷 ☳ 動窟窿前!?

Collision theory

Collision theory is a theory proposed independently by Max Trautz in 1916[1] and William Lewis in 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions.[2] The collision theory states that when suitable particles of the reactant hit each other, only a certain fraction of the collisions cause any noticeable or significant chemical change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as activation energy, at the moment of impact to break the preexisting bonds and form all new bonds. This results in the products of the reaction. Increasing the concentration of the reactant particles or raising the temperature – which brings about more collisions and hence more successful collisions – therefore increases the rate of a reaction.

When a catalyst is involved in the collision between the reactant molecules, less energy is required for the chemical change to take place, and hence more collisions have sufficient energy for reaction to occur. The reaction rate therefore increases.

Collision theory is closely related to chemical kinetics.

Reaction rate tends to increase with concentrationphenomenon explained by collision theory

Rate constant

The rate constant for a bimolecular gas-phase reaction, as predicted by collision theory is

\displaystyle k(T)=Z\rho \exp \left({\frac {-E_{\text{a}}}{RT}}\right),

where:

Z is the collision frequency,
\displaystyle \rho is the steric factor,[3]
Ea is the activation energy of the reaction,
T is the temperature,
R is the gas constant.

The collision frequency is

\displaystyle Z=N_{\text{A}}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}},

where:

NA is the Avogadro constant,
σAB is the reaction cross section,
kB is the Boltzmann’s constant,
μAB is the reduced mass of the reactants.

Quantitative insights

Derivation

Consider the bimolecular elementary reaction:

A + B → C

In collision theory it is considered that two particles A and B will collide if their nuclei get closer than a certain distance. The area around a molecule A in which it can collide with an approaching B molecule is called the cross sectionAB) of the reaction and is, in simplified terms, the area corresponding to a circle whose radius (\displaystyle r_{AB}) is the sum of the radii of both reacting molecules, which are supposed to be spherical. A moving molecule will therefore sweep a volume \displaystyle \pi r_{AB}^{2}c_{A} per second as it moves, where \displaystyle c_{A} is the average velocity of the particle. (This solely represents the classical notion of a collision of solid balls. As molecules are quantum-mechanical many-particle systems of electrons and nuclei based upon the Coulomb and exchange interactions, generally they neither obey rotational symmetry nor do they have a box potential. Therefore, more generally the cross section is defined as the reaction probability of a ray of A particles per areal density of B targets, which makes the definition independent from the nature of the interaction between A and B. Consequently, the radius \displaystyle r_{AB} is related to the length scale of their interaction potential.)

From kinetic theory it is known that a molecule of A has an average velocity (different from root mean square velocity) of \displaystyle c_{A}={\sqrt {\frac {8k_{\text{B}}T}{\pi m_{A}}}} , where \displaystyle k_{\text{B}} is Boltzmann constant, and \displaystyle m_{A} is the mass of the molecule.

The solution of the two-body problem states that two different moving bodies can be treated as one body which has the reduced mass of both and moves with the velocity of the center of mass, so, in this system \displaystyle \mu _{AB} must be used instead of \displaystyle m_{A} .

Therefore, the total collision frequency,[4] of all A molecules, with all B molecules, is

\displaystyle N_{A}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}[A][B]=N_{A}\pi r_{AB}^{2}{\sqrt {\frac {8k_{\text{B}}T}{\pi \mu _{AB}}}}[A][B]=Z[A][B].

From Maxwell–Boltzmann distribution it can be deduced that the fraction of collisions with more energy than the activation energy is \displaystyle e^{\frac {-E_{\text{a}}}{k_{\text{B}}T}} . Therefore, the rate of a bimolecular reaction for ideal gases will be

\displaystyle r=Z\rho [A][B]\exp \left({\frac {-E_{\text{a}}}{RT}}\right),

Where:

Z is the collision frequency,
\displaystyle \rho is the steric factor, which will be discussed in detail in the next section,
Ea is the activation energy (per mole) of the reaction,
T is the absolute temperature,
R is the gas constant.

The product is equivalent to the preexponential factor of the Arrhenius equation.

Validity of the theory and steric factor

Once a theory is formulated, its validity must be tested, that is, compare its predictions with the results of the experiments.

When the expression form of the rate constant is compared with the rate equation for an elementary bimolecular reaction, \displaystyle r=k(T)[A][B] , it is noticed that

\displaystyle k(T)=N_{A}\sigma _{AB}{\sqrt {\frac {8k_{\text{B}}T}{\pi m_{A}}}}\exp \left({\frac {-E_{\text{a}}}{RT}}\right) .

This expression is similar to the Arrhenius equation and gives the first theoretical explanation for the Arrhenius equation on a molecular basis. The weak temperature dependence of the preexponential factor is so small compared to the exponential factor that it cannot be measured experimentally, that is, “it is not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted T½ dependence of the preexponential factor is observed experimentally”.[5]

Steric factor

If the values of the predicted rate constants are compared with the values of known rate constants, it is noticed that collision theory fails to estimate the constants correctly, and the more complex the molecules are, the more it fails. The reason for this is that particles have been supposed to be spherical and able to react in all directions, which is not true, as the orientation of the collisions is not always proper for the reaction. For example, in the hydrogenation reaction of ethylene the H2 molecule must approach the bonding zone between the atoms, and only a few of all the possible collisions fulfill this requirement.

To alleviate this problem, a new concept must be introduced: the steric factor ρ. It is defined as the ratio between the experimental value and the predicted one (or the ratio between the frequency factor and the collision frequency):

\displaystyle \rho ={\frac {A_{\text{observed}}}{Z_{\text{calculated}}}},

and it is most often less than unity.[3]

Usually, the more complex the reactant molecules, the lower the steric factor. Nevertheless, some reactions exhibit steric factors greater than unity: the harpoon reactions, which involve atoms that exchange electrons, producing ions. The deviation from unity can have different causes: the molecules are not spherical, so different geometries are possible; not all the kinetic energy is delivered into the right spot; the presence of a solvent (when applied to solutions), etc.

Experimental rate constants compared to the ones predicted by collision theory for gas phase reactions
Reaction A, s−1 Z, s−1 Steric factor
2ClNO → 2Cl + 2NO 9.4×109 5.9×1010 0.16
2ClO → Cl2 + O2 6.3×107 2.5×1010 2.3×10−3
H2 + C2H4 → C2H6 1.24×106 7.3×1011 1.7×10−6
Br2 + K → KBr + Br 1.0×1012 2.1×1011 4.3

Collision theory can be applied to reactions in solution; in that case, the solvent cage has an effect on the reactant molecules, and several collisions can take place in a single encounter, which leads to predicted preexponential factors being too large. ρ values greater than unity can be attributed to favorable entropic contributions.

Experimental rate constants compared to the ones predicted by collision theory for reactions in solution[6]
Reaction Solvent A, 1011 s−1 Z, 1011 s−1 Steric factor
C2H5Br + OH ethanol 4.30 3.86 1.11
C2H5O + CH3I ethanol 2.42 1.93 1.25
ClCH2CO2 + OH water 4.55 2.86 1.59
C3H6Br2 + I methanol 1.07 1.39 0.77
HOCH2CH2Cl + OH water 25.5 2.78 9.17
4-CH3C6H4O + CH3I ethanol 8.49 1.99 4.27
CH3(CH2)2Cl + I acetone 0.085 1.57 0.054
C5H5N + CH3I C2H2Cl4 2.0 10×10−6

 

滿漢席上放眼瞧☺

※ 註︰

樹莓派 chempy 雖無法安裝

最好還是有

  • pygslodeiv2: solving initial value problems, requires GSL. (>=1.16).

的好。

sudo apt-get install libgsl-dev

sudo pip3 install pygslodeiv2

Arrhenius equation

The Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van ‘t Hoff who had noted in 1884 that Van ‘t Hoff’s equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining rate of chemical reactions and for calculation of energy of activation. Arrhenius provided a physical justification and interpretation for the formula.[1][2][3] Currently, it is best seen as an empirical relationship.[4]:188 It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally-induced processes/reactions. The Eyring equation, developed in 1935, also expresses the relationship between rate and energy.

A historically useful generalization supported by Arrhenius’ equation is that, for many common chemical reactions at room temperature, the reaction rate doubles for every 10 degree Celsius increase in temperature.[5]

Equation

In almost all practical cases, Ea>>RT{\displaystyle E_{a}>>RT}{\displaystyle E_{a}>>RT} and k increases rapidly with T.

Mathematically, at very high temperatures so that \displaystyle E_{a}<<RT , k levels off and approaches A as a limit, but this case does not occur under practical conditions.

Arrhenius’ equation gives the dependence of the rate constant of a chemical reaction on the absolute temperature, a pre-exponential factor and other constants of the reaction.

\displaystyle k=Ae^{\frac {-E_{a}}{RT}}

Where

Alternatively, the equation may be expressed as

\displaystyle k=Ae^{\frac {-E_{a}}{k_{B}T}}

Where

The only difference is the energy units of Ea: the former form uses energy per mole, which is common in chemistry, while the latter form uses energy per molecule directly, which is common in physics. The different units are accounted for in using either the gas constant, R, or the Boltzmann constant, kB, as the multiplier of temperature T.

The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction. If the reaction is first order it has the units: s−1, and for that reason it is often called thefrequency factor or attempt frequency of the reaction. Most simply, k is the number of collisions that result in a reaction per second, A is the number of collisions (leading to a reaction or not) per second occurring with the proper orientation to react[6] and \displaystyle e^{{-E_{a}}/{(RT)}} is the probability that any given collision will result in a reaction. It can be seen that either increasing the temperature or decreasing the activation energy (for example through the use of catalysts) will result in an increase in rate of reaction.

Given the small temperature range of kinetic studies, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the \displaystyle \exp(-E_{a}/(RT)) factor; except in the case of “barrierless” diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.