Question

Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of final pressure of gas in B to that of gas in A is ______.

Options

• `2^(γ - 1)`

• `(1/2)^(γ - 1)`

• `(1/(1 - γ))^2`

• `(1/(γ - 1))^2`

Answer 1

Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of final pressure of gas in B to that of gas in A is `underline(2^(γ - 1))`.

Explanation:

According to the P-V diagram is shown for container A (which is going through an isothermal process) and for container B (which is going through an adiabatic process).

              V →container A

               V →container A

Both processes involve compression of the gas.

(i) Isothermal compression (gas A) (during 1→ 2)

P₁V₁ = P₂V₂

⇒ P₀(2V₀)^γ = P₂(V₀)^γ

⇒ P₀(2V₀) = P₂(V₀)

(ii) Adiabatic compression, (gas B) (during 1→ 2)

P₁V₁^γ = P₂V₂^γ

⇒ P₀(2V₀)^γ = P₂(V₀)^γ

⇒ P₂ = `((2V_0)/V_0) P_0 - (2)^γ P_0`

Hence `((P_2)_B)/((P_2)_A)` = ratio of final pressure = `((2)^γ P_0)/(2P_0) = 2^(γ - 1)` where, γ is ratio of specific heat capacities for the gas.