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Variation of chemical potential with T and P 
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#1
Oct2313, 05:44 PM

P: 692

So the expression for Gibb's free energy is:
dG = SdT + VdP + μdN, Here, we see that the Gibb's free energy changes with temperature (dT), change in pressure (dP) and change in chemical potential (as a result of change in particle number). My question is: we know chemical potential varies with both change in temperature and pressure. So if we don't add/remove particles from the system, the chemical potential does change with variation of P and T...so is that already included in the above equation? (That is, in the above equation, are we accounting for the change in Gibb's free energy as a result of change in chemical potential as a result of variation of T and P, in addition to the change in chemical potential due to change in particle number). Further, when the number of particles changes, there might be a number of chemical reactions that take place, so the temperature T might change because of that also, which would change the sdT term at the beginning, right? I guess I'm just having problems understanding chemical potential :/ 


#2
Oct2313, 06:32 PM

Sci Advisor
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Thanks
PF Gold
P: 5,175

[tex]G=G(T,P,N_1,...,N_m)[/tex] An infinitecimal change in G can be represented using the chain rule for partial differentiation: [tex]dG=\frac{\partial G}{\partial T}dT+\frac{\partial G}{\partial P}dP+\frac{\partial G}{\partial N_1}dN_1+...+\frac{\partial G}{\partial N_m}dN_m[/tex] Each of the partial derivatives in this equation is a function of T, P, and the N's, with [tex]\frac{\partial G}{\partial T}=S[/tex] [tex]\frac{\partial G}{\partial P}=V[/tex] and [tex]\frac{\partial G}{\partial N_i}=μ_i[/tex] I hope this helps. 


#3
Oct2513, 03:11 PM

Sci Advisor
P: 3,593

Of course mu is a function of T and P, also.
Given that ##\mu=\partial G/\partial N## we have ##(\partial\mu /\partial T)_P=\partial^2 G/\partial N \partial T=\partial^2 G /\partial T \partial N =(\partial S/\partial N)_P = S_m## i.e. the partial molar entropy and analogously ##(\partial \mu/\partial P)_T=V_m ## the partial molar volume. So for fixed N, ##d\mu=S_mdT+V_m dP## 


#4
Nov2213, 01:26 PM

P: 692

Variation of chemical potential with T and P
Can I just go on to ask what the difference between chemical potential and chemical affinity is? They seem to , intuitively, mean the same thing but chemical potential is +ve for a reaction that's progressing and affinity is negative!
Also, is A (affinity) always the same sign as the rate of reaction? 


#5
Nov2413, 04:22 PM

Sci Advisor
P: 3,593

##A=\Delta G_r=\sum \nu_i \mu_i##
were ##\nu_i## are the stochiometric coefficients of the reaction taking place. So basically A is a weighed sum of chemical potentials. 


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