What is the coefficient of thermal expansion of real gases in terms of temperature and pressure

Question

Is there an equation for the coefficient of thermal expansion of real gases. If there is, what is it then? because I tried to use Peng Robinson equation of state, by substituting it's equation in terms of volume, I was getting into a mess.

Answer 1

All equations of state (Van-der-Wall, clapeyron, the one you cited) are particular cases of the most general equation, the virial equation.

https://en.wikipedia.org/wiki/Virial_expansion (1)

Notice that there are an infinite number of gases, with different chemical properties. What you want, a general equation for all gases, is pratically impossible. You need to adjust the equations you know for each gas, comparing the set of parameters that best fit the behaviour of a real gas with the predict by (1)

Answer 2

Is there an equation for the coefficient of thermal expansion of real gases.

If you have an expression for the volume $V$, you can use the definition of the thermal expansion coefficient $\alpha$ as the relative increase in volume with increasing temperature $T$ at constant pressure $P$:

$$\alpha\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P.$$

If you have an expression for the entropy $S$, you can use a Maxwell relation:

$$\alpha\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P=-\frac{1}{V}\left(\frac{\partial S}{\partial P}\right)_T.$$

If you have an energy potential such as the Gibbs free energy, you can relate that to the volume and proceed:

$$\alpha\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P=\frac{1}{V}\left(\frac{\partial^2 G}{\partial P\,\partial T}\right)=\frac{1}{V}\left(\frac{\partial^2 G}{\partial T\,\partial P}\right),$$

where the last term of course is the source of the Maxwell relation above, as $S\equiv-\left(\frac{\partial G}{\partial T}\right)_P$.

If you have the compressibility factor $Z=\frac{nRT}{PV}$, you can apply that:

$$\alpha\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P=\frac{1}{T}\left[1+\frac{T}{Z}\left(\frac{\partial Z}{\partial T}\right)_P\right]=\frac{1}{T}\left[1+T\left(\frac{\partial \ln Z}{\partial T}\right)_P\right],$$

which is exactly the approach used in Pratt, "Thermodynamic Properties Involving Derivatives Using the Peng-Robinson Equation of State," Chemical Engineering Education, 35(2), 112-115 (2001), to find the thermal expansion coefficient for the Peng–Robinson gas (too intricate an expression to bother retyping).