Consider the sound wave travelling in ideal gases of He, CH₄, and CO₂. All the gases have the same ratio P/ρ, where P is the pressure and ρ is the density. The ratio of the speed of sound through the gases V_(He): V_(CH₄): V_(CO₂) is given by

Solution

Physics sir · 3 steps

With the same P/ρ in every gas, speed depends only on γ, so this is really about degrees of freedom.

1. Speed of sound. In an ideal gas the speed of sound is v = √(γ P/ρ). All three gases have the same P/ρ, so v ∝ √γ.

   γ = 1 + 2/f

2. Count degrees of freedom. Treat each molecule as rigid and ignore vibration. A linear molecule has 3 translational and 2 rotational degrees of freedom. A non-linear molecule has 3 of each.

   Gas	Shape	f	γ
   He	monatomic	3	5/3
   CH₄	non-linear (tetrahedral)	6	4/3
   CO₂	linear (O=C=O)	5	7/5

3. So. v_(He) : v_(CH₄) : v_(CO₂) = √(5/3) : √(4/3) : √(7/5), which is option (c).

The idea to keep. When P/ρ is the same, compare speeds of sound through γ alone. Find γ from the shape of the molecule: linear means 5 degrees of freedom, non-linear means 6.

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