A heat-conducting piston can move freely inside a closed, thermally insulated cylinder with an ideal gas ( γ = 3 5 ). At equilibrium the piston divides the cylinder into two equal parts, the gas tem

Question

A heat-conducting piston can move freely inside a closed, thermally insulated cylinder with an ideal gas ( γ = 3 5 ​ ). At equilibrium the piston divides the cylinder into two equal parts, the gas temperature being 300 K. The piston is slowly displaced by an external agent. Find the gas temperature when the volume of the greater section is seven times the volume of the smaller section.

TestHub · Accepted Answer

The conducting piston keeps both parts at the same temperature and the whole cylinder is thermally insulated, so the process is reversible-adiabatic and total entropy is conserved: Δ S = 2 C v ​ ln T 0 ​ T ​ + R ln V 0 2 ​ V 1 ​ V 2 ​ ​ = 0 , giving T = T 0 ​ ( V 1 ​ V 2 ​ V 0 2 ​ ​ ) 2 γ − 1 ​ . With V 1 ​ + V 2 ​ = 2 V 0 ​ and V 2 ​ = 7 V 1 ​ : V 1 ​ = 4 V 0 ​ ​ , V 2 ​ = 4 7 V 0 ​ ​ , so V 1 ​ V 2 ​ V 0 2 ​ ​ = 7 16 ​ . With γ = 3 5 ​ , T = 300 ( 7 16 ​ ) 1/3 = 300 × 1.317 = 395 K

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