Wilson Correction Application To Raoult's Law Explained

Raoult's law, established by French chemist François-Marie Raoult in 1887, states that the partial vapour pressure of each component of an ideal mixture of liquids is equal to the vapour pressure of the pure component multiplied by its mole fraction in the mixture. In other words, the vapour pressure of a solution is the mole-weighted mean of the individual vapour pressures.

However, many solutions deviate from Raoult's law, which only applies to ideal solutions. To address this, Wilson's correction can be applied. Wilson's correction, also known as Wilson's equation, is an adaptation of Raoult's law that accounts for non-ideal solutions by incorporating two factors that consider the interactions between molecules of different substances.

Wilson's equation is particularly useful for flash calculations, which are used to determine the number of phases present in a mixture. By applying Wilson's correction to Raoult's law, it is possible to more accurately estimate the vapour pressure of a solution and improve the understanding of its behaviour.

The Wilson correction: an overview

The Wilson correction is applied to Raoult's law, a principle in physical chemistry established by French chemist François-Marie Raoult in 1887. Raoult's law concerns the relationship between the vapour pressure of an ideal mixture of liquids and that of its pure components. According to Raoult, the partial vapour pressure of each component in an ideal mixture is equal to the vapour pressure of the pure component multiplied by its mole fraction in the mixture.

Mathematically, for a single component in an ideal solution, Raoult's law can be written as:

> pi = pi^* xi

Where:

• Pi is the partial pressure of component i in the gaseous mixture above the solution
• Pi^ is the equilibrium vapour pressure of the pure component i
• Xi is the mole fraction of component i in the liquid or solid solution

Raoult's law assumes ideal behaviour, where intermolecular forces between unlike molecules are equal to those between similar molecules, and their molar volumes are the same. This is analogous to the ideal gas law, which is valid when interactive forces between molecules are negligible.

However, in reality, many solutions deviate from Raoult's law as they are not ideal. This is because different chemical components must be chemically identical for an ideal solution to exist. In most cases, the liquids in a mixture do not exhibit the same uniformity in attractive forces, and so the vapour pressure of the solution will be higher or lower than predicted by Raoult's law.

The Wilson correction is used to account for these non-ideal solutions. It is applied when the K-values (equilibrium ratios) are initially assumed to be ideal, following Raoult's law and Dalton's law of partial pressures. The Wilson correction then provides a better estimate by considering the critical pressure and temperature of the component, and the acentric factor, which accounts for the deviation from ideal behaviour.

This corrected K-value can then be used to determine the number of phases present in the system, and subsequently, the compositions of each phase and the mole fraction of each phase.

Raoult's Law: ideal vs non-ideal solutions

Raoult's Law, established in 1887 by French chemist François-Marie Raoult, is a relation of physical chemistry with implications in thermodynamics. The law states that the partial vapour pressure of a solvent in a solution is equal to the vapour pressure of the pure solvent multiplied by its mole fraction in the solution.

Mathematically, this can be written as:

> Psolution = Χsolvent x P0solvent

Where:

• Psolution = vapour pressure of the solution
• Χsolvent = mole fraction of the solvent
• P0solvent = vapour pressure of the pure solvent

Raoult's Law applies to ideal solutions, which are rare. In ideal solutions, the intermolecular interactions between solute-solute (A-A) and solvent-solvent (B-B) are similar to the interaction between solute-solvent (A-B). An ideal solution obeys Raoult's Law for all concentration and temperature ranges, and exhibits zero enthalpy and volume of mixing (ΔHmix = 0 and ΔVmix = 0). Examples of ideal solutions include benzene and toluene, hexane and heptane, and bromoethane and chloroethane.

Non-ideal solutions, on the other hand, do not obey Raoult's Law for all concentration and temperature ranges. Non-ideal solutions can be further classified into two types:

• Non-ideal solutions showing positive deviation from Raoult's Law: These solutions have a total vapour pressure higher than that calculated from Raoult's equation. The interaction between solute-solvent (A-B) is weaker than that between pure components (A-A or B-B). Examples include ethanol and acetone, and carbon disulphide and acetone.
• Non-ideal solutions showing negative deviation from Raoult's Law: These solutions have a total vapour pressure lower than that calculated from Raoult's equation. The interaction between solute-solvent (A-B) is stronger than that between pure components (A-A or B-B). Examples include phenol and aniline, and chloroform and acetone.

Raoult's Law can be adapted for non-ideal solutions by incorporating two factors: a correction for gas non-ideality, or deviations from the ideal gas law (fugacity coefficient); and a correction for interactions in the liquid phase between different molecules (activity coefficient). This modified Raoult's Law equation is:

> yip x p = x i γ i p i *

Where:

• Yi = mole fraction in the gas phase
• P = pressure
• X = mole fraction of component i in the solution
• Γi = activity coefficient
• Pi = equilibrium vapour pressure of the pure component i

Applying Wilson Correction to Raoult's Law

Wilson's equation can be used as an initial estimate for the K-values when calculating flash calculations for two-phase mixtures. While Raoult's Law assumes ideal behaviour, Wilson's equation is a better estimate for mixtures that are far from ideal:

> y = exp [−ΔvapHmix(1−T/Tc) /(RT)]

Where:

• Y = K-value
• ΔvapHmix = excess enthalpy of mixing of the components in the vapour phase
• T = absolute temperature
• Tc = critical temperature of the mixture
• R = gas constant

Limitations of Raoult's Law

Raoult's Law is a relation of physical chemistry, with implications in thermodynamics. It is considered a valid law for ideal solutions, but such solutions are hard to find and rare. Here are some of the limitations of Raoult's Law:

• Raoult's Law is only applicable to ideal solutions. It assumes that the intermolecular forces between different molecules and similar molecules are equal. However, in reality, many liquids in a mixture do not have the same uniformity in terms of attractive forces and will deviate from Raoult's Law. For example, a mixture of chloroform and acetone or a solution of water and hydrochloric acid will show a negative deviation from the law, while mixtures of benzene and methanol or ethanol and chloroform will show a positive deviation.
• Raoult's Law is only applicable to solutions containing non-volatile solutes.
• Raoult's Law is not applicable to solutes that dissociate or associate in a particular solution. For instance, if NaCl (salt) is added to a solution, it will dissociate into Na+ and Cl-.
• Raoult's Law is only valid for very dilute solutions.

Raoult's Law is a useful concept in chemistry, but it is important to be aware of its limitations and the fact that most solutions deviate from ideal behaviour.

How to calculate the vapour pressure of a solution

Raoult's Law states that the vapour pressure of an ideal solution is directly related to the vapour pressure of each chemical component and their mole fractions in the solution. In other words, the vapour pressure of a solution is the mole-weighted mean of the individual vapour pressures.

Mathematically, Raoult's Law for a single component in an ideal solution is:

> p_i = p*_i * x_i

Where:

• P_i is the partial pressure of component i in the gaseous mixture above the solution
• P_i is the equilibrium vapour pressure of the pure component i
• X_i is the mole fraction of component i in the liquid or solid solution

When dealing with a solution that has a non-volatile solute (a solute that does not tend to form vapour at the temperature of the solution), the vapour pressure of that solution is:

> P_(solution) = chi_(solvent) * P_(solvent)^0

Where:

• P_(solution) is the vapour pressure of the solution
• Chi_(solvent) is the mole fraction of the solvent
• P_(solvent)^0 is the vapour pressure of the pure solvent

The mole fraction is the ratio between the solvent moles and the total number of moles in the solution.

If you are dealing with a solution that contains a volatile solute, the vapour pressure of that solution is:

> P_(solution) = chi_(solvent) * P_(solvent)^0 + chi_(solute) * P_(solute)^0

Where:

• P_(solute)^0 is the vapour pressure of the pure solute
• Chi_(solute) is the mole fraction of the solute

Raoult's Law can be applied to non-ideal solutions by incorporating factors that account for the interactions between molecules of different substances. The first factor is a correction for gas non-ideality, or deviations from the ideal gas law, called the fugacity coefficient. The second, the activity coefficient, is a correction for interactions in the liquid phase between different molecules.

This modified or extended Raoult's Law is then written as:

> y_i * phi_p,i * p = x_i * gamma_i * p*_i

Where:

• Y_i is the mole fraction of component i in the gas phase
• Phi_p,i is the fugacity coefficient
• Gamma_i is the activity coefficient
• P_i is the equilibrium vapour pressure of the pure component i

Example Calculation

Let's work through an example to illustrate how to calculate the vapour pressure of a solution.

Say we want to find the vapour pressure of simple syrup, which is traditionally one part sugar dissolved in one part water. So, sugar is our solute and water is our solvent. The current temperature of the simple syrup is 298 K (about 25 degrees Celsius).

First, we need to find the solvent's vapour pressure. Water at 25 degrees Celsius has a vapour pressure of 23.8 mm Hg.

Next, we need to find the mole fraction of our solvent. This is calculated as follows:

> mole fraction = moles of solvent / total number of moles in the solution

We'll say that our recipe for simple syrup uses 1 litre of water and 1 litre of sucrose (sugar).

Mass (1 L of water): 1,000 grams

Mass (1 L of raw sugar): 1,056.7 grams

Moles (water): 1,000 grams * 1 mol / 18.015 g = 55.51 moles

Moles (sucrose): 1,056.7 grams * 1 mol / 342.2965 g = 3.08 moles

Total moles: 55.51 + 3.08 = 58.59 moles

Mole fraction of water: 55.51 / 58.59 = 0.947

Finally, we can plug our values into the Raoult's Law equation:

> Psolution = PsolventXsolvent

> Psolution = (23.8 mm Hg)(0.947)

> Psolution = 22.54 mm Hg

Wilson's Correction

Wilson's equation can be used to estimate K-values (equilibrium ratios) for mixtures that are far from ideal at reservoir conditions. Wilson's equation is as follows:

> ln P = ln P^0 - (A * (1 - T^-B)) / (T - theta)

Where:

• P is the absolute pressure
• P^0 is the vapour pressure of the pure component
• T is the absolute temperature
• A and B are substance-dependent constants
• Theta is the boiling point of the component

Wilson's equation can be used to estimate the initial vapour pressure of a solution, which can then be used as input for Raoult's Law to calculate the vapour pressure of the solution.

Real-world applications of the Wilson correction

The Wilson correction is applied to Raoult's Law to account for non-ideal solutions, which are solutions where the intermolecular forces between unlike molecules are not equal to those between similar molecules. Raoult's Law assumes ideal behaviour and is valid only for ideal solutions.

Wilson's correction is particularly useful in the following real-world applications:

Distillation

Distillation is a common process used to purify solvents, and it relies on the distribution of the constituents between the liquid and vapour phases being at equilibrium. While simple distillation involves minimal equipment, fractional distillation is a more efficient process that uses a distillation column with a series of plates to achieve equilibrium between the liquid and vapour phases. This process is used to recover solvents such as acetonitrile or methanol from azeotropic mixtures.

Chemical Equilibrium in Solution

Raoult's Law, combined with Wilson's correction, can be used to deduce the expression for the equilibrium constant in solution equilibria. This is particularly useful in chemical reactions that readily proceed to equilibrium without a catalyst. By studying the equilibrium constant at different temperatures, chemists can estimate the enthalpy and entropy of bond formation between reactants. This information is valuable for developing models of molecular interactions and complex formation.

Vapour Pressure

Vapour pressure is exhibited by all solids and liquids and depends on the type of liquid and its temperature. Raoult's Law, with Wilson's correction, can be used to understand the vapour pressure of solutions, especially those that deviate from ideal behaviour. For example, a mixture of chloroform and acetone exhibits a negative deviation from Raoult's Law, indicating an attractive interaction between the two components. On the other hand, mixtures like ethanol and water or benzene and methanol show positive deviations, where the cohesion between similar molecules is greater than the adhesion between dissimilar molecules.

Molecular Mass Determination

Raoult's Law can be applied to calculate the molecular mass of an unknown solute. This is achieved by measuring the vapour pressure of the solution and the pure solvent, along with knowledge of the temperature.

Frequently asked questions