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.

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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