The relation between the internal energy `U` adiabatic constant `gamma` is

Options

A. `U = (PV)/(gamma -1)`
B. `U = (PV^(gamma))/(gamma -1)`
C. `U = (PV)/(gamma)`
D. `U = (gamma)/(PV)`

Answer

A

Explanation

a. Change in internal energy
`Delta U = mu C_(v) Delta T implies U_(2) - U_(1) = mu C_(v) (T_(2) - T_(1))`
Let initially `T_(1) = 0` so `U_(1) = 0` and finally `T_(2) = T` and `U_(2) = U`
`U = mu C_(v) T = mu T xx C_(v) = (PV)/(R ) xx (R )/(gamma - 1) = (PV)/(gamma - 1)`
(As `PV = mu RT, :. mu T = PV//R` and `C_(v) = R//(gamma - 1))`