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 constants. What is the effective value of adiabatic constant \(\gamma\) ? 

\(\frac{b+1}{b}\)

For an adiabatic process : 0 = dU + PdV

\(\Rightarrow d(a + bPV) + PdV = 0\)

\(\Rightarrow b P dV + b V dP + PdV = 0\)

\(\Rightarrow (b + 1)PdV + bVdP = 0\)

\(\Rightarrow (b+1)\frac{dV}{V} + b\frac{dP}{P} = 0\)

\(\Rightarrow (b+1) ln V + b ln P =\) constant

\(\Rightarrow V^{b+1} P^b = constant\)

\(\Rightarrow PV^{\frac{b+1}{b}} =\) constant

\(\Rightarrow d(a + bPV) + PdV = 0\)