Two thermodynamical processes are shown in the figure. The molar heat capacity for process A and B are C_A and C_B. The molar heat capacity at constant pressure and constant volume are represented by C_P and C_V, respectively. Choose the correct statement.

Solution

Physics sir · 3 steps

Turn each slope into a polytropic index, then compare heat capacities with C = C_V + R/(1-x).

1. Read the graph. A straight line on a log P against log V graph with slope m means log P = m log V + constant, so PV⁻ᵐ = constant. Line A makes angle tan⁻¹ γ, so its slope is +γ. Line B makes 45°, so its slope is +1.
2. Polytropic index. Compare with PVˣ = constant, which gives x = -m. Process A has x = -γ, so P ∝ V^γ, and process B has x = -1, so P ∝ V. For one mole of ideal gas, any such process has the molar heat capacity below.

   C = C_V + R/(1 - x)

   Process	Index x	Molar heat capacity
   Isochoric	∞	C_V
   A: P ∝ V^γ	-γ	C_A = C_V + R/(1+γ)
   B: P ∝ V	-1	C_B = C_V + R/2
   Isobaric	0	C_P = C_V + R

3. So. Since γ > 1, we have 1 + γ > 2, so R/(1+γ) < R/2 < R. This gives C_P > C_B > C_A > C_V, which is option (a).

The idea to keep. A straight line on a log P against log V graph is a polytropic process with x = -(slope). Then C = C_V + R/(1-x) ranks the heat capacities for you.

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