Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is √2 times the speed of sound, then the value of n will be :

Solution

Physics sir · 4 steps

The rms and sound speeds share √(RT/M), so their ratio fixes γ of the mixture.

1. Compare the speeds. In the mixture, vᵣₘₛ = √(3RT/M) and v_(sound) = √(γ RT/M), with the same T and the same mean molar mass M. Dividing gives vᵣₘₛ/v_(sound) = √(3/γ) = √2, so γₘᵢₓ = 3/2.

   γₘᵢₓ = (n₁ Cₚ₁ + n₂ Cₚ₂)/(n₁ Cᵥ₁ + n₂ Cᵥ₂) = (2 × 5/2R + n × 7/2R)/(2 × 3/2R + n × 5/2R)

2. Use the heat capacities. Helium is monatomic: Cᵥ = 3/2R, Cₚ = 5/2R. Hydrogen is diatomic: Cᵥ = 5/2R, Cₚ = 7/2R. The R cancels, leaving γₘᵢₓ = (5 + 3.5n)/(3 + 2.5n).
3. Solve for n. Set (5 + 3.5n)/(3 + 2.5n) = 3/2. Cross-multiplying gives 10 + 7n = 9 + 7.5n, so 0.5n = 1.
4. So. n = 2 moles of hydrogen, which is option (b).

The idea to keep. For a gas mixture, never average γ directly. Add up nCₚ and nCᵥ separately, then divide. Ratios of molecular speeds in the same gas depend only on their numerical factors.

Answer from the NTA final answer key. Solution by PrepWiser.