Advertisement Remove all ads

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. - Physics


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 Pα. 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.

Advertisement Remove all ads


In time dt, heat transfer through the bottom of the cylinder is given by
`"dQ"/"dt" = "KA(T_s - T_0)"/x`
For a monoatomic gas, pressure remains constant.
∴ `dQ = nC_pdT`

∴ `(nC_pdT)/ dt = "KA(T_2 - T_0)"/x`
For a monoatomic gas,
`C_p = 5/2 R`

`⇒ "n5RdT"/"2dt" = KA(T_s - T_0)/x`

`⇒ "5nR"/2 "dT"/dt = (KA(t_s - T_0))/x`

`⇒ "dT"/(T_s - T_0) = "-2KAdt"/"5nRx"`

Integrating both the sides,

`(T_s - T_0)_"T_0"^"T" = "-2KAt"/"5nRx"`

`⇒ In  ((T_s - T) /(T_s - T_0)) = - "-2KAt"/"5nRx"`

`⇒ T_s - T = (T_s - T_0)e ^("-2KAt"/"5nRx")`
`⇒ T = T_s - (T_s - T_0) =e ^(-"-2KAt"/"5nRx")`
`⇒ T - T_0 = (T_s - T_0) - (T_s - T_0)e^(-"2KAt"/"5nRx"`
`⇒ T- T_0 = (T_s - _0) [l - e^(-"-2KAt"/"5nRx")]`
From the gas equation,

`(P_(a)Al)/(nR) = T - T_0`

∴ `(P_(a)Al)/(nR)= (T_s - T_0) [1 - e^(-"-2KAt"/"5nRx")]`

`⇒ l = (nR)/(P_aA) (T_s - T_0)[ 1 - e^(-"-2KAt"/"5nRx")]`

Concept: Thermal Expansion of Solids
  Is there an error in this question or solution?
Advertisement Remove all ads


HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 6 Heat Transfer
Q 38 | Page 101
Advertisement Remove all ads
Advertisement Remove all ads

View all notifications

      Forgot password?
View in app×