Question

Gases $X,Y,Z,P$ and $Q$ have their van der Waals' constants ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$ (in CGS units) as shown below:
$\begin{array}{cccccc}& X& Y& Z& P& Q\\ a& 6& 6& 20& 0.05& 30\\ b& 0.025& 0.15& 0.1& 0.02& 0.2\end{array}$
The gas with the highest critical temperature is:

• A. $P$
• B. $Q$
• C. $Y$
• D. $Z$
• E. $X$

Answer 1

The correct option is E $X$

For any gas, critical temperature $\left(T_c\right)$ is the temperature at or above which the gas cannot be liquefied, $T_c=\frac{8a}{27Rb}$, where a ,b are van der Waals' constant and R is the gas constant.
$\therefore$ Greater the value of $\left(\frac{a}{b}\right)$, more will be the critical temperature of a gas.
a) For gas $P$, value of $\frac{a}{b}$ = $\frac{0.05}{0.02}=2.5$
b) For gas $Q$, value of $\frac{a}{b}$ = $\frac{30}{0.2}=150$
c) For gas $Y$, value of $\frac{a}{b}$ = $\frac{6}{0.15}=40$
d) For gas $Z$, value of $\frac{a}{b}$ = $\frac{20}{0.1}=200$
e)For gas $X$, value of $\frac{a}{b}$ = $\frac{6}{0.025}=240$
For gas $X$, $\left(\frac{a}{b}\right)$ has the maximum value. Hence it has the highest critical temperature $T_c$ .