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Question
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.
Solution
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")]`
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