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Standard XII

Physics

Dimensional Analysis

The equation ...

Question

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

[M L^{8} T^{- 2}]

and

[L^{3 / 2}]

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[M L^{5} T^{- 2}]

and

[L^{3}]

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[M L^{5} T^{- 2}]

and

[L^{6}]

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[M L^{6} T^{- 2}]

and

[L^{3 / 2}]

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Solution

The correct option is B $[M L^{8} T^{- 2}]$ and $[L^{3 / 2}]$
Given, $[P + \frac{a}{V^{3}}] (V - b^{2}) = c T$
Dimension of $\frac{a}{V^{3}}$ = Dimensions of $P$
$∴$ Dimensions of a = dimensions of $P V^{3}$
$[a] = [\frac{F}{A} V^{3}]$ $(∴ P = \frac{F}{A})$
$= \frac{[M L T^{- 2}]}{L^{2}} \times [L^{3}]^{3} = [M L^{8} T^{- 2}]$
Dimensions of $b^{2}$ = dimensions of V
$∴ [b] = [V]^{1 / 2} = [L^{3}]^{1 / 2}$ or $[b] = [L^{3 / 2}]$

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The equation of state of a gas is given by (P+aV3)(V−b2) = cT, where P, V, Tare pressure, volume and temperature respectively, and a, b, c are constants. The dimensions of a and b are respectively:

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