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The gas law for an isothermal process is PV = constant so the
polytropic exponent is gamma=1. The obvious problem is that
gamma/(gamma-1) is infinite in this case.
What I would like comment on is the following: In the constant
enthalpy form of Bernoulli's Eqn we have
-VdP + mgdh + d(mv^2/2) = 0.
I would use PV=nRt to replace V with nRT/P and use that when
integrating -VdP. That would make the pressure term of the Bernoulli
Eqn
-nRTln(P2/P1)
and the complete equation
nRTln(P1/P2) + mg(h2-h1) + mv2^2/2 - mv1^2/2 = const
Am I making any unjustified assumptions when doing this?