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

b. For an adiabatic process,
`0 = dU + PdV`
or `d (a+ bPV) + PdV = 0`
or `bP dV + dV dP + p dV = 0`
or `(b + 1) PdV + bV dP = 0`
or `(b + 1) (dV)/(V) + (dP)/(P) = 0`
or `(b + 1) log V + b log P = ` constant
`V^(b + 1) p^(b) =` constant
or `PV^((b + 1)/(b)) =` constant
`:. gamma = (b + 1)/(b)`