Question

Denoting the solvent and solute in a dilute binary liquid solution at temperature \(T\) and pressure \(p\) by the subscripts 1 and 2 , respectively, show that if the fugacity of the solute is proportional to its mole fraction in the solution: \(\bar{f}_{2}=\kappa y_{2}\), where \(\kappa\) is a constant (Henry's rule), then the fugacity of the solvent is \(\bar{f}_{1}=y_{1} f_{1}\), where \(y_{1}\) is the solvent mole fraction and \(f_{1}\) is the fugacity of pure 1 at \(T, p\).

Short answer

The fugacity of the solvent is \( \bar{f}_{1} = y_{1} f_{1} \) as shown using Raoult's Law and the relationship \( f_{1} = P_{1}^{*} \).

Step-by-step solution

Understand the Given Information

Given a dilute binary liquid solution with solvent (1) and solute (2) at temperature \(T\) and pressure \(p\), it is stated that the fugacity of the solute is proportional to its mole fraction in the solution, represented as \(\bar{f}_{2} = \kappa y_{2}\), where \(\kappa\) is a constant (Henry's rule).

Fugacity of the Solvent

To demonstrate that the fugacity of the solvent can be expressed as \(\bar{f}_{1} = y_{1} f_{1}\), where \(y_{1}\) is the mole fraction and \(f_{1}\) is the fugacity of pure solvent 1 at the given temperature and pressure.

Apply Raoult's Law for the Solvent

Raoult's law for the solvent states that the partial vapor pressure of the solvent in a solution is equal to the mole fraction of the solvent times the vapor pressure of the pure solvent. Mathematically, it is expressed as: \( P_{1} = y_{1} P_{1}^{*} \), where \( P_{1}^{*} \) is the vapor pressure of the pure solvent.

Fugacity of the Solvent Using Raoult's Law

In a solution, the fugacity \(\bar{f}_{1}\) of the solvent can be approximated as: \( \bar{f}_{1} = P_{1} \phi_{1} \), where \( \phi_{1} \) is the fugacity coefficient. Because the solution is dilute, \(\phi_{1}\) can be approximated to be the same for the pure solvent, thus \( \bar{f}_{1}\approx P_{1} = y_{1} P_{1}^{*} \).

Relating Fugacity to Pure Solvent

Recognize that the fugacity of the pure solvent, when it is alone, can be represented as \( f_{1} = P_{1}^{*} \). Hence, we substitute \( f_{1}\) into the previous result: \( \bar{f}_{1} \approx y_{1} f_{1} \).

Conclusion of the Proof

We have shown through the use of Raoult's Law and the relationship between pressure and fugacity that the fugacity of the solvent in a dilute binary liquid solution can indeed be expressed as \( \bar{f}_{1} = y_{1} f_{1} \).

Key concepts

Raoult's Law

Raoult's Law is fundamental in understanding the behavior of solutions. It states that the partial vapor pressure of a solvent in a solution is directly proportional to its mole fraction. Essentially, if you have a solution of water and salt, the vapor pressure of water above the solution is a fraction of what it would be if the water were pure. This fraction is equal to the mole fraction of water in the solution. For example, if the mole fraction of water is 0.8, then the vapor pressure of water above the solution will be 80% of its vapor pressure if it were pure. This law can be mathematically expressed as: \( P_{1} = y_{1} P_{1}^{*} \) where \( P_{1} \) is the partial vapor pressure of the solvent, \( y_{1} \) is the mole fraction of the solvent, and \( P_{1}^{*} \) is the vapor pressure of the pure solvent.

Raoult's Law is very useful for calculating how the addition of a solute to a solvent will change the vapor pressure and, consequently, boiling and freezing points. However, it is crucial to note that Raoult's Law holds accurately mostly for ideal solutions, where interactions between different molecules are similar to interactions between molecules of the same kind.

Henry's Law

Henry's Law deals with the solubility of gases in liquids. Specifically, it states that at a constant temperature, the amount of gas that dissolves in a liquid is directly proportional to the partial pressure of that gas above the liquid. The law can be formulated as: \( C = kP \) where \( C \) is the concentration of the dissolved gas, \( k \) is Henry's constant, and \( P \) is the partial pressure of the gas.

In the context of the problem, Henry's Law helps us understand why the fugacity of the solute in a dilute binary solution is proportional to its mole fraction. When we say \(\bar{f}_{2} = \kappa y_{2} \), we are effectively applying Henry's Law.

This law is particularly important in situations involving gases dissolving in liquids under low pressures, like carbonated beverages or gases in natural water bodies. Given its simplicity, the law helps us calculate how varying pressures influence gas solubility.

Fugacity Coefficient

The fugacity coefficient helps bridge the gap between real gases and ideal gas behavior. It is a correction factor that accounts for deviations from ideal gas behavior due to molecular interactions. The fugacity coefficient, \( \phi \), is defined as: \( \bar{f} = P \phi \) where \( \bar{f} \) is the fugacity, \( P \) is the pressure, and \( \phi \) is the fugacity coefficient. In an ideal scenario, \( \phi \) is exactly 1, meaning the gas behaves ideally. As real gases deviate from ideal behavior, the fugacity coefficient deviates from unity. The fugacity coefficient provides a way to 'correct' the ideal gas law and can be particularly significant in high-pressure conditions.

In the exercise, we approximate the fugacity coefficient for the solvent as being equal to that of the pure solvent. This helps in simplifying the calculation and shows how real gases can be approximated in thermodynamic equations.

Dilute Solution

A dilute solution contains a small amount of solute compared to the solvent. This simplification is useful in many calculations because it allows us to approximate behaviors that would otherwise be more complex. In the given problem, the dilution approximation helps us apply Raoult's Law and the fugacity concept effectively.

In the context of thermodynamics, analyzing a dilute solution typically assumes that the interactions between molecules of solute and solvent are negligible. This makes it easier to assume the solvent behaves nearly ideally, leading to simplified expressions for properties like vapor pressure and fugacity.

For example, the fugacity of the solute, \( \bar{f_{2}} = \kappa y_{2} \), works neatly under the assumption of a dilute solution because the constant \( \kappa \) remains approximately the same regardless of slight changes in solute amount.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It lays the foundation for understanding how energy transfers in the form of work and heat influence different systems. Key concepts in thermodynamics include: - *First Law of Thermodynamics*: Energy can neither be created nor destroyed, only transferred or converted from one form to another.
- *Second Law of Thermodynamics*: Entropy, a measure of disorder, increases in an isolated system.
- *Third Law of Thermodynamics*: As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.

In the context of our problem, thermodynamics helps us understand why principles like Raoult's Law and Henry's Law operate as they do. It offers insights into how solutions behave under different temperature and pressure conditions.

For example, when dealing with fugacity, we leverage thermodynamic principles to explain why a substance's tend to escape or 'flee' from a phase. This understanding helps in predicting how systems evolve, making thermodynamics crucial for both academic and practical applications in science and engineering.