The vapor pressure of a solvent containing a non-volatile solute in a closed system directly varies with the mole fraction of the solvent. This phenomenon is explained by Raoult's law, which states that the vapor pressure of a solvent in a solution is proportional to its mole fraction in the solution.

Explanation:

1. Raoult's Law:
Raoult's law describes the relationship between the vapor pressure of a solvent in a solution and its mole fraction. According to this law, the partial pressure of a solvent in a solution is equal to the product of its vapor pressure in its pure state and its mole fraction in the solution.

2. Mole Fraction:
The mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles of all components in the solution. Mathematically, the mole fraction (X) is calculated as the moles of the component divided by the total moles in the solution.

3. Non-volatile Solute:
A non-volatile solute is a component that does not readily evaporate or escape from the solution. It remains in the liquid phase and does not significantly contribute to the vapor pressure of the solution.

4. Vapor Pressure and Mole Fraction of Solvent:
In a solution, the vapor pressure of the solvent is reduced by the presence of a non-volatile solute. The solute molecules occupy some of the space on the surface of the solvent, making fewer solvent molecules available to evaporate. Therefore, the vapor pressure of the solvent in the solution is lower than its pure state.

5. Relationship with Mole Fraction of Solvent:
The mole fraction of the solvent in the solution determines the extent to which the vapor pressure is reduced. As the mole fraction of the solvent increases, the number of solvent molecules at the surface of the solution increases, leading to a higher vapor pressure. Conversely, if the mole fraction of the solvent decreases, fewer solvent molecules are available at the surface, resulting in a lower vapor pressure.

6. Conclusion:
In summary, the vapor pressure of a solvent containing a non-volatile solute in a closed system directly varies with the mole fraction of the solvent. As the mole fraction of the solvent increases, the vapor pressure of the solvent in the solution also increases, according to Raoult's law.