The vapor pressure of a solvent decreased by 10 mm of Hg when a non-volatile solute was added to it. The mole fraction of solute in solution is 0.2. What would be the mole fraction... The vapor pressure of a solvent decreased by 10 mm of Hg when a non-volatile solute was added to it. The mole fraction of solute in solution is 0.2. What would be the mole fraction of solute if the vapor pressure is 20 mm of Hg?
Understand the Problem
The question is asking about the relationship between the vapor pressure of a solvent and the addition of a non-volatile solute. Specifically, it seeks to determine the mole fraction of the solute after a specific decrease in vapor pressure.
Answer
The mole fraction of the solute is approximately $0.333$.
Answer for screen readers
The calculated mole fraction of the solute is approximately $0.333$.
Steps to Solve
- Understand Vapor Pressure and Raoult's Law
Vapor pressure of a solvent decreases when a non-volatile solute is added. According to Raoult's Law, the change in vapor pressure ($\Delta P$) is directly proportional to the mole fraction of the solute ($X_{solute}$) and can be described as:
$$ \Delta P = P_{0} \cdot X_{solute} $$
where $P_{0}$ is the vapor pressure of the pure solvent.
- Identify the Known Values
In this case, it’s given that the vapor pressure decreased by 10 mm Hg, and the resulting mole fraction of the solute is 0.2. The initial vapor pressure of the solvent ($P_{0}$) which corresponds to the 20 mm Hg final pressure needs to be calculated:
$$ P = P_{0} - \Delta P $$
where $P = 20 , \text{mm Hg}$.
- Calculate the Initial Vapor Pressure
Rearranging the formula to find $P_{0}$:
$$ P_{0} = P + \Delta P $$
Substituting the known values:
$$ P_{0} = 20 , \text{mm Hg} + 10 , \text{mm Hg} = 30 , \text{mm Hg} $$
- Set Up the Mole Fraction Equation
From Raoult's Law, set up the equation for the mole fraction of the solute:
$$ \Delta P = P_{0} \cdot X_{solute} $$
Substituting the known values into this equation, we have:
$$ 10 = 30 \cdot X_{solute} $$
- Solve for Mole Fraction of Solute
Rearranging the equation to find $X_{solute}$:
$$ X_{solute} = \frac{10}{30} = \frac{1}{3} \approx 0.333 $$
- Compare with Given Answers
Now, compare $0.333$ with the provided options to determine which one is the closest or correct value.
The calculated mole fraction of the solute is approximately $0.333$.
More Information
This problem demonstrates the application of Raoult's Law in determining how the addition of a non-volatile solute affects the vapor pressure of a solvent, as well as how to manipulate equations to isolate specific variables.
Tips
- Misunderstanding the relationship between the vapor pressure decrease and the mole fraction.
- Failing to correctly apply Raoult's Law or to rearrange it properly to find the mole fraction.
- Not keeping track of units, which can lead to confusion in pressure measurements.
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