Gibbs-Duhem equation

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The Gibbs-Duhem equation in thermodynamics describes the relationship between changes in chemical potential for components in a thermodynamical system ^[1] :

$\sum_iN_i\mathrm{d}\mu_i = - S\mathrm{d}T + V\mathrm{d}P \,$

where $N_i\,$ is the number of moles of component i, $\mathrm{d}\mu_i\,$ the incremental increase in chemical potential for this component, $S\,$ the entropy, $T\,$ the absolute temperature, $V\,$ volume and $P\,$ the pressure. It shows that in thermodynamics intensive properties are not independent but related. When pressure and temperature are variable one i-1 of i components have independent values for chemical potential and Gibbs' phase rule follows. The law is named after Josiah Gibbs and Pierre Duhem.

1 Proof
2 Applications
3 External links
4 References

[edit] Proof

The Internal energy $U\,$ stated as:

$U=TS-PV+\sum_i \mu_i N_i\,$

is differentiated with respect to the internal energy:

$\mathrm{d}U=T\mathrm{d}S+S\mathrm{d}T-P\mathrm{d}V-V\mathrm{d}P+\sum_i(\mu_i \mathrm{d}N_i+N_i\mathrm{d}\mu_i)\,$

But the fundamental equation for U states that

$\mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V+\sum_i\mu_i \mathrm{d}N_i\,$

Subtracting yields the Gibbs-Duhem relation:

$0=S\mathrm{d}T-V\mathrm{d}P+\sum_iN_i\mathrm{d}\mu_i\,$

[edit] Applications

The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example.

The relation is particularly useful when considering (binary) solutions. At constant P (isobaric) and T (isothermal) it becomes:

$0=\sum_iN_i\mathrm{d}\mu_i\,$

$N_1\mathrm{d}\mu_1=-N_2\mathrm{d}\mu_2\,$

Dividing by N_total:

$\mathrm{d}\mu_1= {-x_2 \over x_1} \mathrm{d}\mu_2\,$

is obtained

This expression makes it possible to translate information measured for one component into information for the other. For example if the vapor pressure is measured for component 1 as a function of x₁ the vapor pressure for the other component can be derived using Gibbs-Duhem.

(There should be an example here)