20 Jun 2025
Thermodynamic Foundations and Response Functions
Equilibrium, thermodynamic variables, zeroth and first laws, heat, work, and the general heat-capacity relation.
A thermodynamic system is separated from its surroundings by a boundary. A closed system exchanges energy but not matter; an isolated system exchanges neither. The equilibrium states considered here are simple compressible states of fixed composition, described by pressure $p$, volume $V$, and absolute temperature $T$.
Variables and equilibrium
An extensive variable scales with the amount of matter: $V$, internal energy $U$, entropy $S$, and particle number $N$ are examples. An intensive variable does not scale: $p$, $T$, and density are examples. A ratio of two extensive variables, such as $U/N$, is intensive.
Thermodynamic equilibrium requires simultaneously:
- thermal equilibrium: no temperature gradient that drives heat transfer;
- mechanical equilibrium: no unbalanced pressure difference;
- chemical equilibrium: no net reaction or diffusion of components.
The zeroth law states that if systems $A$ and $B$ are separately in thermal equilibrium with system $C$, then $A$ and $B$ are in thermal equilibrium with each other. This transitive relation makes temperature a well-defined state variable and permits thermometry.
State functions, heat, and work
A state function has a change fixed by the end states; thus $dU$ is an exact differential and $\oint dU=0$. Heat and work are modes of energy transfer, not stored properties, so their infinitesimal amounts are written $\delta Q$ and $\delta W$.
Use the sign convention
\[\delta Q>0\quad\text{into the system},\qquad \delta W>0\quad\text{done by the system}.\]The first law is then
\[\boxed{dU=\delta Q-\delta W}.\]For boundary work in a quasistatic change,
\[\delta W=p_{\mathrm{ext}}\,dV.\]Only for a mechanically reversible change does $p_{\mathrm{ext}}$ differ infinitesimally from the system pressure, allowing $\delta W=p\,dV$. Expansion has $dV>0$ and positive work by the system; compression has negative work by the system. Over a cycle, $\Delta U=0$, so
\[\oint\delta Q=\oint\delta W.\]Internal energy is the state function required to make this balance path independent. Enthalpy is
\[H=U+pV.\]When $pV$ boundary work is the only work mode, constant volume gives $\delta W=0$ and $\delta Q=dU$. At constant pressure with only reversible $pV$ work,
\[dH=dU+p\,dV=\delta Q,\]so the corresponding heat capacities are
\[C_V=\left(\frac{\partial U}{\partial T}\right)_V, \qquad C_P=\left(\frac{\partial H}{\partial T}\right)_p.\]These are heat capacities of the whole sample, with units $\mathrm{J\,K^{-1}}$.
Compressibility and expansion coefficient
Define the isothermal compressibility and volume expansion coefficient by
\[\boxed{\kappa_T=-\frac1V\left(\frac{\partial V}{\partial p}\right)_T}, \qquad \boxed{\alpha=\frac1V\left(\frac{\partial V}{\partial T}\right)_p}.\]$\kappa_T$ has units $\mathrm{Pa^{-1}}$ and $\alpha$ has units $\mathrm{K^{-1}}$. Their signs are chosen so that a mechanically stable ordinary material has $\kappa_T>0$. The total differential
\[dV=V\alpha\,dT-V\kappa_T\,dp\]gives, at constant $V$,
\[\left(\frac{\partial p}{\partial T}\right)_V =\frac{\alpha}{\kappa_T}.\]General relation between $C_P$ and $C_V$
For a reversible change, $dS=\delta Q_{\mathrm{rev}}/T$ and the first law gives $dU=T\,dS-p\,dV$. Exactness of $dS$ yields the thermodynamic identity
\[\left(\frac{\partial U}{\partial V}\right)_T =T\left(\frac{\partial p}{\partial T}\right)_V-p.\]Now differentiate $H=U+pV$ at constant pressure:
\[C_P=C_V+ \left[\left(\frac{\partial U}{\partial V}\right)_T+p\right] \left(\frac{\partial V}{\partial T}\right)_p.\]Substitution of the two response identities gives
\[\boxed{C_P-C_V=\frac{TV\alpha^2}{\kappa_T}}.\]The units are $\mathrm{K\,m^3(K^{-2})/(Pa^{-1})=J\,K^{-1}}$. In a stable one-phase region $\kappa_T>0$, so $C_P\geq C_V$. For $n$ moles of ideal gas, $\alpha=1/T$, $\kappa_T=1/p$, and $pV=nRT$; hence
\[\boxed{C_P-C_V=nR}.\]The general identity is re-derived from Maxwell relations in Unit III.
Discussion