Heat Transfer · Thermal Relaxation

Follow heat through the partition until the temperature gap halves

The fixed conducting partition P₁ lets heat flow from the hotter section to the colder one. The insulated piston P₂ moves freely, so section S₂ remains at atmospheric pressure while the temperature difference decays exponentially.

2026 Paper 1 Q11 · conduction through P₁
Relaxation view
Time in units of xR/KA 0.00
Conducting partition
The heat current through P₁ is proportional to the temperature difference between S₁ and S₂.
Relationd(ΔT)/dt = −(KA/xR)ΔT
Resultn = ln 2 ≈ 0.7
Hotter sectionS₁ starts at higher temperature
Cooler sectionS₂ warms at atmospheric pressure
Conducting wallheat current through P₁

Derivation path

01 / HEAT CURRENT

Fourier conduction through P₁

The partition has conductivity K, area A and width x, so heat current is proportional to ΔT.

dQ/dt = KA(T₁ − T₂)/x
02 / TEMPERATURE GAP

The gap decays exponentially

The gas sections convert this heat flow into a first-order relaxation of the temperature difference.

ΔT(t) = ΔT₀ exp[−KAt/(xR)]
03 / HALF VALUE

Set ΔT = ΔT₀/2

The half-time is obtained by taking the natural logarithm of 2.

t₁/₂ = (xR/KA) ln 2
Final value
The given form is t = n xR/KA.
Since t₁/₂ = (xR/KA) ln 2 and ln 2 ≈ 0.7, n = ln 2.
Answer: n ≈ 0.7