Question

Cigars are best stored in a "humidor" at \(18^{\circ} \mathrm{C}\) and \(55 \%\) relative humidity. This means the pressure of water vapor should be \(55 \%\) of the vapor pressure of pure water at the same temperature. The proper humidity can be maintained by placing a solution of glycerol \(\left[\mathrm{C}_{3} \mathrm{H}_{5}(\mathrm{OH})_{3}\right]\) and water in the humidor. Calculate the percent by mass of glycerol that will lower the vapor pressure of water to the desired value. (The vapor pressure of glycerol is zero.)

Short answer

The percent by mass of glycerol needed is approximately 13.8%.

Step-by-step solution

Understand the Problem

We need to calculate the percent by mass of glycerol in a solution that reduces the vapor pressure of water to 55% of its pure state at 18°C. Glycerol does not contribute to the vapor pressure. Thus, we use Raoult's law to determine the amount of glycerol needed.

Apply Raoult's Law

Raoult's Law states that the partial vapor pressure of each component in a solution is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution. The equation is: \[ P_{ ext{solution}} = X_{ ext{water}} imes P_{ ext{pure water}} \] where \(X_{\text{water}}\) is the mole fraction of water.

Calculate Mole Fraction of Water

Given that the pressure of water vapor is 55% of its vapor pressure in pure form, we write: \[ X_{\text{water}} = 0.55 \] This tells us that the mole fraction of water in the solution must be 0.55.

Relate Mole Fraction to Mass Percent

The mole fraction \(X_{\text{water}}\) is calculated using \[ X_{\text{water}} = \frac{n_{\text{water}}}{n_{\text{water}} + n_{\text{glycerol}}} \] We solve for the amount of each component in terms of moles, and then convert to mass using the molecular weights.

Set-Up Equations Using Molecular Weights

The molecular weight of water (H₂O) is approximately 18 g/mol, and glycerol (C₃H₅(OH)₃) is approximately 92 g/mol. Let \(m_{\text{glycerol}}\) and \(m_{\text{water}}\) denote masses. Use \[ n_{\text{water}} = \frac{m_{\text{water}}}{18} \] and \[ n_{\text{glycerol}} = \frac{m_{\text{glycerol}}}{92} \] in the mole fraction equation from Step 4.

Solve for Mass Percent of Glycerol

Rearrange the mole fraction equation for \( X_{\text{water}} = 0.55 \): \[ 0.55 = \frac{\frac{m_{\text{water}}}{18}}{\frac{m_{\text{water}}}{18} + \frac{m_{\text{glycerol}}}{92}} \] Solving this gives:\[ \frac{m_{\text{glycerol}}}{m_{\text{water}}} = \frac{0.45 \cdot 18}{0.55 \cdot 92} \] Calculate the values and convert to a percent by mass. Let total mass be \(m_{\text{total}} = m_{\text{water}} + m_{\text{glycerol}}\).

Final Calculation

Using the ratio from Step 6:\[ \frac{m_{\text{glycerol}}}{m_{\text{water}}} = \frac{8.1}{50.6} \approx 0.16 \]Convert to mass percent glycerol:\[ \% \text{Glycerol} = \frac{m_{\text{glycerol}}}{m_{\text{total}}} \times 100 \approx \frac{0.16 \cdot m_{\text{water}}}{1.16 \cdot m_{\text{water}}} \times 100 = \frac{0.16}{1.16} \times 100 \approx 13.8\% \]

Key concepts

Vapor Pressure

Vapor pressure is a key concept when dealing with solutions and their components. It refers to the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. In simpler terms, it's the measure of a liquid's tendency to evaporate. For instance, in our problem, to maintain the correct humidity for cigars stored in a humidor, the vapor pressure of water needs to be carefully controlled. The vapor pressure of a pure substance like water will differ when mixed with another substance, as it depends on the interactions between molecules. When we introduce glycerol into the solution with water, it alters the water's vapor pressure due to these interactions. In this case, we seek to reduce the vapor pressure of water to 55% of its pure state by adding glycerol, which itself doesn't exert any vapor pressure.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It's defined as the ratio of the number of moles of a component to the total number of moles of all components in the mixture. In mathematical terms, the mole fraction of water (\(X_{water}\)) in our solution is crucial for applying Raoult's Law. Raoult's Law connects the mole fraction of a component in a solution to its vapor pressure, allowing for the calculation of how a solute, like glycerol, impacts the vapor pressure of water. Here, we need the mole fraction of water to be 0.55 to achieve the desired vapor pressure in the humidor, meaning 55% of each mole in the solution should be water.

Glycerol Solution

Glycerol, a viscous liquid frequently used for its moisture-retaining properties, is central to this exercise. When mixed with water, glycerol effectively lowers the overall vapor pressure of the system. Unlike water, glycerol has an effectively zero vapor pressure at the relevant conditions. Thus, by adding glycerol to water, we reduce the vapor pressure to the desired level for cigar preservation. The absence of vapor pressure in glycerol means it's solely the dilution effect on the water that achieves the desired humidity level. Understanding glycerol's role in this scenario helps us strategically achieve the required conditions inside the humidor.

Percent by Mass

The concept of percent by mass is crucial in determining the concentration of glycerol needed. Percent by mass, sometimes called mass percent, is defined as the mass of a specific component divided by the total mass of the solution, multiplied by 100 to express it as a percentage. In this exercise, you're calculating what percent of the total solution mass consists of glycerol, necessary to lower the water's vapor pressure to 55% of its pure state. By using molecular weights and the mole fraction equation, you can convert the ratio of moles into a mass percentage. This assists in comprehending and controlling the composition of the glycerol-water mixture to effectively manage the vapor pressure in a practical setting.