# Gibbs-Duhem equation

The Gibbs-Duhem equation (according to Josiah Willard Gibbs and Pierre Duhem ) describes the relationship between the changes in the chemical potentials of the components in a thermodynamic system .

## formulation

${\ displaystyle \ sum _ {i} n_ {i} \ mathrm {d} \ mu _ {i} = - S \ mathrm {d} T + V \ mathrm {d} p}$

Here referred to

• ${\ displaystyle n_ {i}}$the amount of substance of the system component  i
• ${\ displaystyle \ mathrm {d} \ mu _ {i}}$the total differential of the chemical potential of the system component${\ displaystyle \ mu _ {i}}$${\ displaystyle i}$
• S the entropy
• T is the absolute temperature
• V is the volume
• p the pressure .

The Gibbs-Duhem equation is often used with isothermal and isobaric process management at the same time . Then follows:

${\ displaystyle \ mathrm {d} T = 0; \; \ mathrm {d} p = 0 \ Rightarrow \ sum _ {i} n_ {i} \ mathrm {d} \ mu _ {i} = 0}$

In such a process, the sum of the products from the amount of substance of the individual components and the change in their chemical potential disappears${\ displaystyle n_ {i}}$${\ displaystyle \ mu _ {i}.}$

## meaning

The Gibbs-Duhem equation is of great interest for thermodynamics because it shows that in a thermodynamic system not all intensive variables (variables such as temperature, pressure, chemical potential, which do not depend on the amount of a substance) are mutually variable .

If you take z. If, for example, the temperature and the pressure are variable, only the components can have independent chemical potentials. From this follows Gibbs' phase rule , which specifies the number of possible degrees of freedom for this system. ${\ displaystyle i-1}$${\ displaystyle i}$

## Derivation

The Gibbs energy is a positively homogeneous function of the degree in the quantities of matter ; that is for each and : true . Therefore, Euler's homogeneity relation applies to the Gibbs energy: ${\ displaystyle 1}$${\ displaystyle k}$${\ displaystyle n_ {1}, ..., n_ {k}}$${\ displaystyle \ lambda \ in \ mathbb {R}}$${\ displaystyle \ lambda> 0}$${\ displaystyle \ lambda G (T, p, n_ {1}, ..., n_ {k}) = G (T, p, \ lambda n_ {1}, ..., \ lambda n_ {k}) }$

${\ displaystyle G (T, p, n_ {1}, ..., n_ {k}) = \ sum _ {i} n_ {i} {\ frac {\ partial G (T, p, n_ {1} , ..., n_ {k})} {\ partial n_ {i}}} = \ sum _ {i} n_ {i} \ mu _ {i}}$

Thus applies to the total differential

${\ displaystyle dG = \ sum _ {i} n_ {i} d \ mu _ {i} + \ sum _ {i} \ mu _ {i} dn_ {i}}$

On the other hand, because of the definition of ${\ displaystyle G}$

${\ displaystyle \! \, \ mathrm {d} G = -S \ mathrm {d} T + V \ mathrm {d} p + \ sum _ {i} \ mu _ {i} \ mathrm {d} n_ {i }.}$

By comparing the two expressions, the Gibbs-Duhem equation follows:

${\ displaystyle \ sum _ {i} n_ {i} d \ mu _ {i} = - S \ mathrm {d} T + V \ mathrm {d} p}$