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One mole of an ideal gas whose pressure ...

One mole of an ideal gas whose pressure changes with volume as `P=alphaV` , where `alpha` is a constant, is expanded so that its volume increase `eta` times. Find the change in internal energy and heat capacity of the gas.

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The process equation of the thermodynamic process in which gas is undergoing is given as
`P=alphaV`…(3.91)
Rearranging we can write
`PV^(-1)=alpha` (a constant) …(3.92)
Equation-(3.92) shows that this is a polytropic process with the value of polytropic constant n=-1. Thus the molar heat capacity of gas in this process can be given as
`C=R/(gamma-1)+R/(1-n)`
or `=R/(gamma-1)+R/2` [n=-1]
`=R/2[(gamma+1)/(gamma-1)]`
If during the process, temperature of gas changes from `T_1` to `T_2` then change in internal energy of gas is given as
`DeltaU=C_VDeltaT` [As n=1 mole]
`=R/(gamma-1)(T_2-T_1)` [ As `C_V=R/(gamma-1)` ] ...(3.93)
Here it is given that initial and final volumes of the gas are `V_0` and `etaV_0`. Thus the respective pressures in initial and final states are
and `P=alpha V_0`
`P=alpha eta V_0`
Thus from gas law we can find initial and final temperature as
`T_1=(P_1V_1)/R=(alphaV_0.V_0)/R`
and `T_2=(P_2V_2)/R=(alphaetaV_0. etaV_0)/R` [As n=1 mole]
Now from equation-(3.93), change in internal energy of gas is given as
`DeltaU=R/(gamma-1)[(alphaeta^2V_0^2)/R-(alphaV_0^2)/R]`
`=(alphaV_0^2)/(gamma-1)[eta^2-1]`
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