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 \(c_i\).