An amount n (in moles) of a monatomic gas at an initial temperature T 0 is enclosed in a cylindrical vessel fitted with a light piston. The surrounding air has a temperature T s (T 0 ) and the atmospheric pressure is p a . 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.

Question

An amount n (in moles) of a monatomic gas at an initial temperature T 0 is enclosed in a cylindrical vessel fitted with a light piston. The surrounding air has a temperature T s (>T 0 ) and the atmospheric pressure is p a . 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.

Philoid · Accepted 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=nC_pdT (2)

where,

dQ=change in heat

n = number of molecules

dT = change in temperature

C_p = 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, C_p=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 = )