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One mole of an ideal gas with heat capac...

One mole of an ideal gas with heat capacity at constant pressure `C_p` undergoes the process `T = T_0 + alpha V`, where `T_0` and `alpha` are constants. Find :
(a) heat capacity of the gas as a function of its volume ,
(b) the amount of heat transferred to the gas, if its volume increased from `V_1` to `V_2`.

Text Solution

Verified by Experts

As volume of gas increases from `V_1` to `V_2` , the corresponding temperatures ofthe gas are
`T_1=T_0+alphaV_1`
and `T_2=T_0+alphaV_2`
The amount of heat supplied to the gas can be given as
`Q=intdQ=int_(T_1)^(T_2)CdT` [ As n=1 mole]
or `Q=int_(T_1)^(T_2)[C_P+(RT_0)/(alphaV)]dT`
From the given relation we have
`V=(T-T_0)/alpha`
Thus `Q=int_(T_1)^(T_2)[C_P+(RT_0)/(T-T_0)]dT`
`=C_P(T_2-T_1)+RT_0 ln((T_2-T_0)/(T_1-T_0))`
`=C_P alpha (V_2-V_1)+RT_0 ln (V_2/V_1)`
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