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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.
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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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