Wednesday, March 13, 2019

thermodynamics - Extra 5/2 tau term in chemical potential of a monoatomic ideal gas?




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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