Question

The equation of state for a gas is given by $(P+\frac{a}{V^2})(V-b)=RT$,
where $P$, $V$ and $T$ are pressure, volume and temperature respectively and $R$ is the universal gas constant. The dimensions of the constant $a$ is

• A. $\left[M^{5}L{T}^{-2}\right]$
• B. $\left[M{L}^5{T}^{-2}\right]$
• C. $\left[M{L}^2{T}^{-2}\right]$
• D. $\left[M{L}^5{T}^2\right]$

Answer 1

The correct option is B $\left[M{L}^5{T}^{-2}\right]$

Given equation is
$(P+\frac{a}{V^2})(V-b)=RT$
From the principle of homogeneity of dimensions,
$P$ and $\frac{a}{V^2}$ have same dimension.
$\therefore \left[a\right]=\left[P{V}^2\right]=[\frac{ML{T}^{-2}}{L^2}\times (L^3{)}^2]=\left[M{L}^5{T}^{-2}\right]$

Why this question? Tip: Principle of homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another.