Raoult's Law - Relationship To Thermodynamics

Relationship To Thermodynamics

Raoult’s law was originally discovered as an approximate experimental law. Using Raoult’s law as the definition of an ideal solution, it is possible to deduce that the chemical potential of each component is given by

,

where is the chemical potential of component i in the pure state. This equation for the chemical potential may then be used to derive other thermodynamic properties of an ideal solution. (see Ideal solution)

However a more theoretical thermodynamic definition of an ideal solution is one in which the chemical potential of each component is given by the above formula. It is then possible to re-derive Raoult’s law as follows:

If the system is at equilibrium, then the chemical potential of the component i must be the same in the liquid solution and in the vapor above it. That is,

Assuming the liquid is an ideal solution, and using the formula for the chemical potential of a gas, gives:

\mu _{i,{\rm liq}}^{\star} + RT\ln x_i = \mu_{i,{\rm vap}}^\ominus + RT\ln \frac{{f_i }} {{p^\ominus }}

where ƒi is the fugacity of the vapor of and indicates reference state.

The corresponding equation for pure in equilibrium with its (pure) vapor is:

\mu _{i,{\rm liq}}^{\star} = \mu _{i,{\rm vap}}^\ominus + RT\ln \frac{{f_i^{\star}}} {{p^\ominus }}

where * indicates the pure component.

Subtracting both equations gives us

which re-arranges to

The fugacities can be replaced by simple pressures if the vapor of the solution behaves ideally i.e.

which is Raoult’s Law.

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