Concepts Of Physics MCQ Edition [Volume 2]PhysicsSpecific Heat Capacities of Gases
An adiabatic cylindrical tube having a volume V₀ is separated into two parts by a frictionless adiabatic separator, as shown in the figure. Initially, the separator is held in the middle. The left part is filled with an ideal gas at pressure p₁ and temperature T₁ , while the right part contains another ideal gas at pressure p₂ and temperature T₂ . The specific heat ratio, C_p / C_V = , is identical for both gases. Th
Options
- Ap₁^ 1/ V₀ p₁^ 1/ + p₂^ 1/ and p₂^ 1/ V₀ p₁^ 1/ + p₂^ 1/
- Bp₂^ 1/ V₀ p₁^ 1/ + p₂^ 1/ and p₁^ 1/ V₀ p₁^ 1/ + p₂^ 1/
- Cp₁ V₀ p₁ + p₂ and p₂ V₀ p₁ + p₂
- Dp₁^ V₀ p₁^ + p₂^ and p₂^ V₀ p₁^ + p₂^
Correct answer
A. p₁^ 1/ V₀ p₁^ 1/ + p₂^ 1/ and p₂^ 1/ V₀ p₁^ 1/ + p₂^ 1/
Step-by-step solution
Since the separator is adiabatic and the process is slow, the expansion or compression of both gases is reversible and adiabatic. For an adiabatic process, pV^ = constant . Let the final pressure of both gases in equilibrium be p . The initial volume of each part is V₀ 2 . For the left part: p₁ ( V₀ 2 )^ = p V₁^ V₁ = V₀ 2 ( p₁ p )^ 1/ For the right part: p₂ ( V₀ 2 )^ = p V₂^ V₂ = V₀ 2 ( p₂ p )^ 1/ The total volume of the tube remains constant, so V₁ + V₂ = V₀ . Substituting the expressions for V₁ and V₂ : V₀ 2 ( p₁