Question

Write the van der Waal's equation of state for a real gas. Obtain the expressions for critical volume, critical pressure and critical temperature for this gas. Calculate the critical
temperature of He if the critical pressure is
2.5 atm and critical density is 0.069 g cm -3.

Answer 1

Solution.

The van der Waals equation is a model equation of state of an imperfect gas.

$(P+\dfrac{N^2a}{V^2})(V-Nb)=Nk_bT;$

where P is the pressure, V is the volume, N is the number of molecules, T is the temperature, k_B is the Boltzmann constant, and a and b are characteristic of each real steel gas, which will be defined below.

$P_k=\dfrac{a}{27b^2}; V_k=3b; T_k=\dfrac{8a}{27bR};$

$P_k=2.5atm=2.5\sdot10^{5}Pa;$

$\rho_k=0.069g/cm^3=69kg/m^3;$

$\mu(He)=4\sdot10^{-3}kg/mol;$

$T_k-?;$

$V_k=\dfrac{3}{8}\dfrac{m}{\mu}\dfrac{RT_k}{P_k}\implies$

$T_k=\dfrac{8\mu P_kV_k}{3mR_{He}}= \dfrac{8\mu P_k}{3\rho_kR_{He}};$

$R_{He}=\dfrac{R}{\mu};$

$R_{He}=\dfrac{8.31J/molK}{4\sdot10^{-3}kg/mol}=2.0775\sdot10^3J/(kgK)$ ;

$T_k=\dfrac{8\sdot4\sdot10^{-3}\sdot2.5\sdot10^5}{3\sdot69\sdot2.0775\sdot10^3}=0.019K;$

Answer: $T_k=0.019K.$