An ideal gas with adiabatic exponent `gamma` undergoes a process in which internal energy depends on volume
as `U=aV^(alpha)` then select the correct statement .

Options

A. Change in internal energy is`(gamma-1)/(alpha) DeltaT`
B. Molar heat capacity of process is `(R)/(1-gamma)+(R)/(alpha)`
C. Heat exchange in the process is given by `DeltaU[1+(1-gamma)/(alpha)]`
D. Equation of process is given is ` PV^(x)`=constant , where `xgt 0 if alpha lt 1`.

Answer

D

Explanation

`U = aV^(alpha)" "rArrnC_(v) T=aV^(alpha)`
`TV^(-alpha)= a`
On comparing with, `TV^(n-1)`= constant
Polytropic index, `n = 1- alpha`
Molar heat capacity , `= (R)/(gamma-1 )-(R)/(n-1)= - (R)/(1-gamma)+(R)/(1-(1-alpha))=- (R)/(1-gamma)+(R)/(alpha)`
` q =C DeltaT`
`=[-C_(V) +(R)/(alpha)]DeltaT = - C_(V) DeltaT +(R)/(alpha)DeltaT = - DeltaU + (R)/(alpha)DeltaT," "=-DeltaU +(gamma-1)/(gamma-1)xx (RDeltaT)/(alpha)`
`= -DeltaU + (R)/(gamma-1 )DeltaT xx (gamma-1)/(alpha)," "= -DeltaU + DeltaU. (gamma-1)/(alpha)=DeltaU[(gamma-1)/(alpha)-1]`