Question

The gas equation for $n$ moles of a real gas is $(P+\frac{a}{V^2})(V-b)=nRT$, where $P$ is the pressure, $V$ is the volume, $T$ is the absolute temperature, $R$ is the molar gas constant and $a,b$ are arbitrary constants. Which of the following have the same dimensions as those of $PV$ ?

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

Answer 1

Answer: (A), (B), (C), (D)

$(P+a/V^2)(V-b)=nRT$

$\Rightarrow \left[P\right]=\left[a/V^2\right]$ and $\left[V\right]=\left[b\right]$

$\Rightarrow \left[P\right]\left[V\right]=\left[a/V^2\right]\left[V\right]$

$\Rightarrow \left[PV\right]=\left[a/V\right]$

Similarly $\Rightarrow \left[PV\right]=\left[P\right]\left[b\right]\left[P\right]=\left[a/V^2\right]$ [Since, $\left[V\right]=\left[b\right]$]

$\Rightarrow \left[P\right]\left[V\right]=\left[a/V^2\right]\left[b\right]$ ;

⇒[P] [V]=$\left[ab/V_2\right]$

Conclusion:

The correct options are

(A) $nRT$

(B) $\frac{a}{V}$

(C) $Pb$

(D) $\frac{ab}{V^2}$