Question

One way of writing the equation of state for real gases is, $P\overline{V}=RT[1+\frac{B}{V}+......]$ where $B$ is constant. An approximate expression for '$B$' in terms of van der Waal's constant 'a' and 'b' is :

• A. $b+a/RT$
• B. $b-a/RT$
• C. $b+a/2RT$
• D. $a+b/RT$

Answer 1

The correct option is B $b-a/RT$

Van der Waal's equation:

$(P+\frac{a}{{\overline{V}}^2})(\overline{V}-b)=RT$

On rearranging:

$P=\frac{RT}{\overline{V}-b}-\frac{a}{RT\overline{V}}$

$\Rightarrow \frac{P\overline{V}}{RT}=\frac{1}{1-\frac{b}{\overline{V}}}-\frac{a}{RT\overline{V}}$

As we know, using binomial expansion,

$\frac{1}{1-x}=1+x+x^2+x^3+......$

Expanding $\frac{1}{1-\frac{b}{\overline{V}}}$using binomial expansion,

$\frac{P\overline{V}}{RT}=(1+\frac{b}{\overline{V}}+\frac{b^2}{{\overline{V}}^2}+........)-\frac{a}{RT\overline{V}}$

$\Rightarrow \frac{P\overline{V}}{RT}=1+(b-\frac{a}{RT})\frac{1}{\overline{V}}+\frac{b^2}{{\overline{V}}^2}+......$

Comparing with the given equation:

$\Rightarrow B=b-\frac{a}{RT}$

Hence, the correct answer is option $\text{B}$.