In Chapter 6, we derived an expression for the total differential of the Gibbs function:
\[ dG = V\,dp - S\,dT \]
This relationship describes how the Gibbs function changes when pressure and temperature change. However, many spontaneous processes occur even when both temperature and pressure remain constant.
For example:
Since these processes occur at constant temperature and pressure, something must be missing from the expression above.
The missing variable is composition.
The Gibbs function depends not only on temperature and pressure, but also on the amounts of the various components present in the system.
Consider a binary mixture containing components A and B.
The Gibbs function can be written as
\[ G = G(T,p,n_A,n_B) \]
The total differential is therefore
\[ dG = \left( \frac{\partial G}{\partial T} \right)_{p,n_A,n_B}dT + \left( \frac{\partial G}{\partial p} \right)_{T,n_A,n_B}dp + \left( \frac{\partial G}{\partial n_A} \right)_{T,p,n_B}dn_A + \left( \frac{\partial G}{\partial n_B} \right)_{T,p,n_A}dn_B \]
From Chapter 6,
\[ \left( \frac{\partial G}{\partial T} \right)_p = -S \]
\[ \left( \frac{\partial G}{\partial p} \right)_T = V \]
Substituting these results gives
\[ dG = -S\,dT + V\,dp + \left( \frac{\partial G}{\partial n_A} \right)_{T,p,n_B}dn_A + \left( \frac{\partial G}{\partial n_B} \right)_{T,p,n_A}dn_B \]
The final two terms describe how the Gibbs function changes when the composition of the system changes.
The quantity
\[ \boxed{ \mu_A = \left( \frac{\partial G} {\partial n_A} \right)_{T,p,n_B} } \]
is called the chemical potential of component A.
Similarly,
\[ \boxed{ \mu_B = \left( \frac{\partial G} {\partial n_B} \right)_{T,p,n_A} } \]
is the chemical potential of component B.
Substituting these definitions into the total differential yields
\[ \boxed{ dG = V\,dp - S\,dT + \mu_A\,dn_A + \mu_B\,dn_B } \]
For a system containing many components, this becomes
\[ \boxed{ dG = V\,dp - S\,dT + \sum_i \mu_i\,dn_i } \]
This is one of the most important equations in thermodynamics because it extends the Gibbs function to systems whose composition can change.
The chemical potential tells us how much the Gibbs function changes when a small amount of a component is added to a system.
Since systems tend to move toward lower Gibbs functions, the chemical potential provides the driving force for changes in composition.
Suppose that component A has a lower chemical potential than component B:
\[ \mu_A < \mu_B \]
Converting a small amount of B into A will decrease the total Gibbs function of the system.
Nature therefore tends to favor processes that move material from regions of high chemical potential toward regions of low chemical potential.
In this sense, chemical potential plays a role analogous to:
Many equilibrium conditions can be expressed in terms of chemical potentials.
| System | Equilibrium Condition |
|---|---|
| Thermal equilibrium | Equal temperature |
| Mechanical equilibrium | Equal pressure |
| Diffusion equilibrium | Equal chemical potential |
| Phase equilibrium | Equal chemical potential |
| Chemical equilibrium | Equal chemical potential |
For example, if component A can move between two phases, equilibrium requires
\[ \boxed{ \mu_A^{(1)} = \mu_A^{(2)} } \]
If the chemical potentials are not equal, material will move until they become equal.
The chemical potential is the most important thermodynamic quantity for systems whose composition can change.
It governs:
Big picture: The chemical potential is the partial molar Gibbs function. It describes how the Gibbs function changes when composition changes and serves as the driving force behind mixing, diffusion, phase transitions, and chemical reactions.
Consider a binary system containing components A and B at constant temperature and pressure.
The chemical potentials are
\[ \mu_A = -25.0\ \mathrm{kJ\,mol^{-1}} \]
\[ \mu_B = -18.0\ \mathrm{kJ\,mol^{-1}} \]
Suppose that \(0.100\ \mathrm{mol}\) of B is converted into A. Calculate the change in the Gibbs function.
At constant temperature and pressure,
\[ dG = \mu_A\,dn_A + \mu_B\,dn_B \]
Since \(0.100\ \mathrm{mol}\) of B is converted into A,
\[ dn_A = +0.100\ \mathrm{mol} \]
\[ dn_B = -0.100\ \mathrm{mol} \]
Therefore,
\[ \Delta G = \mu_A\,\Delta n_A + \mu_B\,\Delta n_B \]
\[ \Delta G = \left(-25.0 \frac{kJ}{mol}\right)(0.100\ mol) + \left(-18.0\frac{kJ}{mol}\right)(-0.100\ mol) \]
\[ \Delta G = -2.50\ kJ + 1.80\ kJ \]
\[ \boxed{ \Delta G = -0.70\ \mathrm{kJ} } \]
Interpretation: The Gibbs function decreases when B is converted into A. Since systems tend toward lower Gibbs functions, this process is spontaneous.
Notice that component A has the lower chemical potential:
\[ \mu_A < \mu_B \]
Therefore, matter tends to move toward A because doing so lowers the total Gibbs function of the system.
Big picture: Chemical potential extends thermodynamics to systems whose composition can change. Just as temperature drives heat flow and pressure drives mechanical change, chemical potential drives the movement of matter. It is one of the most important concepts in thermodynamics because it governs mixing, phase equilibrium, and chemical equilibrium.