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:

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

See also

• solubility