24 Jul 2025

T-dS Equations, Heat Capacities, and Joule-Kelvin Effect

Maxwell-relation response identities and throttling coefficients for ideal and van der Waals gases.

bsc semester-iv mj-6 heat-and-thermodynamics joule-kelvin-effect heat-capacities

For a fixed-composition simple compressible system, the Maxwell relations turn entropy differentials into measurable response functions.

The two $T\,dS$ equations

Using $S=S(T,V)$,

\[dS=\left(\frac{\partial S}{\partial T}\right)_VdT +\left(\frac{\partial S}{\partial V}\right)_TdV.\]

Since $(\partial S/\partial T)_V=C_V/T$ and $(\partial S/\partial V)_T=(\partial p/\partial T)_V$,

\[\boxed{T\,dS=C_V\,dT +T\left(\frac{\partial p}{\partial T}\right)_VdV}.\]

With $(\partial p/\partial T)_V=\alpha/\kappa_T$,

\[\boxed{T\,dS=C_V\,dT+\frac{T\alpha}{\kappa_T}\,dV}.\]

Using $S=S(T,p)$ and $(\partial S/\partial p)_T=-(\partial V/\partial T)_p=-V\alpha$ gives

\[\boxed{T\,dS=C_P\,dT-TV\alpha\,dp}.\]

General value of $C_P-C_V$

Evaluate the first $T\,dS$ equation at constant pressure. Then $dV=V\alpha\,dT$ and $T\,dS=C_P\,dT$, so

\[C_P=C_V+T\left(\frac{\partial p}{\partial T}\right)_VV\alpha.\]

Therefore

\[\boxed{C_P-C_V=\frac{TV\alpha^2}{\kappa_T}}.\]

For the van der Waals molar equation

\[p=\frac{RT}{V_m-b}-\frac{a}{V_m^2},\]

direct differentiation gives

\[\left(\frac{\partial p}{\partial T}\right)_{V_m} =\frac{R}{V_m-b},\] \[\left(\frac{\partial p}{\partial V_m}\right)_T =-\frac{RT}{(V_m-b)^2}+\frac{2a}{V_m^3}.\]

Using $C_{P,m}-C_{V,m}=-T(p_T)^2/p_V$,

\[\boxed{C_{P,m}-C_{V,m} =\frac{R}{1-\dfrac{2a(V_m-b)^2}{RTV_m^3}}}.\]

The ideal-gas limit $a,b\to0$ is $C_{P,m}-C_{V,m}=R$.

Joule-Kelvin coefficient

In steady throttling through a porous plug or valve, assume adiabatic walls, no shaft work, and negligible changes of bulk kinetic and gravitational potential energy. The steady-flow energy equation then gives

\[H_1=H_2.\]

The Joule-Kelvin coefficient is

\[\mu_{\mathrm{JT}}=\left(\frac{\partial T}{\partial p}\right)_H.\]

From

\[dH=C_P\,dT+ \left[V-T\left(\frac{\partial V}{\partial T}\right)_p\right]dp,\]

set $dH=0$ to obtain

\[\boxed{\mu_{\mathrm{JT}} =\frac{T(\partial V/\partial T)_p-V}{C_P} =\frac{V}{C_P}(\alpha T-1)}.\]

Its SI unit is $\mathrm{K\,Pa^{-1}}$. A pressure drop has $dp<0$: the gas cools when $\mu_{\mathrm{JT}}>0$ and warms when $\mu_{\mathrm{JT}}<0$.

For an ideal gas, $V=nRT/p$ and $\alpha=1/T$, so

\[\boxed{\mu_{\mathrm{JT}}=0}.\]

For one mole of van der Waals gas, implicit differentiation at constant pressure gives

\[\left(\frac{\partial V_m}{\partial T}\right)_p =\frac{R/(V_m-b)} {RT/(V_m-b)^2-2a/V_m^3}.\]

Hence the exact coefficient is

\[\boxed{ \mu_{\mathrm{JT}}= \frac1{C_{P,m}} \left[ \frac{TR/(V_m-b)}{RT/(V_m-b)^2-2a/V_m^3}-V_m \right]}.\]

In the dilute, low-pressure limit $V_m\gg b$,

\[\boxed{\mu_{\mathrm{JT}}\simeq \frac1{C_{P,m}}\left(\frac{2a}{RT}-b\right)}.\]

This approximation predicts an inversion temperature $T_i\simeq2a/(Rb)=2T_B$: below it attractions dominate and throttling cools; above it excluded volume dominates and throttling warms.

The linked Unit III Maxima worksheet verifies the van der Waals $C_P-C_V$ and exact Joule-Kelvin expressions. Every displayed symbolic residual is zero.

Solved Problems

1. Derive $(\partial V/\partial T)_p$ from an equation of state

Let the equation of state be written explicitly as $p=p(T,V)$. Its total differential is

\[dp=\left(\frac{\partial p}{\partial T}\right)_VdT +\left(\frac{\partial p}{\partial V}\right)_TdV.\]

Along an isobar $dp=0$, so

\[\boxed{ \left(\frac{\partial V}{\partial T}\right)_p =-\frac{(\partial p/\partial T)_V} {(\partial p/\partial V)_T}}.\]

For a stable ordinary fluid, the denominator is negative and the numerator is usually positive, giving positive thermal expansion. The derivative has units $\mathrm{m^3\,K^{-1}}$, as required.

2. Derive the dilute van der Waals Joule-Kelvin coefficient

Expand the molar equation through first order in density:

\[p\simeq\frac{RT}{V_m}+\frac{RTb-a}{V_m^2}.\]

Solving perturbatively for volume at fixed $p$ gives

\[V_m\simeq\frac{RT}{p}+b-\frac{a}{RT}.\]

Therefore

\[T\left(\frac{\partial V_m}{\partial T}\right)_p-V_m =T\left(\frac Rp+\frac{a}{RT^2}\right) -\left(\frac{RT}{p}+b-\frac{a}{RT}\right) =\frac{2a}{RT}-b.\]

Hence

\[\boxed{\mu_{\mathrm{JT}}\simeq \frac{1}{C_{P,m}}\left(\frac{2a}{RT}-b\right)}.\]

Both terms in parentheses have molar-volume units; division by $C_{P,m}$ gives $\mathrm{K\,Pa^{-1}}$. This is a low-pressure approximation and fails near condensation or the critical region.

Descriptive Questions

Derive both $T\,dS$ equations and state the natural independent variables used in each.
Explain why $C_P-C_V$ is nonnegative in a stable single phase.
State the steady-flow assumptions that make a throttling process isenthalpic.
Explain physically why an ideal gas has zero Joule-Kelvin coefficient.

Numerical Problems

Find the molar value of $C_P-C_V$ for an ideal gas.

Final answer: $C_{P,m}-C_{V,m}=R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}$.
A system has $T=300\ \mathrm K$, $V=0.010\ \mathrm{m^3}$, $\alpha=3.0\times10^{-3}\ \mathrm{K^{-1}}$, and $\kappa_T=1.0\times10^{-5}\ \mathrm{Pa^{-1}}$.

Final answer: $C_P-C_V=TV\alpha^2/\kappa_T=2.70\ \mathrm{J\,K^{-1}}$.
For carbon dioxide use $a=0.364\ \mathrm{Pa\,m^6\,mol^{-2}}$, $b=4.27\times10^{-5}\ \mathrm{m^3\,mol^{-1}}$, $C_{P,m}=37.1\ \mathrm{J\,mol^{-1}K^{-1}}$, and $T=300\ \mathrm K$ in the dilute formula.

Final answer: $\mu_{\mathrm{JT}}=6.72\times10^{-6}\ \mathrm{K\,Pa^{-1}}=0.672\ \mathrm{K\,bar^{-1}}$.
Estimate the dilute inversion temperature for those van der Waals constants.

Final answer: $T_i=2a/(Rb)=2.051\times10^3\ \mathrm K$.
A gas with $\mu_{\mathrm{JT}}=0.25\ \mathrm{K\,bar^{-1}}$ is throttled through a pressure drop of $20\ \mathrm{bar}$ within a range where $\mu_{\mathrm{JT}}$ is constant.

Final answer: $\Delta T=\mu_{\mathrm{JT}}\Delta p=-5.0\ \mathrm K$.

The $T\,dS$ and Joule-Kelvin Maxima worksheet verifies the implicit derivative, van der Waals limits, response identity, and all five numerical answers.

References

Joule-Thomson effect, Wikipedia.
M. W. Zemansky and R. H. Dittman, Heat and Thermodynamics, 7th ed., McGraw-Hill, 1997, chapters “Thermodynamic relations” and “Real gases.”
P. Atkins, J. de Paula, and J. Keeler, Atkins’ Physical Chemistry, 11th ed., Oxford University Press, 2018, chapters “The first law” and “The properties of gases.”