The chemical potential of an ideal monoatomic gas should be: μ=τlnnnQ http://web.mit.edu/ndhillon/www/Teaching/Physics/bookse5.html
I get this result if I derive it using the definition of Gibbs Free Energy: G=U+PV-TS
However, if I use the differential equation of enthalpy, H: dH=TdS+VdP+μdN
I get the result: μ=τln(nnQ)−52τ
This is the proof, which is where I'm trying to figure where the error is:
dH=52NkBdT = TdS+VdP+μdN
At constant T and P, -TdS =μdN
μ = -TdSdN
For a monoatomic gas (Sackur Tetrode), S = NkBln(VNλ3)+52NkB
Since PkBT=NV
μ=-TdSdN = -kBTln(kBTPλ3)−52kBT at constant P and T.
Rearrange, μ=τln(nnQ)−52τ
Just trying to figure out what I am doing wrong with this proof. Shouldn't the answers be the same if I start from G vs. H?
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