Question

The equation of state of a gas is given by $\left(\begin{array}{c}P+\frac{a}{v^3}\end{array}\right)(V-b^2)=cT$, where P, V, T are pressure, volume and temperature respectively, and a, b, c are constants. The dimensions of a and b are respectively

• A. $M{L}^8{T}^{-2}$ and $L^{3/2}$
• B. $M{L}^5{T}^{-2}$ and $L^3$
• C. $M{L}^5{T}^{-2}$ and $L$
• D. $M{L}^6{T}^{-2}$ and $L^{3/2}$

Answer 1

The correct option is A $M{L}^8{T}^{-2}$ and $L^{3/2}$

The dimensions of $\frac{a}{v^3}$ must be that of pressure.

Dimension of pressure are $\left[M{L}^{-1}{T}^{-2}\right]$

Hence dimension of $a$ is $\left[M{L}^{-1+3\ast 3}{T}^{-2}\right]=\left[M{L}^8{T}^{-2}\right]$

The dimension of $b^2$ must be that of volume.

Hence $\left[b^2\right]=\left[L^3\right]$

$⟹\left[b\right]=\left[L^{3/2}\right]$