Question

The equation of state of a real gas can be expressed as $(P+\frac{a}{V_2})(V-b)=cT$, where P is the pressure, V the volume, T the absolute temperature and a, b, c are constants. What are the dimensions of 'a'-

• A. $M^0{L}^3{T}^{-2}$
• B. $M{L}^{-2}{T}^5$
• C. $M{L}^{5}T-2$
• D. $M^0{L}^3{T}^0$

Answer 1

The correct option is C $M{L}^{5}T-2$

$\left(p+\frac{a}{V_2}\right)\left(v-b\right)=cT$

$p$ is pressure, $V$ is volume and $T$ is temperature

$\begin{array}{c}p=\frac{F}{A}=\frac{ML{T}^{-2}}{L^2}=M{L}^{-1}{T}^{-2}\\ V=L^3\end{array}$

We cannot add or subtract quantities of different dimensions.

$\begin{array}{c}\therefore p=\frac{a}{V^2}\\ \Rightarrow a=\frac{p}{V^2}=\frac{M{L}^{-1}{T}^{-2}}{{\left(L^3\right)}^2}=M{L}^5{T}^{-2}\end{array}$

Hence, Option $C$ is correct.