Past exam of the quantum condensed matter course of the University of Cambridge / 2004 / Section B / B7

This is the Joule-Thomson coefficient, ${\mu }_{JT}=(\partial T/\partial P{)}_H$, describing a temperature change during an isenthalpic pressure change. Here $H$ is enthalpy, $C_P=(\partial H/\partial T{)}_P$ is the heat capacity at constant pressure, and $V$ is volume, all for the same chosen amount of substance. Starting with $dH=TdS+VdP$ and using the Gibbs free-energy Maxwell relation $(\partial S/\partial P{)}_T=-(\partial V/\partial T{)}_P$ givesSetting $dH=0$ provesIn a steady insulated throttle or porous plug, negligible kinetic- and potential-energy changes make the upstream and downstream enthalpies equal. A positive coefficient gives cooling on a pressure decrease; a negative one gives warming. For an ideal gas, $V\propto T$ at fixed $P$, so the coefficient is zero. Real-gas interactions permit cooling and an inversion curve, making the effect useful in gas refrigeration and liquefaction.

The measurement of the Joule-Thomson effect with a porous plug compares pressures and thermometers on the two sides of an insulated flow restriction. Suppressing external heat exchange and separating the readings from high-speed jets are part of making the enthalpy argument applicable; an arbitrary adiabatic expansion need not be isenthalpic.

The Clapeyron equation gives the slope of a two-phase phase coexistence curve. Let $s_l,s_v$ and $v_l,v_v$ be entropy and volume per molecule, or consistently per mole. Equality of the two chemical potentials and $d\mu =-sdT+vdP$ implyAt reversible coexistence the latent heat in the same amount convention is $\mathcal{L}=T(s_v-s_l)$. ThereforeThe relation assumes thermodynamic equilibrium between two bulk phases of a pure substance, neglecting curvature and other work terms. For vaporization both latent heat and volume change are positive, giving a positive vapor-pressure slope. If the vapor is approximately ideal and its volume dominates, the ideal-vapor Clausius-Clapeyron approximation gives $d\ln P/dT\simeq \mathcal{L}/(k_B{T}^2)$ with latent heat per molecule.

Experimentally, a sealed two-phase sample in a temperature-controlled bath supplies the coexistence pressure as temperature changes. Independent calorimetry and phase-density measurements supply $\mathcal{L}$ and $\Delta v$, allowing the pressure slope to test the equation. The formula connects those distinct measurements and is a practical route to latent heats.

This relates an thermodynamic equilibrium variance to the local stiffness of an appropriate thermodynamic free energy. At fixed bath temperature, suppose the effective thermodynamic free energy $A(x)$ includes the constraints and the measure for a chosen coordinate. Its thermodynamic equilibrium probability density function is proportional to $e^{-A(x)/(k_{B}T)}$. Near a stable minimum $x_0$, writeThe resulting normal distribution gives the Gaussian fluctuation approximationThis is exact for a quadratic thermodynamic free energy with the stated flat coordinate measure; in general it is a small-fluctuation result. A smaller curvature means a softer restoring response and larger fluctuations. Near vanishing curvature, higher-order terms become important, so the quadratic approximation cannot alone predict an actual infinite variance.

For example optical trap Langevin calibration uses the Brownian displacement of a trapped bead: $A=\kappa (x-x_0{)}^2/2$ gives $\kappa =k_{B}T/⟨\Delta x^2⟩$. A camera or position detector records the fluctuations. Finite exposure averages rapid motion and can bias the measured variance; the measurement protocol matters as well as the thermodynamic equilibrium identity.