據『歷史典故』上說，東晉時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︰

Auguries of Innocence

To see a world in a grain of sand,
一粒沙裡看世界
And a heaven in a wild flower,
一朵花中見天堂
Hold infinity in the palm of your hand,
掌尺足握無限
And eternity in an hour.
時針能持永恆

…

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，由於潛心研讀洛克和博克的經驗主義哲學著作，於是對這個大千世界有了深刻的認識。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【壯美與優美的觀念起源之哲學探究】，布莱克用來觀察『飛鳥之姿』── Auguries ──，體驗預示的藝術與參與，果然恰當！！就像『掌尺』的積累可成無限，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_a，單位是千焦耳每摩爾（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]

E_a 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:

N_A is the Avogadro constant,

σ_AB is the reaction cross section,

k_B is the Boltzmann’s constant,

μ_AB 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 section (σ_AB) 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,

E_a is the activation energy (per mole) of the reaction,

T is the absolute temperature,

R is the gas constant.

The product Zρ 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 T^½ 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

pyodeint: solving initial value problems, requires boost (>=1.65.0).
pycvodes: solving initial value problems, requires SUNDIALS (>=2.7.0).

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, $E_{a}>>RT$ 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

$k$ is the rate constant
$T$ is the absolute temperature (in kelvin)
$A$ is the pre-exponential factor, a constant for each chemical reaction. According to collision theory, A is the frequency of collisions in the correct orientation
$E a$ is the activation energy for the reaction (in the same units as R*T)
$R$ is the universal gas constant:^[1]^[2]^[3]

Alternatively, the equation may be expressed as

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

Where

$E a$ is the activation energy for the reaction (in the same units as k_B*T)
$k B$ is the Boltzmann constant

The only difference is the energy units of $E a$ : 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, $k B$ , 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.

FreeSandal

STEM 隨筆︰鬼月談化學︰☳ 動《碰撞》

活化能