In an adiabatic process, the density of a diatomic gas becomes 32 times its initial value. The final pressure of the gas is found to be n times the initial pressure. The value of n is :

Solution

Physics sir · 4 steps

Turn volume into density, and the adiabatic law becomes P ∝ ρ^γ. Then 32 is just 2⁵.

1. Volume to density. For a fixed mass of gas, ρ = m/V, so V ∝ 1/ρ. Put this into PV^γ = constant and it becomes P ∝ ρ^γ.

   P₂/P₁ = (ρ₂/ρ₁)^γ

2. Gamma for diatomic. A diatomic gas has C_V = 5/2R and C_P = 7/2R, so γ = 7/5.
3. Put in numbers. n = 32^(7/5) = (2⁵)^(7/5) = 2⁷ = 128.
4. So. The final pressure is 128 times the initial pressure, which is option (a).

The idea to keep. In adiabatic problems, rewrite PV^γ in whatever variable the question gives you. With density, P ∝ ρ^γ. Remember γ = 5/3 for monatomic and 7/5 for diatomic.

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