One mole of a monoatomicideal gasis contained in a rigidcontainer ofvomlume V_(0) with walls of total surface area A , thickness x and thermal conductivityK . The temperature outsideis T_(s) (T_(s) gt T_(0)).

One mole of a monoatomicideal gasis contained in a rigidcontainer ofvo

1.	One mole of a monoatomicideal gasis contained in a rigidcontainer ofvomlume V_(0) with walls of total surface area A , thickness x and thermal conductivityK . The temperature outsideis T_(s) (T_(s) gt T_(0)).
Answer» Solution :Sincethe volume of thecontainer is fixed , only the TEMPERATUREAND pressure of the gasvarieswith time, Further the outsidetemperature beings MORETHAN that of the gas , heat from surroudingenters into the gas . If P andT bethe pressure and temperatureof the gas at ANYINSTANT , then `(P)/(T) = (P_(0))/(V_(0))"".......(i) ` The heat flowinginto the gas , duringa small time interval dt will be`dQ = (KA(T_(s) - T)dt)/(x) ............(ii)` But `dQ = 1((3)/(2)R) dT [ because dQ = nC_(v)TD and C_(v) = 3R//2` fora monoatomicgas ] `therefore ` EQ.(ii) can be rewritten as `(3RdT)/(2) = (KA)/(x) (T_(s)T)dt (or ) (dT)/((T_(s) - T)) = (2KA)/(3Rx) dt` sinceat `t = 0 , T = T_(0)` and at`t = t , T = T` `thereforeint_(T_(0))^(T) (dT)/((T_(s) - T)) = (2KA)/(3Rx) int_(0)^(1) dt rArrIn ((T_(s) - T)/(T_(s)-T_(0))) = (-2At)/(3Rx)` i.e.,`T_(s) - T = (T_(s)- T) e^((2KA t)/(3Rx)) ` or `T = T_(s) - (T_(s) - T_(0))e^((2KAt)/(3Rx)) therefore "From Eq." (i) P = (P_(0))/(V_(0)) [T_(s) - (T_(s) - T_(0))e^((2KAt)/(3Rx))]`