Physical Chemistry II 物理化学 II 笔记¶

Chapter 1 : Chemical and Electrochemical Equilibrium¶

Gibbs Energy Minimum¶

for rxn \(A \rightleftarrows B\), def. \(\xi\) as rxn extent(mol)

so \(-\dd n_A = \dd n_B = \dd \xi\),

for

def. rxn quotient \(Q = \frac{p_B}{p_A}\)

at Eq. \(0 = \Delta_rG = \Delta_rG^{\circ} + RT\ln{K}\)

General rxns¶

we get:

as \(a_j\) remains activity or fugacity

Since \(\delta _rG^{\circ}\) is defined as a single pressure, we have \((\frac{\partial K}{\partial p})_T = 0\)

for rxn \(A \rightleftarrows 2B\), \(A\) changes from \(n\) to \((1-\beta)n\)

so we have:

as \(p\) increase, \(\beta\) decrease.

Response of equilibria to changes¶

as

that is:

that means: when \(\Delta_rH^{\circ} < 0\), when \(T\) decrease, \(K\) increase. Vice Versa

Equilibrium electrochemistry¶

for rxn \(\ce{H_2O <--> H_2 + O_2}\) we got half rxns:

Anode: \(\ce{H_2 - 2e^-<--> 2H^+}\)

Cathode: \(\ce{0.5O_2 + 2H^+ + 2e^- <--> H_2O}\)

for non-electrochemistry states:

consider the Electromotive Force(EMF)

this is called Nerst Equation.

At eq: \(0 = E^{\circ} - \frac{RT}{\nu F}\ln{K}\), so

Lattice gas¶

def. \(N\) as indistinguishable particles, \(N_0\) as indistinguishable sites

as Boltzman Entropy \(S = k_B \ln{\Omega}\), since \(\displaystyle \Omega = \frac{N_0!}{(N_0 - N)!(N)!}\)

let \(x = N/N_0\), using Stiring Approximation:

for general cases:

Electrochemical Potential¶

so as \(\displaystyle \mu = (\frac{\Delta G}{\Delta N})_{T,p,\varphi} = (\frac{\partial g}{\partial x})_{T, p, \varphi}\), def \(g = G/N, h = H/N, S = S/N\)

For a lattice as:

so:

in solution, when \(x \rightarrow 0\),

that is called Electrochemical Potential

Standard Hydrogen Potential¶

define Standard Hydrogen Potential as 0 V as the Standard Hydrogen Electrode:

whereas happens Pt electrode and every species are at the standard state (形成铂黑结构增加催化位点，促进反应快速达到平衡。)

eg: \(\ce{2H_2 + O_2 -> 2H_2O}\), \(E = \pu{1.23V}\)

so \(\ce{0.5O_2 + 2H^+ + 2e^- <--> H_2O}\), \(E = \pu{1.23V}\)

eg2: \(\ce{AgCl + e <--> Ag + Cl^-}\)

where \(b\) stands molality(质量摩尔浓度), \(\gamma\) stands activity coefficient

as Debye-Huckle Formula \(\ln{\gamma} = cb^{1/2}\):

Reversible Hydrogen Electrode (RHE)¶

Eg: Water Electrolysis

where \(\displaystyle V_{RHE} = E^{\circ} + \frac{RT}{4F} \ln(p(\ce{O_2})/p^{\circ})\)

In real case, if we want to drive the rxn, need let \(E > E^{\circ} = 1.23 \mathrm{V}\)

Chapter 2 : Fuel Cells, Batteries and Electric Double Layer¶

Determining TD functions¶

Energy Efficiency¶

MEA: 膜电极组件

Rechargable batteries¶

Charging Mode¶

• Positive Electrode: supply e^-(Sluggish) Anode
• Negative Electrode: receive e^-(Energetic) Cathode

Electric Double Layer(EDL)¶

Gouy-Chapman-Stern(GCS) Model¶

The Gibbs adsorption isotherm¶

Pure Phase(A) | …… | Pure Phase(B)

Reference System: （省略了偏摩尔量）

Actual system:

where \(A\) indicates interfacial area, we call \(\displaystyle(\frac{\partial G_S}{\partial A})_{p,T,n} = \gamma\) as Surface Tension

As p&T are const., 认为所有物质均为纯态，即\(\mu_i^R = \mu_i^S\)，we have:

\(n_i^\sigma\) is difference.

for total differential:

that is Gibbs-Duham Relation :

\(\Gamma\) is called Surface excess concertation(表面过剩浓度)

上文所说的实际上是，相对表面过剩浓度\(\Gamma_i^1\)，即认为组分1的表面过剩为零的平面为划分面，其不依赖于选取的分界面的位置 :

we have：

Electrocapillary Equation¶

for Working Electrode :

we have:

for Ref Electrode :

做全微分：

代入原方程：

where \(\sigma_M = -F\Gamma_e\), \(\sigma_S = -F(\Gamma_{K+} - \Gamma_{Cl-})\) stand for excess charge on WE and EDL

由于电中性：

于是：

由G-D方程：

代入：

that is Lippmann's equation:

Droping Mercury Electrode¶

当且仅当导数为0时达到双电层净电荷平衡，前半段表面富集电子，排斥阴离子而吸引阳离子；后半段表面缺少电子，吸引阴离子。

Adsorption Isotherm¶

Assumptions:

• All surface sites are identical
• No lateral interactions
• A full coverage can be achieved

def \(a_i^A = \frac{\Gamma_i}{\Gamma_s-\Gamma_i}\), \(\theta_i = \frac{\Gamma_i}{\Gamma_s}\)

that is:

this is Langmuir Isotherm

Including interaction among adsorbates: Frumkin isotherm

其中\(-2g\Gamma_i\)代表相互作用项。

Chapter 3 : Chemical Kinetics¶

Kinetic Theory of Gases KTG (Ideal Gases)¶

Assume one molecule in a \(abc\) box:

每次撞击动量改变为\(2mu\)，间隔时间为：

With symmetry \(\langle u_x^2 \rangle = \langle u_y^2 \rangle = \langle u_z^2 \rangle\)

that is :

let \(h(u_x,u_y,u_z)\) be the fraction of molecules with velocity between \(u_x + du_x,u_y + du_y, u_z + du_z\), as independent:

之后利用Boltzman公式，然后归一化概率密度函数：

as polar coordinate，转化为球坐标（即在薄球壳内认为\(\times 4\pi u^2\)）:

求导可得最概然速率：

Collision of gas¶

assume collision in \(dt\) in the oblique cylinder，考虑能撞上面积A的分子数：

柱形内碰撞分子数等于\(柱内分子数 \times 速度概率 \times 方向概率\)

其中\(\xi_{coll}\)代表碰撞频率，即单位面积单位时间的碰撞次数。

Mean Free Path¶

molecules in the cylinder will be collided

(in dealing with two molecules relative motion)

\(Z_A\) is called collision frequency of one molecule

\(l\) is called mean free path

the total collision frequency \(Z\) is:（the average angle is 90° so \(u_r = \sqrt{2}u\)）

Collision theory: \(r \propto Z_{AB}\) and rxn only happen when \(u_r > u_0\)

let \(\epsilon_r = \frac12 \mu u_r^2\) :

即为Arrhenius Equation

Phenomenological Kinetics¶

first/second order reactions ....

Transition State Theory:

• Reactant are in eq. with the activated complex(AC) or transition state
• AC converts to product in a irreversible step

at eq \(\mu_A + \mu_B = \mu_{AB^\ddagger}\)

from statistical mechanics: \(Q = q^N / N!\)

let \(\nu_c\) be the freq. at which AC cross over the barrier

Normal mode for the rxn¶

对于普通分子的三维配分函数：

对于AC，只有一维的平动坐标：

平衡常数

在一维坐标上积分速率得到

代入即可得到：

Arrhenius Equation¶

求导得：

由此即可得：

Chapter 4: Chemical Kinetics (II)¶

Potential Energy Surface¶

the top point of the surface is called Saddle Point

at local minimum:

at local maximum:

KIE¶

Primary KIE¶

Zero Point Energy: \(E_0 = \frac 12 h\nu\) for ground state

Harmonic Approx.: \(\displaystyle{\nu = \frac 1{2\pi}\sqrt{\frac k\mu}}, \tilde{v} = \frac 1{2\pi c}\sqrt{\frac k\mu}\)

Secondary KIE¶

KIE: \(S_N1\) ~ 1.2, \(S_N2\) ~ 1. why?

\(S_N1\) mechanism involves a temporary increase in the bond order of C-H(D)

Early vs. Late TS¶

• Early TS: solvent interacts with IS&TS similarly ⇒ No solvent effect

• Later TS: solvent interacts with IS&TS differently ⇒ expect solvent effects

Postulates based on TST¶

Bronsted-Evans-Polanyi principle (BEP)¶

for a series of similar reactions:

or

where \(0<\alpha<1\)

The Hammond Postulate¶

more reactive rxn will produce AC more resembles the reactant, result in less selectivity (Activity-Selectivity Principle)

The Curtin-Hammett Principle¶

The ratio of products is determined by the relative heights of the highest energy barriers, leading to different products

Microscopic Reversibility¶

The forward and reverse rxns go identical path

Kinetic vs Thermodynamic Control¶

Quasi Steady Approx. (QSSA)¶

Key assumptions:

• Concentrations of intermediates are low (why)

• Variations of concentrations of intermediates with time are negligible.

consider rxn:

we have:

as \(k_2\gg k_1 \Rightarrow t_{max} \to 0, y_{max} \to 0\)

for rxns with more than one intermediates:

all \(\ce{I_n}\) can consider into one steady state.

Pseudo-Equilibrium/ Rate limiting step¶

If one step has a significantly lower rate constant, then it is considered as RLS

Several steps in a row sequence could have low but comparable rate constants ⇒ kinetically relevant not RLS/RDS.

Surface Mediate rxns¶

转换频率（Turnover freq.）：

Most Abundant Reaction Intermediate (MARI):

Chapter 5: Chemical Kinetics II¶

RDS¶

let $k_{-2} =0 $

$$ \begin{aligned} \frac{d[I]}{dt} = 0 &= k_1[R] - k_{-1}[I] - k_2 [I] + k_{-2}[P] \ &= A\exp(-\frac{\Delta G_1^\ddagger}{RT}) [R] - A\exp-\frac{\Delta G_1^\ddagger+\Delta G_1}{RT} - A\exp-\frac{\Delta G_2^\ddagger}{RT}

\end{aligned} $$

when $\Delta G_1^\ddagger + \Delta G_1 \ll \Delta G_2^\ddagger $ :

Only the highest barrier determines the rate.

Degree of Rate Control (DRC)¶

define the DRC of elementary step i :

for RDS on step i :

from TST :

$$ \begin{aligned} X_{RC,i} &= (\frac{\partial \ln r}{\partial \ln k_i}){k \ &= (\frac{\partial \ln r}{\partial \ln (\frac{k_BT}{h} \exp(\frac{-\Delta G_i^\ddagger}{RT}))})}, K_i{k \ &= (\frac{\partial \ln r}{\partial (\frac{-\Delta G_i^\ddagger}{RT})})}, K_i{k}, K_i

\end{aligned} $$

we can also define Degreee of thermodynamic controlled ：

for eg:

$$ \begin{gathered} \ce{A + \ast <--> A\ast} &K_1 \ \ce{B + \ast <--> B\ast} &K_2 \ \ce{A\ast + B\ast -> AB\ast + \ast} &RDS, k_3\ \ce{AB\ast <--> AB + \ast}

\end{gathered} $$

若A为MARI物种： [ \theta_* = \frac{1}{P_A K_1} \quad \theta_B = \frac{K_2 P_B}{K_1 P_A} ]

可得\(X_{A,RC} = -2\)，为毒化剂作用

\(X_{\ddagger,RC} = 1\)意味着RDS控制反应速率

Enzymatic Catalysis¶

Michaelis Manton kinetics

Reaction with 2 substates:

solve :

Assuming \([S_2]\) is large :

Kinetics in thermal dynamically non-ideal systems¶

应用艾琳方程：

we have Broused-Bjerrum relation

Reaction is dilute electrolytes¶

if \(Z_A\) or \(Z_B = 0\) ，\(k = k_0\)

环己烯的氢气加成¶

on Pt：

H₂(g) ⇌ H₂(l)          // 溶解
H₂(l) ⇌ H₂(p)          // p: 物理吸附于表面
H₂(p) + ** → 2H* (k₃)  // **RDS：氢分子解离**

由于放热步骤，可认为过渡态和H2活度系数相近。于是\(r \propto K_H c^l_{H2}\)依赖于溶剂。

on Pd:

H₂(g) ⇌ H₂(l) ⇌ H₂(p) ⇌ 2H* (快速平衡)
RH* + H* ⇌ X^\neq → RH₂** (k₆ₐ, **RDS：表面加氢**)

RH为MARI物种，其物理状态受金属表面作用支配而与溶剂无关。

补偿效应¶

• 分解反应活化熵增加而活化焓也增加
• 缩合反应活化熵减小而活化焓也减小

焓熵相互补偿，导致\(k\)变化不大。考虑水合：

反应器¶

间歇式反应器 Batch¶

• 系统类型：封闭系统。
• 混合方式：投料时瞬间混合，反应过程中无混合。
• 浓度特征：所有组分的浓度随时间不断变化。不存在真正的稳态。
• 应用：常用于小批量、多品种的生产，如制药、精细化工。

连续搅拌釜式反应器 Continuously stirred tank reactor¶

• 系统类型：开放系统。
• 混合方式：瞬间、全混。进料一进入CSTR，其组成立即与釜内及出口物料完全相同。
• 浓度特征：釜内各处浓度均匀，且不随时间变化（稳态），但等于出口的低浓度。
• 应用：适用于对返混要求不高的反应，易于控制温度和连续操作。

活塞流反应器l Pug-flow reaction¶

• 系统类型：开放系统。
• 混合方式：在垂直于流动的方向上无混合；在沿流动的方向上无混合。流体像“活塞”一样向前推进。
• 浓度特征：反应物浓度沿反应器长度方向逐渐降低。在任一特定截面上，组成是均匀的。
• 应用：适用于要求高转化率的反应，因为它避免了返混导致的效率下降，是工业上最常用的反应器类型之一。

Chapter 6 Electrode Kinetics¶

Electro Kinetics¶

理想可极化电极：不允许电子通过，只能积累电荷，电极本身不发生电化学反应，如惰性电极

理想不可极化电极：允许电子通过，反应快且电位基本不变，如参比电极

实际存在过电位：

Semi-empirical Steady-State Treatment¶

扩散层：假定电极附近有一段无法搅拌的，厚度为\(\delta\)的停滞层（stagnant layer），其浓度是线性减小的：

where \(m = D / \delta\) stands for 质量传递系数（mass transfer coef.）. 若认为单位时间的扩散通量可用电流表示：

当\(C_O = 0\)时达到极限电流：

if R initially absent:

令半波电位\(i = i_l/2\)：

可反映相对扩散速率

id R is isolable:

when i=0, \(C_O = C_O^b\):

于是过电位：

Electrode kinetics¶

Tafel Equation:

b：Tafel 斜率 (mV/dec)，表示电流增一数量级所需的电位变化

Bulter-Volmer Kinetics

Chapter 5: Phase Diagrams¶

Canonical ensemble 正则系综¶

define

\[ \alpha = -\frac{\mu}{kT}, \beta = \frac 1{kT} \]

we have mixing entropy(用概率代替浓度):

and two condtrains:

when reaching maximum entropy, we can derive:

where \(Z(V,T,N)\)是正则配分函数

在这里用离散和是不精确的，我们改成积分形式。by using H, for single partical:

and for many partials:

if particals are 全同：

这些即认为最小的单位空间是\(\frac 1{h^3}\)，把N个粒子分配到这些空间里。

通过配分函数计算能量的期望值：

进而还可得出自由能的关系，我们有：

巨配分函数\(\Xi\)：

where \(\lambda = e^{\beta\mu}\) represent absolute acivity（绝对活度）

回到气体，对单个粒子，由于动能项积分可以从麦克斯韦分布推导，我们有：

定义构型积分：

于是：

于是我们有：

压强在恒体积下：

这里我们引入集团展开（Cluster Expansion）。我们知道维利展开：

两边系数对比：

事实上，在这里泰勒硬展开就可以得到 \(b_j\) 的各个解，但这样太麻烦了。不过比较第一项就可以得到：

这是由于此时没有任何外场。

引入Mayer函数：

于是对于稀薄气体展开：

现在假设我们认为：

对照上面的式子：

回到：

于是：

再努努力！

现在我们考虑 \(Q_3\) ，这时候就要考虑三体作用了。

于是：

经过一堆化简可以得到：

其中 \(\triangle\) 表示这是三个粒子两两作用的结果

对于第四位力系数：

其中对于四粒子系统，有3种 \(\square\) 作用，6种 \(\oslash\) 作用，1种 \(\boxtimes\) 作用。

for 维利展开：

hard-sphere :

square-well potential :

Lennard-Jones potential :

Chapter 6¶

Liquid cases: simple liquid theory¶

for 傻逼 case, He liquid 不考虑量子效应，直观上看就是解一个原子和周围的核和电子的关系，这里就要用B-O近似。

一个粒子包含六个信息：

将N个位置出现粒子的概率用向量表示：

现在我们假设只关心n个粒子的位置状态，其他的通过积分平均掉：

消掉两边

密度分布函数：在给定位置\(r_1, ...,r_n\)同时找到n个粒子的数密度

对均匀体系，显然找到一个粒子的密度分布是是密度常数 \(\rho\)

correlation function n粒子相关函数 （找到粒子\(r\)的相对概率）:

for sphere-symmetry molecules, \(g^{(2)}\) only depend on \((r_1 - r_2)\)

对这里的 \(g(r)\) 而言，归一化积分，得到的应该是除了参考粒子外得到所有其他粒子数 \(N-1\) ：

对于相关函数：

这里液体呈现出类似1D-crystal的关系，这是由于液体由于熵效应，在半径\(\sigma\)的地方出现粒子的概率很大，相应的在中间出现的概率很小。

引入结构因子 Structrural factor:

for pair-wise additive potential :

经过一番推导：

so \(\overline E = ...\)

where

作变量替换

于是

解构一下：

Chapter 7 Polymer¶

different levels¶

• primary structure
• secondary structure
• tertiary structure
• quantinary structure

simple example: random coil¶

for freely joint chain:

\(N\) as number of residues, \(l\) as monomer length, \(nl\) as end-to-end length.

define \(R_c = Nl\) as counter length:

so:

probability of observing on end-to-end distance of \(nl\) is:

so

for 1D cases:

for 3D cases:

so RMSD

conformation entropy:

and we know:

def \(\nu = n/N\), we got

calculation of chemical potential¶

Free energy:

and

这里我们需要计算一个比较难算的量 构型积分之比。我们引入一个参数 \(\xi\) 称为耦合参数，现在我们假设第 \(N\) 个粒子是一个 ，它占据一部分空间但不与其他粒子发生作用。当其从0到1的过程相当于时粒子从完全没作用线性变化到与正常作用一样。

so

where

so 原始转化为

于是就可以表达出化学势。

perturbation free energy calculation¶

然而这样很多情况会失败，原因在于据大多数情况，对于偏离平衡态的 \(H_y\) 能量会特别高，导致这个数特别小甚至就是0.

为了避免这一点，我们定义一个耦合参数 \(\lambda\) 进行插值：

这样我们就可以积分计算：

于是

在实际计算中，通常 \(P(F) = e^{-\beta H}/Q\) 是一个突跃值，实际上我i们可以通过测量及其多个温度下的曲线并作其抛物线：

Chapter 8 Brownian motion¶

Lagevin Equation¶

assume a motion satisfied Lagevin Equation:

where \(A(t)\) stands a random force.

assume \(u = c(t)e^{-\zeta t}\) ：

integrate:

transform into u:

and we can know a random value's average equals to zero:

for the RMSE：

for derectly case:

what the fucking is \(\ev{ A(t')A(t^{\prime\prime})}\) ?

let \(\ev{ A(t')A(t^{\prime\prime})} = \ev{A(0)A(t^{\prime\prime} - A')} = \phi(t^{\prime\prime}-t')\) , this forces is prop to delta function因为作用时间极短：

as \(t \to \infty\):

with = 0 :

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