A rigid box holds 1 mole of monatomic ideal gas. Walls have total area A, thickness D and conductivity K; surroundings remain at T₀/2. Initially gas is at T₀. Temperature follows T=(T₀/2){1+exp[xKAt/(DR)]}. Find x.
Physics · Mixed Chapter
JEE Main Practice

Question

A rigid box holds 1 mole of monatomic ideal gas. Walls have total area A, thickness D and conductivity K; surroundings remain at T₀/2. Initially gas is at T₀. Temperature follows T=(T₀/2){1+exp[xKAt/(DR)]}. Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Heat conduction loss rate from the gas: dQ/dt=−(KA/D)(T−T₀/2). Step 2: At constant volume, Cᵥ=(3/2)R per mole, so (3R/2)(dT/dt)=−(KA/D)(T−T₀/2). Step 3: Solve dθ/dt=−[2KA/(3DR)]θ for θ=T−T₀/2. Step 4: T=T₀/2+(T₀/2)exp[−(2KA/(3DR))t]; hence x=−2/3≈−0.67.