The viscosity η of a gas depends on the long-range attractive part of the intermolecular force, which varies with molecular separation r according to F = μ r – n where n is a number and μ is a constant. If η is a function of the mass m of the molecules, their mean speed v , and the constant μ then which of following is correct -

Question

The viscosity η of a gas depends on the long-range attractive part of the intermolecular force, which varies with molecular separation r according to F = μ r – n where n is a number and μ is a constant. If η is a function of the mass m of the molecules, their mean speed v , and the constant μ then which of following is correct -, η ∝ m n + 1 v n + 3 μ n - 2, η ∝ m n + 1 n - 1 v n + 3 n - 1 μ - 2 n - 1, η ∝ m n v - n μ - 2, η ∝ m v μ - n

MARKS App · Accepted Answer

D i m e n s i o n o f η ≡ F r A v ≡ M L T - 2 [ L ] L 2 L T - 1 η ≡ M L - 1 T - 1 Dimensions of μ = F r n μ = M L T - 2 L n μ = M L n + 1 T - 2 Let ‘ η ’ depend on mass m mean speed v and constant μ as - η ∝ m a v b μ c M L - 1 T - 1 ∝ M a L T - 1 b M L n + 1 T - 2 c M L - 1 T - 1 ∝ M a + c L b + c n + 1 T - b - 2 c Equating dimensions both sides a + c = 1 ⇒ c = 1 - a b + c n + 1 = - 1 ⇒ b = - 1 + c n + 1 - b + 2 c = - 1 ⇒ b = 1 - 2 c ∴ 1 - 2 c = - 1 + c n + 1 2 c - 1 = 1 + c n + 1 2 c - c n + 1 = 2 c 2 - n - 1 = 2 c 1 - n = 2 = c = - 2 n - 1 a = 1 - c a = 1 + 2 n - 1 = n - 1 + 2 n - 1 a = n + 1 n - 1 b = 1 - 2 c = 1 + 4 n - 1 = n - 1 + 4 n - 1 b = n + 3 n - 1 ∴ η ∝ m n + 1 n - 1 v n + 3 n - 1 μ - 2 n - 1

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