Question

A modified form of Vander Waals' equation of state for $1$ mole is:
$\left(P+\frac{a}{V^2}\right)\left(V-b\right)=RT$
Report Boyle's Temperature $T_b$ .

• A. $T_b=\sqrt{\frac{\alpha }{R_b}}$
• B. $T_b=\sqrt{\frac{\alpha \beta }{R}}$
• C. $T_b=\sqrt{\alpha R\beta }$
• D. None of these

Answer 1

Answer: (A)

$\text{P}=\frac{\text{RT}}{\text{V}-b}-\frac{a}{{\text{TV}}^2}$
$\frac{\text{PV}}{\text{RT}}=\frac{\text{V}}{\text{V}-b}-\frac{a}{{\text{RT}}^2\text{V}}=\frac{1}{1-\frac{b}{\text{V}}}-\frac{a}{{\text{RT}}^2\text{V}}$
$\text{Z}={\left(1-\frac{b}{\text{V}}\right)}^{-1}-\frac{a}{{\left(\text{RT}\right)}^2\text{V}}$
$=1+\frac{b}{\text{V}}+\frac{b^2}{{\text{V}}^2}+\text{...}-\frac{a}{{\text{RT}}^2\text{V}}$
$=1+\left(b-\frac{a}{{\left(\text{RT}\right)}^2}\right)\frac{1}{\text{V}}+\frac{b^2}{{\left(\text{V}\right)}^2}+\text{...}$
$\therefore$ Second virial co efficient $B=b-\frac{\alpha }{{\text{RT}}^2}$
Third virial co efficient $\text{C}=b^2$
At Boyles' Temperature second virial co efficient = o
$\therefore$ $b=\frac{a}{{\text{RT}}_{{\text{b}}^2}}$
$\therefore$ ${\text{T}}_{\text{b}}=\sqrt{\frac{a}{\text{R}b}}$