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Question

A monoatomic ideal gas is contained in a rigid container of volume V with walls of total surface area A, thickness x and thermal conductivity K. The gas is at an initial temperature To and pressure Po. The temperature of the surrounding air is Ts. All the temperatures are in absolute scale. If the temperature of the gas at any time t can be expressed as T(t)=Ts(TsTo)eλt, find the value of λ.
  1. λ=2KA3Rx
  2. λ=5KA3Rx
  3. λ=4KA3Rx
  4. λ=2KA3R


Solution

The correct option is A λ=2KA3Rx
For monoatomic gas,
CV=3R2,CP=5R2,Ts>To,
Assume 1 mole of gas is present in the container. 


Let at any time t, the temperature of gas be T.
The heat transfer to the container is
dQdt=KA(TsT)x ... (i)
Heat gained by the gas is
dQdt=nCVdTdt  ... (ii)
From equation (i) and (ii),
nCVdTdt=KA(TsT)x
(3R2)dTdt=KA(TsT)x (n=1)
Integrating on both sides,
3R2TTodTTsT=KAxt0dt
3R2[ln(TsT)1]TTo=KAx[t]t0
TsTTsTo=e2KAt3Rx
T(t)=Ts(TsTo)eλt where λ=2KA3Rx

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