Gibbs energy=chem potential (not convinced)

In summary, the conversation discusses the relationship between Gibbs energy and chemical potential. It is explained that Gibbs energy is an extensive quantity and is related to chemical potential through a partial derivative with respect to the number of particles. The conversation also delves into the importance of considering other variables, such as volume and entropy, in determining the chemical potential. Finally, the possibility of proving this relationship rigorously without relying on macroscopic thermodynamics is discussed.
  • #1
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Gibbs energy=chem potential (solved)

my thermal book gives a hand-waving argument saying the followings:
firstly, Gibbs energy, defined by:
[tex]G\equiv U+PV-TS[/tex]

is an extensive quantity (proportional to N), and also
[tex]\left (\frac{\partial G}{\partial N}\right ) _{T,P}=\mu[/tex]

so far so good, but then it says:

if P and T are held constant then [itex]\mu[/itex] is also constant, which implies whenever a particle is added to the system, G is increased by [itex]\mu[/itex].

Thus,
[tex]G=N\mu[/tex]

But why must [itex]\mu[/itex] be solely dependent on T and V? why can't [itex]\mu[/itex] depend on.. let's say N? is there any algebraic prove for that?

edit: oh yeah I see... the book skips a very Very important reason of why it works!
since V, S and U are also extensive,
[tex]V\sim N[/tex]
[tex]S\sim N[/tex]
[tex]U\sim N[/tex]

Thus,
[tex]\left (\frac{\partial G}{\partial N}\right ) _{T,P}=\mu=
\frac{\partial U}{\partial N}+P\frac{\partial V}{\partial N}-T\frac{\partial S}{\partial N}[/tex]

and each of the three partial derivatives is independent of N since V, S and U are directly related to N...

don't you just hate it when books make some non-rigorous arguments, left out the important details and act as if the things are obvious and trivial?!
 
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  • #2
But what they did is entirely correct. I can always rewrite the chemical potential as a function of other intensive/extensive variables because of the existence of equations of state.
 
  • #3
you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding [tex]<N>\mu[/tex] in the grand canonical ensemble.
 
  • #4
really...? I'm interested... can you provide more details please? I would really love a rigorous argument on this problem.

so, how would you go from the definition of G and mu??
 
  • #5
quetzalcoatl9 said:
you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding [tex]<N>\mu[/tex] in the grand canonical ensemble.

I'm intrigued, since I've never seen this done before. I've always seen, starting from the microcanonical ensemble, a derivation that leads to something that we recognize as F or some such, and then that's the connection to thermodynamics.
 

What is Gibbs energy?

Gibbs energy, also known as Gibbs free energy, is a thermodynamic quantity that measures the amount of energy available to do work in a chemical system. It takes into account both the enthalpy (heat content) and entropy (disorder) of a system.

What is chemical potential?

Chemical potential is a measure of the energy needed to add one mole of a substance to a system, while keeping the temperature, pressure, and number of moles of other substances constant. It is closely related to Gibbs energy and is often used in the study of thermodynamics and chemical reactions.

Why is Gibbs energy equal to chemical potential?

This relationship is rooted in the definition of Gibbs energy and chemical potential. Gibbs energy is defined as the energy available to do work, while chemical potential is defined as the energy needed to add one mole of a substance. In a system at equilibrium, the Gibbs energy must be equal to the chemical potential for all substances in the system.

How is Gibbs energy used in chemistry?

Gibbs energy is a useful tool in predicting whether a chemical reaction will occur spontaneously. If the Gibbs energy change for a reaction is negative, it indicates that the reaction will occur spontaneously in the forward direction. Additionally, Gibbs energy is used in the study of phase transitions, such as melting and vaporization.

Is Gibbs energy the only factor determining chemical potential?

No, there are other factors that can affect chemical potential, such as temperature, pressure, and the concentration of reactants and products. However, Gibbs energy is an important factor in determining chemical potential and plays a key role in predicting the spontaneity of chemical reactions.

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