Question

Which of the following does not possess the same dimensional formula as that for $RT$:

• A. $PV$
• B. $Pb$
• C. $\frac{a}{V^2}$
• D. $\frac{ab}{V^2}$

Answer 1

Answer: (C)

$\frac{a}{V^2}$

The vander Waals gas equation is $\left(\begin{array}{c}P+\frac{a}{V^2}\end{array}\right)(V-b)=RT$, where $P$ is the pressure, $V$ is molar volume and $T$ is the temperature of the given sample of gas. $R$ is called mola gas constant, $a$ and $b$ are called vander Walls constants.

$\left(\begin{array}{c}P+\frac{a}{V^2}\end{array}\right)(V-b)=RT$,

$PV+a/V-Pb-ab/V^2=RT$

so,

$\left[PV\right]=\left[RT\right],\left[Pb\right]=\left[PV\right]=\left[RT\right]$
$\left[\begin{array}{c}\frac{a}{V^2}\end{array}\right]=\left[\begin{array}{c}\frac{P{V}^2}{V^2}\end{array}\right]=\left[P\right]\ne \left[RT\right]$ and
$\left[\begin{array}{c}\frac{ab}{V^2}\end{array}\right]=\left[\begin{array}{c}\frac{\left(P{V}^2\right)V}{V^2}\end{array}\right]=\left[PV\right]=\left[RT\right]$