Raoult's law
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Once the components in the solution have reached chemical equilibrium, the total vapor pressure of the solution is:
- [\ P_= (P_)_ X_1 + (P_)_ X_2 \cdots]
- [\ P_=(P_)_ X_i ]
- (Pi)pure is the vapor pressure of the pure component
- Xi is the mole fraction of the component in solution
This law is strictly valid only under the assumption that the chemical bond between the two liquids is equal to the bonding within the liquids, the conditions of an ideal solution. Therefore, comparing actual measured vapor pressures to predicted values from Raoult's law allows information about the relative strength of bonding between liquids to be obtained. If the measured value of vapor pressure is less than the predicted value, fewer molecules have left the solution than expected. This is put down to the strength of bonding between the liquids being greater than the bonding within the individual liquids, so fewer molecules have enough energy to leave the solution. Conversely, if the vapor pressure is greater than the predicted value more molecules have left the solution than expected, due to the bonding between the liquids being less strong than the bonding within each.
Deduction of Raoult’s Law (or Raoult’s Equation)
We define an ideal solution, as the solution for which the chemical potential of the component [i] is[\mu _i = \mu _i ^\circ + RT\ln x_i]
Where the reference state is the pure substance at work [P] and [T].
Experimentally, an ideal solution has a zero enthalpy of solution and verifies volume additivity.
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,
[\mu _ = \mu _\,]
If the liquid is an ideal solution, and using the formula for a gas’ chemical potential
[\mu _ ^\circ + RT\ln x_i = \mu _} ^\circ + RT\ln \frac] (1)
(where [f] is the fugacity of the vapor of [i])
If we study the component i in its pure state, we would have
[\mu _^* = \mu _^*\,]
Where * indicates that we study a pure component.
[\mu _ ^\circ + RT\ln x_i = \mu _}} + RT\ln \frac]
But now, [x_i=1], so
[\mu _ ^\circ = \mu _}} + RT\ln \frac] (2)
Substracting (1)-(2) gives us
[(\mu _^\circ - ^\circ }) + RT\ln x_i = RT\ln \frac]
which can be written as
[\frac + \ln x_i = \ln \frac]
or
[e^x_i = \frac]
This equation is valid for the ideal solution.
Now, let’s suppose the vapor of the solution behaves as an ideal gas. In this case, fugacity and pressure are identical, and we get
[e^ x_i = \frac]
For most substances we have [\Delta \mu \approx 0], and then
[x_i \approx \frac]
Finally,
[P_i \approx x_i P_i^*]
This last equality is what is known as Raoult’s Law. However, in physical chemistry and other sciences, a scientific law is a generalization of an empiric observation, and must be therefore verifiable in every experiment.
However, this equation is not verified by most solutions, and then cannot be considered a law. We can clearly see that not only the solution must be ideal; we also have to suppose that the vapor has an ideal gas behavior and that for the substance it is true that [\Delta \mu \approx 0]. Raoult’s Law should be called Raoult’s Equation, as its range of validity is strictly limited.
Moreover, an ideal solution cannot be defined as the one that follows Raoult's Equation. As shown above, an ideal solution follows approximately the equation.
See also
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