Question

Find dimensions of constants $a$ and $b$ in equation.
$(P+\frac{a}{V^2})(V-b)=RT$, where $P$ is pressure and $V$ is volume, $R$ is universal gas constant, $T$ is temperature.

Answer 1

We can add and structure only like quantities.
$\Rightarrow$ Dimensions of $P$ = Dimensions of $\frac{a}{V^2}$ ...... (i)
and dimension of $V$ = Dimensions of b ............ (ii)
From (i),
Dimensions of a = Dimensions of $P\times$ Dimensions of $V^2\left[a\right]$

$=\left[M^1{L}^{-1}{T}^{-2}\right]\times [L^3{]}^2$

$=\left[M^1{L}^5{T}^{-2}\right]$

Unit of $a=$ Unit of $p\times Unitof{v}^2$

$=\frac{N}{m^2}\times m^6=N{m}^4$

From (ii), [b] = $\left[V\right]=\left[M^0{L}^3{T}^0\right]$
So unit of b = Unit of V = $m^3$