Q. 38

An amount n (in moles) of a monatomic gas at an initial temperature T0 is enclosed in a cylindrical vessel fitted with a light piston. The surrounding air has a temperature Ts(>T0) and the atmospheric pressure is pa. Heat may be conducted between the surrounding and the gas through the bottom of the cylinder. The bottom has a surface area A, thickness x and thermal conductivity K. Assuming all changes to be slow, find the distance moved by the piston in time t.

Answer :

Given,


In time dt, heat transfer through the bottom of the cylinder is given by-


= (1)


In case of monoatomic gas, pressure remains constant.


Hence the heat content at constant pressure(enthalpy) is given by


dQ=nCpdT (2)


where,


dQ=change in heat


n = number of molecules


dT = change in temperature


Cp = amount of heat required to raise the temperature of a substance of 1Kg mass by one degree Celsius at constant pressure.


Comparing above equations-


=


For a monoatomic gas, Cp=52 R


=


=


=-


Integrating both the sides, we get


=


=-


ln( =-


Taking antilog


=


T =


Rewriting


T- = ) (1)


Now, we know the gas equation given by


=


Substituting in (1)


= T- = )


Solving for the length/distance,


l = )


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