​What is the dimensional formula of ab ^{-1} in the equation ​(P + a/V^2) ( V - b) = RT,

Question

What is the dimensional formula of \( a b^{-1}\) in the equation \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}\), where letters have their usual meaning.

(1)\( \left[\mathrm{M}^{0} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]\)

(2) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)

(3) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{5} \mathrm{~T}^{3}\right]\)

(4)\( \left[\mathrm{M}^{6} \mathrm{~L}^{7} \mathrm{~T}^{4}\right]\)   

Answer 1

Correct option is : (2) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)

\(\because[\mathrm{V}]=[\mathrm{b}]\)

\(\therefore\) Dimension of \( \mathrm{b}=\left[\mathrm{L}^{3}\right]\)

\(\&[\mathrm{P}]=\left[\frac{\mathrm{a}}{\mathrm{V}^{2}}\right]\)

\([\mathrm{a}]=\left[\mathrm{PV}^{2}\right]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\left[\mathrm{L}^{6}\right]\)

Dimension of \(\mathrm{a}=\left[\mathrm{ML}^{5} \mathrm{~T}^{-2}\right]\)

\(\therefore \ \mathrm{ab}^{-1}=\frac{\left[\mathrm{ML}^{5} \mathrm{~T}^{-2}\right]}{\left[\mathrm{L}^{3}\right]}=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)