Question

The vapor pressure of a solution containing 53.6 \(\mathrm{g}\) glycerin \(\left(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3}\right)\) in 133.7 \(\mathrm{g}\) ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) is 113 torr at \(40^{\circ} \mathrm{C}\) . Calculate the vapor pressure of pure ethanol at \(40^{\circ} \mathrm{C}\) assuming that glycerin is a nonvolatile, nonelectrolyte solute in ethanol.

Short answer

The vapor pressure of pure ethanol at \(40^{\circ}C\) is approximately 135.6 torr, calculated using the given vapor pressure of the solution (113 torr) and the mole fraction of ethanol in the solution (0.833) with Raoult's Law.

Step-by-step solution

Calculate the moles of glycerin and ethanol in the solution

First, let's determine the moles of glycerin and ethanol in the solution, using their molecular weights (glycerin: 92 g/mol and ethanol: 46 g/mol). Moles of glycerin = \( \frac{53.6 \text{ g}}{92 \text{ g/mol}} = 0.582 \text{ mol}\) Moles of ethanol = \( \frac{133.7 \text{ g}}{46 \text{ g/mol}} = 2.91 \text{ mol}\) Total moles = 3.492 mol

Calculate the mole fraction of ethanol in the solution

Next, we need to calculate the mole fraction of ethanol in the solution. Mole fraction is defined as the ratio of the moles of one component to the total moles of the solution. Mole fraction of ethanol (X_ethanol) = \( \frac{\text{moles of ethanol}}{\text{total moles}} = \frac{2.91 \text{ mol}}{3.492 \text{ mol}} = 0.833\)

Determine the relationship between vapor pressures using Raoult's Law

Raoult's Law states that the partial vapor pressure of each component in a solution is equal to the product of its mole fraction and the vapor pressure of the pure component. \( P_{solution} = X_{ethanol}P_{ethanol} \) We have the vapor pressure of the solution (P_solution = 113 torr) and the mole fraction of ethanol (X_ethanol = 0.833). We need to find the vapor pressure of pure ethanol (P_ethanol).

Calculate the vapor pressure of pure ethanol

To find the vapor pressure of pure ethanol, rearrange the equation from Step 3 to solve for P_ethanol, and then plug in the values for P_solution and X_ethanol to get the result. \( P_{ethanol} = \frac{P_{solution}}{X_{ethanol}} = \frac{113 \text{ torr}}{0.833} = 135.6 \text{ torr}\) Thus, the vapor pressure of pure ethanol at 40°C is approximately 135.6 torr.

Key concepts

Vapor Pressure

Vapor pressure is a fundamental concept in solution chemistry. It refers to the pressure exerted by the vapor in equilibrium with its liquid phase in a closed system. When we add a nonvolatile solute, like glycerin in this exercise, to a solvent such as ethanol, the vapor pressure is lowered. This occurs because the solute molecules occupy space at the liquid's surface, diminishing the number of solvent molecules that can escape into the vapor phase.

In simpler terms, the presence of more particles at the surface means fewer solvent molecules can evaporate at any given time. As a result, the overall vapor pressure of the solution is reduced compared to that of the pure solvent.

This principle is governed by Raoult's Law, which provides a mathematical basis for predicting the vapor pressure changes due to solutes in solutions.

Mole Fraction

The mole fraction is a crucial component in calculations involving solutions. It is a way to express the concentration of a component in a mixture. The mole fraction is defined as the ratio of the moles of one component to the total moles of all components in the solution.

In the exercise provided, the mole fraction of ethanol is calculated to determine how much it contributes to the vapor pressure. We find this by dividing the moles of ethanol by the total moles in the solution. This fraction is essential when applying Raoult's Law, as it directly influences the partial vapor pressure of the ethanol in the solution.

Understanding mole fractions helps us see how the quantities of different components interact within a mixture, influencing properties such as vapor pressure.

Solution Chemistry

Solution chemistry focuses on the behavior and interactions of solutes and solvents in a solution. A solution is a homogeneous mixture made of two or more substances. In our exercise, ethanol acts as the solvent and glycerin as a solute.

The properties of a solution, such as boiling point, freezing point, and vapor pressure, are influenced by the nature and quantity of the solute added. For instance, adding glycerin, a nonvolatile solute, to ethanol lowers the solution's vapor pressure, which aligns with what we observe using Raoult's Law.

This understanding allows chemists to predict how adding different solutes at varying concentrations will change the characteristics of the solution, which is essential in fields ranging from creating everyday products to complex pharmaceutical formulations.