Determine the average molar heat capacity of an ideal gas under going a process shown in fig.

Options

A. `((gamma-1)/(gamma-1))R`
B. `((3gamma-2)/(gamma-1))R`
C. `((gamma-2)/(gamma-4))R`
D. `((gamma+1)/(gamma-1))R`

Answer

B

Explanation

Let n be the number of moles of the gas,
then `T_(A)=(2P_(0)V_(0))/(nR) and T_(B)=(3P_(0)V_(0))/(nR)`
Now, work done by the gas during the proces is given by area under curve
`DeltaW=(1)/(2)(P_(0)+2P_(0))(3V_(0)-V_(0))=3P_(0)V_(0)`
work done per mole of the gas per unit rise in temperature.
`W^(1)=(DeltaW)/(nDeltaT)=(DeltaW)/(n(T_(B)-T_(A))` ,
`=(3P_(0)V_(0))/(n[[3P_(0)V_(0))/(nR)-(2P_(0)V_(0))/(nR)])=3R`
Molar heat capacity of the gas for the above process will be
`C=C_(v)+W'=(R)/(gamma-1)+3R=((3gamma-2)/(gamma-1))R`