Essential Thermodynamics

Notes Chemistry

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 2019/11/30   Share

Essential Thermodynamics - summary of Cambridge IA chemistry TD course.

Laws

First Law:

$dU =\delta q+ \delta w \label{td1}\tag{1}$

Second Law: entropy of the Universe increases in a spontaneous process.

Entropy is classically defined as

$dS=\frac{\delta q_{rev}}{T}. \tag{2}\label{entropy}$

Boltzmann’s entropy relates to the number of microsystems:

$S=k_B \ln \Omega.$

Master equations

1. U: Internal energy

Consider reversible gas expansion, then $(\ref{td1})$ becomes

$dU=\delta q-p\,dV$

If the heat is reversible,

$dU=T\,dS-p\,dV \label{ms1}$

2. H: Enthalpy

$\begin{align} H&=U+pV \nonumber\\ dH&=dU+p\,dV+V\,dp \nonumber\\ &=T\,dS-p\,dV+p\,dV+V\,dp\nonumber\\ &=T\,dS+V\,dp \end{align}$

3. G: Gibbs energy

$\begin{align} G&=H-TS \nonumber\\ dG&=dH-T\,dS-S\,dT \nonumber \\ &=T\,dS+V\,dp-T\,dS-S\,dT\nonumber\\ &=V\,dp-S\,dT \end{align}$

A: Hemholtz energy

$\begin{align} A&=U-TS \nonumber\\ dA&=dU-T\,dS-S\,dT \nonumber\\ &=-p\,dV-S\,dT \end{align}$

Processes

Gas expansion

Ideal gas law: $pV=nRT$ .

The work done is against the external pressure,

$\delta w=-p_\mathrm{ext}\,dV.$

Hence by integration,

$\begin{align} w&=-p_\mathrm{ext}(V_f-V_i)&&\mathrm{constant\ external\ pressure}\\ w&=-nRT\ln\frac{V_f}{V_i}&&\mathrm{isothermal,\ reversible}. \end{align}$

The work done in a reversible process is maximum.

If the process is reversible, than $q_\mathrm{rev}=-w$ , so the entropy is

$\Delta S=nR\ln\frac{V_f}{V_i}.$

Constant Volume process

Since $dV=0$ ,

$dU = \delta q_\rm{const.\ V}.$

Define constant volume heat capacity

$C_V=\left ( \frac{\partial U}{\partial T}\right )_V.$

Constant Pressure process

Substitute $dU$ into $dH$ ,

$\begin{align} dH&=\delta q+Vdp\\ dH&=\delta q_\rm{const\ p} \end{align}$

Define the constant pressure heat capacity

$C_p=\left ( \frac{\partial H}{\partial T}\right )_V. \label{H-cp} \tag{3}$

Also at constant pressure, substitute into $(\ref{entropy})$ ,

$\frac{dS}{dT}=\frac{C_p}{T}. \label{S-cp} \tag{4}$

Gibbs energy

At constant pressure, maximizing the entropy of the universe is equivalent to minimizing the Gibbs energy. At equilibrium, $dG=0$ .

Gibbs-Helmholtz equation

Consider the derivative

$\begin{align} \frac{d}{dT}\left(\frac{G}{T}\right)&=\frac{1}{T}\frac{dG}{dT}-\frac{1}{T^2}G\\ &=-\frac{1}{T}S-\frac{1}{T^2}(H-TS)\\ &=-\frac{H}{T^2}. \end{align}$

This is the Gibbs-Helmholtz equation.

The analogue for the Helmholtz free energy is $\left(\frac{\partial (A/T)}{\partial T}\right)_V=-\frac{E}{T^2}$ .

Mixtures

Ideal Gas

The Gibbs free energy of a component of ideal gas is

$G_{m,i}=G_{m,i}^\circ + RT\ln\frac{p_i}{p^\circ}.$

Chemical potential

The Gibbs energy of a system can be written in terms of chemical potentials of components by

$G=n_A\mu_A+n_B\mu_B+n_C\mu_C+...$

Then,

$\mu_i(c_i)=\mu_i^\circ+RT\ln\frac{c_i}{c^\circ}$

in an ideal solution.

The standard concentration is $\pu{1 mol dm-3}$ , and the standard pressure is $\pu{1 bar}$ .

Chemical changes

During an reaction,

$\Delta_r G=-\nu_A\mu_A-\nu_B\mu_B+\nu_C\mu_C+\nu_D\mu_D$

At equilibrium,

$\Delta_rG=0\\ \implies -\nu_A\mu_A-\nu_B\mu_B+\nu_C\mu_C+\nu_D\mu_D=0.$

Equilibrium constant

$\Delta_r G^\circ=-RT\ln K$

Put into G-H equation to get van’t Hoff isochore

$\frac{d \ln K}{d T}=\frac{\Delta_r H^\circ}{RT^2}$

Temperature dependence

Define the reaction heat capacity change to be

$\Delta_r C_p^\circ=-\nu_AC_{p,A}^\circ-\nu_BC_{p,B}^\circ+\nu_CC_{p,C}^\circ+\nu_DC_{p,D}^\circ,$

then the variation of enthalpy and entropy of reaction with temperature can be worked out (from $(\ref{H-cp}$ ) and $(\ref{S-cp})$ ).

$\Delta_rH^\circ(T_2)=\Delta_rH^\circ(T_1)+(T_2-T_1)\Delta_r C_p^\circ\\ \Delta_rS^\circ(T_2)=\Delta_rS^\circ(T_1)+\ln\frac{T_2}{T_1}\Delta_r C_p^\circ.$

Electrochemistry

From $\Delta G = nFE$ ,

$\Delta S = nF\left(\frac{\partial E}{\partial T}\right)_P.$

And then, everything can be dealt with throught Nernst equation.

$E=E^\circ-\frac{RT}{nF}\ln\frac{a_P^{\nu_P}a_Q^{\nu_Q}}{a_A^{\nu_A}a_B^{\nu_B}}$

where the $a_i$ represents the activities. In ideal solutions, these terms are represented by the concentrations $c_i$ .

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Published on：30th November 2019, 5:32 pm

Updated on：9th June 2020, 3:22 pm

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CATALOG

1. Laws
2. Master equations
3. Processes
4. Mixtures
1. 4.1. Ideal Gas
2. 4.2. Chemical potential
5. Chemical changes

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