An ideal gas whose adiabatic exponent equals `gamma` expands so that the amount of heat transferred to it is equal to the decrease of its internal energy. Find
a. the molar heat capacity of the gas, and
b. the T -V equation for the process.

Explanation

a. `Delta U = int_(T)^(T + Delta T) (m)/(M) C_(v) dT = int_(T)^(T + Delta T) (m)/(M) (R )/(gamma -1) dT ( :' C_(v) = (R )/(gamma -1))`
`implies Delta U = (m)/(M) (R )/(gamma -1) Delta T`
Similarly, `Delta Q = (m)/(M) C Delta T`
Where `C` is the molar heat capacity in the process.
It is given that `DeltaQ = - Delta U`
b. `dQ = dU + dW implies 2dQ = dW`
(`:' dQ = - dU` given )
`implies 2 CdT = pdV` `( :' dQ = CdT`)
`implies -(2R)/(gamma -1) dT = pdV`
`implies (2R)/(gamma -1) dT + pdV = 0`
`implies (2R)/(gamma -1) dT + (RT)/(V) dV = 0`
`implies (dT)/(T) + (gamma -1)/(2) (dV)/(V) = 0`
Integrating, `TV^((gamma - 1)//2)` = constant