1 mole of rigid diatomic gas performs a work of Q/5 when heat Q is supplied to it. The molar heat capacity of the gas during this transformation is xR/8. The value of x is _________. [R = universal gas constant]

Solution

Physics sir · 3 steps

Only 4/5 of the heat stays inside as internal energy, and Δ U always equals nCᵥΔ T.

1. First law. The gas does work W = Q/5, so the change in internal energy is Δ U = Q - Q/5 = 4Q/5.
2. Write both in ΔT. For a rigid diatomic gas Cᵥ = 5R/2, so for 1 mole Δ U = 5R/2Δ T. By the definition of molar heat capacity for this process, Q = C Δ T.

   4/5 C Δ T = 5R/2 Δ T ⇒ C = 25R/8

3. So. Comparing C = 25R/8 with xR/8 gives x = 25.

The common slip: 15. Using the monatomic value Cᵥ = 3R/2 instead of 5R/2 gives C = 15R/8, so x = 15, not 25.

The idea to keep. For any process, Δ U = nCᵥΔ T and Q = nCΔ T. Use the first law to find what fraction of Q becomes Δ U, then divide.

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