Question

One mole of a monatomic ideal gas is contained in a rigid container of volume $V_0$ with walls of total surface area A, thickness x and thermal conductivity K. The gas is at an initial temperature $T_0$ and pressure $P_0$. Find the pressure of the gas as a function of time if the temperature of surrounding air is $T_s$. All the temperatures are in absolute scale.

• A.
• B. Zero
• C.
• D. None

Answer 1

The correct option is A

Because the volume of the gas is kept constant, a heat dQ given to the gas increases its temperature by $dT=\frac{dQ}{C_v}$. Also, for a monatomic gas, $C_v=\left(\frac{3}{2}\right)R$. If the temperature of the gas at time t is T, then rate of flow of heat into the gas is -
$\frac{dQ}{dt}=\frac{KA(T_s-T)}{x}or,C_v\frac{dT}{dt}=\frac{KA(T_s-T)}{x}or,\frac{3}{2}R\frac{dT}{dt}=\frac{KA(T_s-T)}{x}or,\frac{dT}{dt}=\frac{2KA}{3xR}(T_s-T)or,{}_{T_0}^T\frac{dT}{T_s-T}{=}_{T_0}^T\frac{2KA}{3xR}dtor,log\frac{T_s-T_0}{T_s-T}=\frac{2KA}{3xR}tor,T_s-T=(T_s-T_0)e^{\frac{2KA}{3xR}t}orT=T_s-(T_s-T_0)e^{\frac{2KA}{3xR}t}$
As the volume remains constant,
$\frac{P}{T}=\frac{P_0}{T_0}orP=\frac{P_{0}T}{T_0}or,P=\frac{P_0}{T_0}\left(T_s\right)-(T_s-T_0)e^{\frac{2KA}{3xR}t}$