Question

Calculate the vapor pressure at \(25^{\circ} \mathrm{C}\) of a solution containing \(165 \mathrm{g}\) of the nonvolatile solute, glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) in \(685 \mathrm{g} \mathrm{H}_{2} \mathrm{O}\). The vapor pressure of water at \(25^{\circ} \mathrm{C}\) is \(23.8 \mathrm{mmHg}\).

Short answer

The vapor pressure of the solution at \(25^{\circ} C\) is \(23.2 mmHg\).

Step-by-step solution

Calculate the moles of glucose and water

To calculate the mole fraction, it's first required to know the moles of glucose and water. Use the given mass for each and their molar masses for this. For glucose \(C_6H_{12}O_6\), the molar mass is approximately \(180.156 g/mol\). For water \(H_2O\), it's \(18.015 g/mol\). Thus, \n\nMoles of glucose = \(165g / 180.156 g/mol = 0.916 mol\) \n\nMoles of water = \(685g / 18.015 g/mol = 38 mol\)

Calculate the mole fraction of water

The mole fraction is calculated by dividing the moles of water by the total moles in the solution. It is given by \n\nMole fraction of water = Moles of water / (Moles of water + Moles of glucose) \n\nSubstituting the values, \n\nMole fraction of water = \(38 mol / (38 mol + 0.916 mol) = 0.976\)

Calculate the vapor pressure of the solution

Finally, apply Raoult's law, which states that the vapor pressure of a solution is equal to the mole fraction of the solvent times the vapor pressure of the pure solvent. \n\nThe vapor pressure of the solution \( P_{solution}\) is thus given by \n\n\( P_{solution} \) = Mole fraction of water \(\times \) Vapor pressure of pure water \n\nSubstituting the values, \n\n\( P_{solution} \) = \(0.976 \times 23.8 mmHg = 23.2 mmHg \)

Key concepts

Vapor Pressure

Vapor pressure is a fundamental concept in chemistry, referring to the pressure exerted by a vapor in equilibrium with its liquid or solid form. This is an important factor to consider when analyzing solutions, especially in the context of Raoult's law. At a given temperature, such as the scenario provided where we are looking at 25°C, the vapor pressure of a solvent like water can be influenced by the presence of solutes.

In general, when a nonvolatile solute is added to a solvent, it decreases the solvent's vapor pressure. This is because fewer molecules of the solvent are able to escape into the vapor phase, resulting in a lower equilibrium vapor pressure.

Understanding vapor pressure is crucial because it determines how easily a liquid can transition into a gas, affecting phenomena such as boiling points which can have significant practical implications, such as in cooking or industrial processes.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a solution. 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 mixture. This is a unitless number, providing insight into the proportion of the solute versus the solvent.

In our example, we calculated the mole fraction of water, which is a key player in determining the vapor pressure of the solution using Raoult’s law. The mole fraction helps us understand how the presence of a solute like glucose affects the overall composition of the solution.

• Formula: Mole fraction of component A = Moles of A / Total moles in the solution
• It is crucial in calculating other important properties like boiling point elevation and freezing point depression.

By using the mole fraction, we can more accurately predict the behavior of solutions under various conditions.

Nonvolatile Solute

A nonvolatile solute is one that does not easily vaporize and thus does not contribute to the vapor pressure of the solution. In the provided exercise, glucose acts as a nonvolatile solute when dissolved in water, affecting the solution's vapor pressure.

The key feature of a nonvolatile solute is its lack of contribution to the vapor phase. This absence means that when a nonvolatile solute is dissolved in a solvent:

• The boiling point of the solution increases (known as boiling point elevation).
• The freezing point of the solution decreases (known as freezing point depression).
• The overall vapor pressure of the solution is lower than that of the pure solvent.

Nonvolatile solutes are important in many practical applications, ranging from cooking to industry, because they modify the properties of solvents in predictable ways. This predictive ability is crucial in many chemical processes and solutions chemistry.