A monatomic ideal gas is contained in a rigid container of volume V with walls of total inner 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 the surrounding air is `T_(s)` . All temperature are in absolute scale.

Explanation

As the volume of the gas is constant, a heat `DeltaQ given to the gas increases its temperature by `deltaT=DeltaQ//C_(v)` . Also, for a monatomic gas, C_(v)=3/2R` . If the temperature ot the gas at time t is T, the heat current into the gas is
`(DeltaQ)/(Deltat)=(KA(T_(s)-T))/(x)` . or, `(DeltaT)/(Deltat)=(2KA)/(3xR)(T_(s)-T)` . or, `int_(to))^(T)(dT)/(T_(s)-T)=int_(0))^(t)(2KA)/(3xR)t` . or, `T_(s)-T(T_(s)-T_(0))e^(-(2KA)/(3xR)t` . or, `T=T_(s)(T_(s)-T_(0))e^(-(2KA)/(3xR)t` . As the volume remains constant, `P/T=p_(0)/T_(0)` . or, `p=p_(0)/T_(0)T` . `=p_(0)/(T_(0))[T_(s)-(T_(s)-T_(0))e^((2KA)/(3xR)t)]` .