Saturated Fluid - Raoult's Law

Raoult's Law

At boiling and higher temperatures the sum of the individual component partial pressures becomes equal to the overall pressure, which can symbolized as P_tot.

Under such conditions, Dalton's law would be in effect as follows:

P_tot = P₁ + P₂ + ...

Then for each component in the vapor phase:

y₁ = P₁ / P_tot, y₂ = P₂ / P_tot, ... etc.

where P₁ = partial pressure of component 1, P₂ = partial pressure of component 2, etc.

Raoult's law is approximately valid for mixtures of components between which there is very little interaction other than the effect of dilution by the other components. Examples of such mixtures includes mixtures of alkanes, which are non-polar, relatively inert compounds in many ways, so there is little attraction or repulsion between the molecules. Raoult's law states that for components 1, 2, etc. in a mixture:

P₁ = x₁ P o₁, P₂ = x₂ P o₂, ... etc.

where P o₁, P o₂, etc. are the vapor pressures of components 1, 2, etc. when they are pure, and x₁, x₂, etc. are mole fractions of the corresponding component in the liquid.

Recall from the first section that vapor pressures of liquids are very dependent on temperature. Thus the Po pure vapor pressures for each component are a function of temperature (T): For example, commonly for a pure liquid component, the Clausius–Clapeyron relation may be used to approximate how the vapor pressure varies as a function of temperature. This makes each of the partial pressures dependent on temperature also regardless of whether Raoult's law applies or not. When Raoult's law is valid these expressions become:

P₁T = x₁ P o₁T, P₂T = x₂ P o₂T, ... etc.

At boiling temperatures if Raoult's law applies, the total pressure becomes:

P_tot = x₁ P o₁T + x₂ P o₂T + ... etc.

At a given P_tot such as 1 atm and a given liquid composition, T can be solved for to give the liquid mixture's boiling point or bubble point, although the solution for T may not be mathematically analytical (i.e., may require a numerical solution or approximation). For a binary mixture at a given P_tot, the bubble point T can become a function of x₁ (or x₂) and this function can be shown on a two-dimensional graph like a binary boiling point diagram.

At boiling temperatures if Raoult's law applies, a number of the preceding equations in this section can be combined to give the following expressions for vapor mole fractions as a function of liquid mole fractions and temperature:

y₁ = x₁ P o₁T / P_tot, y₂ = x₂ P o₂T / P_tot, ... etc.

Once the bubble point T's as a function of liquid composition in terms of mole fractions have been determined, these values can be inserted into the above equations to obtain corresponding vapor compositions in terms of mole fractions. When this is finished over a complete range of liquid mole fractions and their corresponding temperatures, one effectively obtains a temperature T function of vapor composition mole fractions. This function effectively acts as the dew point T function of vapor composition.

In the case of a binary mixture, x₂ = 1 − x₁ and the above equations can be expressed as:

y₁ = x₁ P o₁T / P_tot, and

y₂ = (1  −  x₁) P o₂T / P_tot

For many kinds of mixtures, particularly where there is interaction between components beyond simply the effects of dilution, Raoult's law does not work well for determining the shapes of the curves in the boiling point or VLE diagrams. Even in such mixtures, there are usually still differences in the vapor and liquid equilibrium concentrations at most points, and distillation is often still useful for separating components at least partially. For such mixtures, empirical data is typically used in determining such boiling point and VLE diagrams. Chemical engineers have done a significant amount of research trying to develop equations for correlating and/or predicting VLE data for various kinds of mixtures which do not obey Raoult's law well.