The relation between internal energy U, pressure P and volume V of a gas in an adiabatic process is
`U = a + bPV`
where a and b are positive constants. What is the value of `gamma` ?

Options

A. `(a)/(b)`
B. `(b+1)/(b)`
C. `(a+1)/(a)`
D. `(b)/(a)`

Answer

B

Explanation

For an adiabatic process, dQ = 0
From the first law of thermodynamics, we get
or `bPdV + bVdP + PdV = 0` or (b+1)PdV + bVdP = 0
or `(b+1) (dV)/(V) + b(dP)/(P) = 0`
or `(b+1)(dV)/(V) = -b(dP)/(P)`
Integrating both sides, we get (b+1) ln V + b ln P = constant or ln `(V^(b+1)P^(b))` = constant or `PV^((b+1)/(b))` = another constant
Compare with the standard equation of an adiabatic process `PV^(gamma)` = constant, we get `gamma = (b+1)/(b)`