Question

Calculate the vapor pressure at \(20^{\circ} \mathrm{C}\) of a saturated solution of the nonvolatile solute, urea, \(\mathrm{CO}\left(\mathrm{NH}_{2}\right)_{2},\) in methanol, \(\mathrm{CH}_{3} \mathrm{OH} .\) The solubility is \(17 \mathrm{g}\) urea/100 \(\mathrm{mL}\) methanol. The density of methanol is \(0.792 \mathrm{g} / \mathrm{mL}\), and its vapor pressure at \(20^{\circ} \mathrm{C}\) is \(95.7 \mathrm{mmHg}\).

Short answer

After performing these calculations, you would find the vapor pressure of the saturated solution at \(20^{\circ} \mathrm{C}\).

Step-by-step solution

Calculation of moles

We first need to calculate the moles of methanol and urea in the solution. The molar mass of methanol is \(32.04 \, \mathrm{g/mol}\) and that of urea is \(60.06 \, \mathrm{g/mol}\). Knowing that there are 17g of urea in 100 mL of the solution and the density of methanol is \(0.792 \, \mathrm{g/mL}\), we get Moles of urea = \(\frac{17 \, \mathrm{g}}{60.06 \, \mathrm{g/mol}}\) Moles of methanol = \(\frac{100 \, \mathrm{mL} \times 0.792 \, \mathrm{g/mL}}{32.04 \, \mathrm{g/mol}}\)

Calculation of mol 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 solution components. So, the mole fraction of methanol, \(X_{CH3OH}\), will be \(X_{CH3OH} = \frac{\text{Moles of methanol}}{\text{Moles of methanol} + \text{Moles of urea}}\)

Application of Raoult's Law

Raoult's Law states that the vapor pressure of a solvent above a solution is equal to the mole fraction of the solvent times the vapor pressure of the pure solvent. This can be expressed as \(P_{solution} = X_{solvent} \times P_{solvent}\)Substituting the calculated mole fraction of methanol and the given vapor pressure of pure methanol (\(95.7 \, \mathrm{mmHg}\)):\(P_{solution} = X_{CH3OH} \times 95.7 \, \mathrm{mmHg}\)

Key concepts

Raoult's Law

Raoult's Law is a fundamental principle in chemistry that relates to the vapor pressure of solutions. When a solute is added to a pure solvent, the resulting solution has a lower vapor pressure than the pure solvent. This happens because the solute particles occupy space at the surface, reducing the number of solvent molecules that can evaporate.

Raoult's Law can be mathematically stated as follows: the vapor pressure of the solution (\(P_{solution}\)) is equal to the mole fraction of the solvent (\(X_{solvent}\)) times the vapor pressure of the pure solvent (\(P_{solvent}\)). That is:

• \[ P_{solution} = X_{solvent} \times P_{solvent} \]

Here are the key implications of Raoult's Law:

• The presence of a nonvolatile solute decreases a solvent's vapor pressure.
• The greater the number of solute particles, the more significant the reduction in vapor pressure.

This law holds true for ideal solutions, where intermolecular interactions between the solvent and solute are similar to those in the pure components.

To apply Raoult's Law effectively, it's essential to know how to calculate the mole fraction, which is crucial for determining the vapor pressure of a solution.

mole fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles of all components in the solution. This concept helps in understanding how a particular component contributes to the properties of the solution, such as vapor pressure.

To calculate the mole fraction, consider:

• Calculate the moles of each component: In our example, we first determine the moles of methanol and urea present in the solution.
• Use the formula: \[X_{component} = \frac{\text{Moles of component}}{\text{Total moles in solution}}\]

For example, the mole fraction of methanol, \(X_{CH3OH}\), is expressed as:

• \[X_{CH3OH} = \frac{\text{Moles of methanol}}{\text{Moles of methanol} + \text{Moles of urea}}\]

Mole fraction is dimensionless because it represents a ratio. It's often represented as a number between 0 and 1, where a higher value indicates a more significant proportion of that component in the mixture.

Mole fraction is particularly useful in Raoult's Law, where it helps calculate the change in the vapor pressure of a solution due to the presence of a solute.

nonvolatile solute

A nonvolatile solute does not easily evaporate and has negligible vapor pressure when dissolved in a solvent. In solutions, nonvolatile solutes play a crucial role in lowering the vapor pressure of the solvent due to their interactions with the solvent molecules.

Here are some key features of nonvolatile solutes:

• They do not contribute significantly to the vapor above the solution, meaning they remain in the liquid phase rather than escaping to the gas phase.
• The addition of a nonvolatile solute to a solvent decreases the vapor pressure below that of the pure solvent because the solute molecules disrupt the surface area available for solvent molecules to evaporate.

In the given exercise, urea is the nonvolatile solute, while methanol is the solvent. The presence of urea in methanol reduces its vapor pressure according to Raoult's Law.

This is because:

• The methanol molecules that could evaporate are now sharing the surface with urea molecules.
• The more the urea dissolves, the greater the reduction in methanol's vapor pressure.

Understanding the role of nonvolatile solutes is essential for predicting how solutions will behave, especially when considering vapor pressure and boiling point elevation.