Physical Chemistry II¶
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
Pressure¶
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 \(\ln{K} = -\frac{\Delta_rG^{\circ}}{RT} = -\frac{1}{R} (\frac{\partial \Delta_rG^{\circ}/T}{\partial T}) = \frac{\Delta_rH^{\circ}}{RT^2}\)
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 \(\Omega = \frac{N_0!}{(N_0 - N)!(N)!}\)
let x = N/N_0, using Stiring Approximation:
for general cases:
Electrochemical Potential¶
so as \(\mu = (\frac{\Delta G}{\Delta N})_{T,p,\varphi} = (\frac{\partial g}{\partial x})_{T, p, \varphi}\)
For a lattice as:
so: $$ \mu = h' + k_BT\ln{\frac{x}{1-x}} + ze\varphi $$
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
eg: \(\ce{2H_2 + O_2 -> 2H_2O}\), \(E = 1.23\mathrm{V}\)
so \(\ce{0.5O_2 + 2H^+ + 2e^- <--> H_2O}\), \(E = 1.23 \mathrm{V}\)
eg2: \(\ce{AgCl + e <--> Ag + Cl^-}\)
where \(b\) stands molality, \(\gamma\) stands activity coefficient
as Debye-Huckle Formula \(\ln{\gamma} = cb^{1/2}\):
Reversible Hydrogen Electrode (RHE)¶
Eg: Water Electrolysis
where \(V_{RHE} = E^{\circ} + \frac{RT}{4F} \ln{p(\ce{O_2})/p^{\circ}}\)
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¶
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)
Ref:
Actual:
we call \((\frac{\partial G_S}{\partial A})_{p,T,n} = \gamma\) as Surface Tension
As p&T are const., we have:
$$ \dd G^\sigma = \dd G_S - \dd G_R = \sum_i \mu_i(\dd n_i^S - \dd n_i^R) + \gamma \dd A= \sum_i \mu_i\dd n_i^\sigma + \gamma \dd A $$ for total differential:
As Gibbs-Duham Relation :
$$ A\dd \gamma + \sum_i n_i^\sigma \dd \mu_i = 0 \ -\dd \gamma = \sum_i \frac{n_i^\sigma}{A} \dd\mu_i = \sum_i \Gamma_i \dd\mu_i $$ \(\Gamma\) is called Surface excess concertation(表面过剩浓度)
Electrocapillary Equation¶
Wire---Cu|Ag|AgCl(Ref)| K+, Cl-, M(ekectrolyte) |(WE)Hg|Ni|Cu---Wire
for Working Electrode :
$$ \begin{aligned} -\dd \gamma &= (\Gamma_{Hg}\dd\mu_{Hg} + \Gamma_{e}\dd\mu_{e})\ &+(\Gamma_{K}\dd\mu_{K} + \Gamma_{Cl}\dd\mu_{Cl})\ &+(\Gamma_{M}\dd\mu_{M} + \Gamma_{H_2O}\dd\mu_{H_2O}) \end{aligned} $$ 天书
Droping Mercury Electrode¶
Adsorption Isotherm¶
Assumptions:
- All surface sites are identical
- No lateral interactions
- A full coverage can be achieved
def \(a_i = \frac{\Gamma_i}{\Gamma_s-\Gamma_i}\), \(\theta = \frac{\Gamma_i}{\Gamma_s}\)
that is: $$ \frac{\theta}{1-\theta} = a_ibe\ \theta_i = \frac{a_i}{RT}be{1+\sum_j^N a_j}{RT}}be $$ this is }{RT}}Langmuir Isotherm
Chapter 3 : Chemical Kinetics¶
Kinetic Theory of Gases KTG (Ideal Gases)¶
$$ p = (\frac{\partial U}{\partial V})_S \quad \text{Pressure is a measurement of energy density} $$ Assume one molecule in a \(abc\) box:
$$
\Delta t = \frac{2a}{u_{1x}}
$$
$$
F_1 = \frac{\Delta (mu_x)}{\Delta t} = \frac{mu_{1x}^2}{a} \
P_1 = \frac{F_1}{bc} = \frac{mu_{1x}^2}{V}\
P = \sum_i P_i = \frac mV \sum_i u_{ix}^2
$$
With symmetry \(\langle u_x^2 \rangle = \langle u_y^2 \rangle = \langle u_z^2 \rangle\)
as polar coordinate: $$ F(u)du = 4\pi \left( \frac{m}{2\pi RT} \right)^\frac 32 u^2 e{-mu2/2RT} du\ \boxed{ \langle u \rangle = 4\pi \left( \frac{m}{2\pi RT} \right)^\frac 32 \int u^3 e{-mu2/2RT} du = \sqrt{\frac{8RT}{\pi M}} }\ \langle u^2 \rangle = 4\pi \left( \frac{m}{2\pi RT} \right)^\frac 32 \int u^4 e{-mu2/2RT} du = \frac{3RT}{M} $$
Collision of gas¶
assume collision in \(dt\) in the oblique cylinder
Mean Free Path¶
molecules in the cylinder will be collided $$ dN_{coll} =\rho\pi d^2 \langle u \rangle dt \ Z_A = \frac{dN_{coll}}{dt} = \rho\pi d^2\sqrt{\frac{8RT}{\pi \mu}} = \rho\pi d^2\sqrt{\frac{16RT}{\pi m}} $$ (in dealing with two molecules relative motion)
\(Z_A\) is called collision frequency of one molecule $$ l = \langle u \rangle dt = \langle u \rangle / Z_A = \frac{1}{\sqrt{2}\rho \pi d^2} = \frac{RT}{\sqrt{2}N_AP \pi d^2} $$ \(l\) is called mean free path
the total collision frequency \(Z\) is:（the average angle is 90° so \(u_r = \sqrt{2}u\)） $$ Z_{AA} = \frac 12 \rho Z_A = \frac12 \pi d^2 \langle u_r \rangle \rho^2 = \frac{1}{\sqrt2} \pi d^2 \langle u \rangle \rho^2\ Z_{AB} = \sigma_{AB} \langle u_r \rangle \rho_A \rho_B = \pi\left( \frac{d_A + d_B}{2} \right)^2 ·\sqrt{\frac{8RT}{\pi \mu}}·\rho_A \rho_B $$ Collision theory: \(r \propto Z_{AB}\) and rxn only happen when \(u_r > u_0\)
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:
let \(\nu_c\) be the freq. at which AC cross over the barrier
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.: \(\displaystyle{\nu = \frac 1{2\pi}\sqrt{\frac k\mu}}\)
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¶
Postulates based on TST¶
Bronsted-Evans-Polanyi principle (BEP)¶
for a series of similar reactions: $$ \Delta G_1^\ddagger - \Delta G_2^\ddagger = \alpha(\Delta G_1-\Delta G_2) $$ or $$ \ln(k_2/k_1) = \alpha\ln(K_2-K_1) $$ 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: $$ \ce{A ->[k_1] B ->[k_2] C} $$ we have: $$ \begin{gathered} \frac{dx}{dt} = k_1x \quad \frac{dy}{dt} = k_1x-k_2y \quad \frac{dz}{dt} = k_2y \ \Rightarrow y = \frac{k_1}{k_2 - k_1}(e^{-k_1t} - e^{-k_2t}) \ \frac{dy}{dt} = 0 \Rightarrow t_{max} = \frac 1{k_2-k_1}\ln{\frac {k_2}{k_1}} \ y_{max} = \frac {k_1}{k_2-k_1}[(\frac {k_2}{k_1})^{\frac {k_1}{k_2-k_1}} - (\frac {k_2}{k_1})^{\frac {k_2}{k_2-k_1}}] \end{gathered} $$ as \(k_2 \gg k_1 \Rightarrow t_{max} \to 0, y_{max} \to 0\)
for rxns with more than one intermediates: $$ \ce{A->[k_1]I_1 ->[k_2]I_2->...->I_n->[k_{n+1}]B} $$ 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.