First slide

Dimensions of physical quantities

Question

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

Moderate

A

$[{ML}^{8} T^{- 2}] and [L^{3 / 2}]$
B

$[{ML}^{5} T^{- 2}] and [L^{3}]$
C

$[{ML}^{5} T^{- 2}] and [L^{6}]$
D

$[{ML}^{6} T^{- 2}] and [L^{3 / 2}]$

Solution

Given, $[P + \frac{a}{V^{3}}] (V - b^{2}) = cT$
Dimensions of $\frac{a}{V^{3}} =$ Dimensions of P
$∴$ Dimensions of a = dimensions of PV³
$[a] = [\frac{F}{A} V^{3}] (∵ P = \frac{F}{A})$
$= \frac{[{MLT}^{- 2}]}{[L^{2}]} \times {[L^{3}]}^{3} = [{ML}^{8} T^{- 2}]$
Dimensions of $b^{2} =$ dimensions of V
$∴ [b] = [V]^{1 / 2} = {[L^{3}]}^{1 / 2} or [b] = [L^{3 / 2}]$

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