Question

The property relation (pu/RT) = Z is known as the compressibility factor for a given substance. For an ideal gas, Z = 1 at any state. (a) Find the compressibility factor Ze for a van der Waals gas at the critical point. (b) Using this value of Ze, rewrite the expressions for the constants a and b ex cluding the property v.

Answer 1

Ideal gas equation

PV=nRT

n=1mol

$Z=\frac{PV}{RT}$

Z=compressibility factor

Z=1 ideal gas

For Vander val equation

Z>1

All temperature

Critical pressure

$P_c=\frac{a}{27b^2}$

$T_c=\frac{8a}{27Rb}$

$\frac{P_cV_c}{RT_c}=\frac{3}{8}=0.375$

Wander val equation

$(P+\frac{an^2}{V^2})(V-nb)=nRT$

$(P+\frac{3}{V^2})(V-\frac{1}{3})=\frac{8t}{3}$

Where

$p=\frac{P}{P_c},v=\frac{V}{V_c}\\t=\frac{T}{T_c}$

(1) compression factor Z = PV / RT versus p (pressure in units of the critical pressure for a van der Waals gas

(2) several values of t (temperature in units of the critical temperature.) For a van der Waals gas the compression factor is greater than 1 for all temperatures greater than

t = 27/8 = 3.375 >1

(3)At this temperature, the compression factor is close to 1 up to p equals approximately ~2