GATE 2021 CH – Question 31
The van der Waals equation of state is given by $P_r=\frac{8T_r}{3v_r-1}-\frac3{v_r^2}$, where $P_r,T_r$ and $v_r$ represent reduced pressure, reduced temperature and reduced molar volume, respectively. The compressibility factor at critical point $(z_c)$ is 3/8. If $v_r=3$ and $T_r=4/3$, then the compressibility factor based on the van der Waals equation of state is _____ (round off to 2 decimal places).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.83 to 0.85
Explanation
First evaluate the reduced pressure:
$$P_r=\frac{8(4/3)}{3(3)-1}-\frac3{3^2}=\frac43-\frac13=1.$$
Since $z=Pv/(RT)$ and $z_c=P_cv_c/(RT_c)$, the reduced variables give $z=z_cP_rv_r/T_r$. Thus
$$z=\frac38\frac{1\times3}{4/3}=\frac{27}{32}=0.84375.$$
Rounded to two decimals, **z = 0.84**.