The Martian Cave Heater · AI Generator #GEN-140C5429

Physics · thermodynamics

Problem

A Mars exploration team discovers a sealed underground cave with volume

V = 2000\;\text{m}^3

containing Martian atmospheric gas at temperature

T_1 = 220\;\text{K}

and pressure

P_1 = 600\;\text{Pa}

. To make the cave habitable, they install a heater that adds thermal energy at a constant rate of

\dot{Q} = 1200\;\text{W}

. The cave walls are excellent insulators, but gas can slowly leak through small cracks. The leak rate follows the kinetic theory prediction: the number of molecules escaping per second is proportional to

\sqrt{T}

, where the proportionality constant

\alpha = 2.0 \times 10^{18}\;\text{s}^{-1}\text{K}^{-1/2}

. After a long time, the system reaches thermal equilibrium where the rate of energy input equals the rate of energy loss due to escaping gas molecules. Find the final equilibrium temperature

T_f

of the gas in the cave. Assume the gas behaves ideally with

C_V = \frac{5}{2}R

(diatomic molecules), and use

k_B = 1.38 \times 10^{-23}\;\text{J/K}

Related theory

thermodynamics

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