Question

The Van der Waal's equation of state for some gases can be expressed as:
$(P+\frac{a}{V^2})(V-b)=RT$
where P is the pressure, V is the molar volume, and T is the absolute temperature of the given sample of gas and a,b and R are constants.
Which of the following does not have the same dimensional formula as that for RT?

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

Answer 1

The correct option is C $\frac{a}{V^2}$

The given equation can be simplified as:

$PV+\frac{a}{V}-Pb-\frac{ab}{V^2}=RT$

From the given equation we can say,

$\left[RT\right]=\left[P\right]\left[V\right]$

$\Rightarrow \left[RT\right]=\left[{ML}^{-1}{T}^{-2}\right]\left[L^3\right]$

$\Rightarrow \left[RT\right]=\left[{ML}^2{T}^{-2}\right]$

The dimensions for $RT$.

$\left[Pb\right]=\left[{ML}^{-1}{T}^{-2}\right]\left[L^3\right]$

$=\left[{ML}^2{T}^{-2}\right]=\left[RT\right]$

The dimensions of $\frac{a}{V^2}$

$\Rightarrow \left[\frac{a}{V^2}\right]=\frac{\left[{ML}^5{T}^{-2}\right]}{L^6}=\left[{ML}^{-1}{T}^{-2}\right]$

So, option $C$ is not correct.