If the equation of state of a gas is expressed as `(P + a/(V^2)) (V - b) = RT` where P is the pressure, V is the volume and T the absolute temperature and a, b , R are constants, then find the dimensions of 'a' and 'b' ?

Explanation

By principle of homogeneity of dimensions P can added to P only. It means `(a)/(V^(2))` also gives pressure. Dimension formulae for pressure `(P)= M^(1) L^(-1) T^(-2)` and Volume (V) `= M^(0) L^(3) T^(0)`.
Since `(a)/(V^(2))`= pressure
`therefore (a)/((M^(0)L^(3)T^(0))^(2))= M^(1)L^(-1) T^(-2)`
`rArr (a)/(M^(0)L^(6)T^(0))= M^(1)L^(-1) T^(-2)`
`therefore a= M^(1) L^(5)T^(-2)`
Similarly, b will have same dimensions as volume `V-b=` volume
`therefore b= M^(0) L^(3) T^(0)`