Thermodynamic Relations

Equations of state

The ideal-gas law, $pv = RT$, fails near condensation (Lesson 3). In 1873 Johannes van der Waals proposed the first equation of state that captured both gas and liquid:

$p = \frac{RT}{v - b} - \frac{a}{v^2}.$

The constant $b$ is the volume the molecules themselves occupy; the $a/v^2$ term is the pull of their mutual attraction, which reduces the pressure on the walls. Both constants can be found from the critical point, and in reduced form — pressure, temperature and volume divided by their critical values — the equation is the same for every fluid. That is the origin of the principle of corresponding states and the generalised compressibility chart.

Modern cubic equations, like the Peng–Robinson equation behind this course's refrigerant and real-gas calculations, refine the attraction term but keep the structure. Below the critical temperature, all of them produce a strange wiggle:

On part of the wiggle, squeezing the fluid would lower its pressure — mechanically unstable. Real fluids don't follow it; they split into liquid and vapour at a constant pressure. Maxwell showed where that pressure must be: where the two areas between the curve and the horizontal line are equal. The equal-area rule comes from requiring the Gibbs function, $g = h - Ts$, to be the same in both phases — a condition we will meet again for chemical equilibrium in Lesson 14.

The web of properties

For a simple compressible substance, two properties fix the state, so every property is a function of two others. Partial derivatives then connect them, and the first and second laws provide four fundamental relations — the T ds equations of Lesson 6 and two more:

$du = T\,ds - p\,dv, \quad dh = T\,ds + v\,dp, \quad d\psi = -p\,dv - s\,dT, \quad dg = v\,dp - s\,dT,$

with the Helmholtz function $\psi = u - Ts$ and the Gibbs functionGibbs function GG = H − TS. At fixed temperature and pressure, a system reaches equilibrium where G is a minimum. $g = h - Ts$. Each is an exact differential, so its mixed second derivatives must be equal. That gives the four Maxwell relationsMaxwell relationsEqualities between partial derivatives — such as (∂s/∂p)_T = −(∂v/∂T)_p — that let unmeasurable properties be computed from measurable ones., for example

$\left(\frac{\partial s}{\partial p}\right)_T = -\left(\frac{\partial v}{\partial T}\right)_p.$

The left side involves entropy, which can't be measured. The right side is just how much a substance expands when heated — measurable with a ruler and a thermometer. Maxwell relations let us compute the unmeasurable from the measurable. Combined with heat-capacity data, they generate entire property tables from an equation of state.

Payoff 1: the Clapeyron equation

Apply the Maxwell relation $(\partial p/\partial T)_v = (\partial s/\partial v)_T$ across a phase change, where pressure depends only on temperature, and you get the Clapeyron equationClapeyron equationdp_sat/dT = h_fg/(T v_fg): the slope of the saturation curve is set by the latent heat.:

$\left(\frac{dp}{dT}\right)_\text{sat} = \frac{h_{fg}}{T\,v_{fg}}.$

The slope of the vapour-pressure curve gives the latent heat, with no calorimeter. Treat the vapour as an ideal gas and neglect the liquid's volume, and it becomes the Clausius–Clapeyron equation, $d\ln p/d(1/T) = -h_{fg}/R$: a straight line on a plot of $\ln p$ against $1/T$.

The same equation applied to melting explains one of water's oddities. Ice is less dense than liquid water, so $v_{fg}$ for melting is negative, and the melting line slopes backwards: squeezing ice lowers its melting point, by about 1 K per 135 bar.

Payoff 2: the Joule–Thomson coefficient

Lesson 4 said an ideal gas doesn't change temperature in a throttle. A real one does. The Joule–Thomson coefficient measures how much:

$\mu_J = \left(\frac{\partial T}{\partial p}\right)_h = \frac{1}{c_p}\left[T\left(\frac{\partial v}{\partial T}\right)_p - v\right].$

The expression on the right — another product of the Maxwell relations — needs only the equation of state and $c_p$. For an ideal gas $T(\partial v/\partial T)_p = v$ and the coefficient is zero. For real gases it can have either sign:

At moderate temperatures, molecular attraction dominates: as the gas expands, its molecules pull against each other, and it cools. At high temperatures, molecular size dominates and it warms. The boundary is the inversion curve. Nitrogen inverts at about 620 K, so it cools in any throttle at room temperature. Hydrogen's maximum inversion temperature is about 200 K, so hydrogen escaping through a leak at room temperature warms; helium must be below about 45 K before it will cool.

Next: back to the plant, and the plume over its cooling tower. Why does a glass of cold water sweat?