热力学能量说明
The partial Gibbs energy and the Gibbs energy
The partial Gibbs energy, or the chemical potential, is the partial quantity of the Gibbs energy, which means that it is independent of the amount of substance. Generally speacking, the Gibbs energy can be described as a function temperature, pressure and compositions of components, and thus the chemical potential can be derived from such a mathematical formula. Sometimes, one might feel confused that there exist several different formula of chemical potential with respect to the Gibbs energy. In this post, two different fomulas are to be discussed, respectively.
In the following derivation, the temperature and pressure are assumed as constant.
For Gibbs energy with independent composition
In this context, the Gibbs energy is a function of the solute components, denoted as
or
where
The chemical potential of the
In CALPHAD method and other physical models, the mole fraction or the composition is the most common state variable for the fraction of the components. And thus, we have a motivation to substitue the amount, i.e.,
As
, we can obtain
Take the last part from the equation above and we have
Note One should keep in mind that we should not sustitue
Futhermore,
Put this result back into the original equation,
Simplify the equation above,
For Gibbs energy with dependent composition
In this context, the Gibbs energy is a function of the solute components, denoted as
or
with a constraint that
The chemical potential of the
In the spirit of the CALPHAD method and other physical models, the mole fraction or the composition is the most common state variable for the fraction of the components. And thus, we have a motivation to substitue the amount, i.e.,
As
with the previous constraint, and we can obtain
Note It is worthy of noticing that the superscript of the summation has been changed from
Take the last part from the equation above and we have
Note One should keep in mind that we should not sustitue
Futhermore,
Put this result back into the original equation,
Simplify the equation above,
Differnce between the two formula of the chemical potential
- For Gibbs energy with independent composition
- For Gibbs energy with dependent composition
Relation between the chemical potential and the diffusion potential
Diffusion potential is one of the most important variables in phase-field modeling. Mathematically, it should be defined as the the partial derivative of the mole Gibbs energy.
Note partial derivative of the mole Gibbs energy and the partial quantity of the Gibbs energy are two completely different things.
Considering that the mole Gibbs energy is discribed with those independent composition variables,
the diffusion potential of the
Note I am writing this to tell that, diffusion potential, is state variable of the solute, which means that a solvent component always exist in this context.
The equation above is really convenient when you have a description of the mole Gibbs energy related to compositions of the solute components. However, CALPHAD technique has adopted the second strategy to demonstrate the relationship between the mole Gibbs energy and composition, which is
with a constraint that
Obviously, there no straight-forward way to compute the derivative of such a formula of the Gibbs energy, as a constraint has been imposed on the function.
Fortunately, we have another definition of the mole Gibbs energy,
Apply the rule of total derivative to the equation above
According to Gibbs-Duhem equation, we have
and thus
Supposing that the
because
Consequently, all the composition variables in the equation above become independent, and we know
according to the properties of the total derivative.