Question

A solution is prepared by mixing 1.000 mole of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) and 3.18 moles of propanol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{OH}\right)\) What is the composition of the vapor (in mole fractions) at \(40^{\circ} \mathrm{C} ? \mathrm{At} 40^{\circ} \mathrm{C},\) the vapor pressure of pure methanol is 303 torr, and the vapor pressure of pure propanol is 44.6 torr.

Short answer

The mole fractions of methanol and propanol in the solution are \(x_{CH_3OH} = \frac{1.000}{4.18}\) and \(x_{CH_3CH_2CH_2OH} = \frac{3.18}{4.18}\), respectively. After calculating the partial pressures using Raoult's Law, and the total vapor pressure, the mole fractions of methanol and propanol in the vapor phase can be found by dividing their respective partial pressures by the total vapor pressure. The mole fractions of methanol and propanol in the vapor phase represent the composition of the vapor at 40°C.

Step-by-step solution

Understand Raoult's Law

Raoult's Law states that the partial pressure of a component in a solution is equal to its mole fraction multiplied by its vapor pressure in the pure state at the same temperature. Mathematically, it can be represented as: \(P_i = x_i P_i^*\) Where: - \(P_i\) is the partial pressure of component i in the solution - \(x_i\) is the mole fraction of component i in the solution - \(P_i^*\) is the vapor pressure of pure component i at the same temperature

Calculate the mole fraction of each component in the solution

We are given 1.000 mole of methanol and 3.18 moles of propanol are mixed together. First, let's find the mole fractions of each component in the solution. Total moles = moles of methanol + moles of propanol Total moles = 1.000 + 3.18 = 4.18 Mole fraction of methanol, \(x_{CH_3OH}\) = \(\frac{1.000}{4.18}\) Mole fraction of propanol, \(x_{CH_3CH_2CH_2OH}\) = \(\frac{3.18}{4.18}\)

Calculate the partial pressure of each component in the solution

Using the mole fractions of methanol and propanol, we can calculate their partial pressures using Raoult's Law: For methanol: \(P_{CH_3OH} = x_{CH_3OH} P_{CH_3OH}^*\) \(P_{CH_3OH} = (\frac{1.000}{4.18})(303\, \mathrm{torr})\) For propanol: \(P_{CH_3CH_2CH_2OH} = x_{CH_3CH_2CH_2OH} P_{CH_3CH_2CH_2OH}^*\) \(P_{CH_3CH_2CH_2OH} = (\frac{3.18}{4.18})(44.6\, \mathrm{torr})\)

Calculate the total vapor pressure and mole fractions in the vapor phase

Now, let's calculate the total vapor pressure of the solution by adding the partial pressures of methanol and propanol: Total vapor pressure = \(P_{CH_3OH} + P_{CH_3CH_2CH_2OH}\) Finally, we can calculate mole fractions in the vapor phase: Mole fraction of methanol in the vapor phase = \(\frac{P_{CH_3OH}}{Total\, vapor\, pressure}\) Mole fraction of propanol in the vapor phase = \(\frac{P_{CH_3CH_2CH_2OH}}{Total\, vapor\, pressure}\) These mole fractions represent the composition of the vapor at 40°C. Make sure to calculate each value using the expressions derived above to get the final numerical results.

Key concepts

Vapor Pressure

Vapor pressure is a critical concept in understanding how substances evaporate and form vapors in equilibrium with their liquid phase. At any given temperature, the vapor pressure of a liquid is the pressure exerted by its vapor when evaporation and condensation occur simultaneously. It reveals how volatile a liquid is—meaning how easily it converts to vapor.

In the exercise, we consider the vapor pressures of pure liquids: methanol at 303 torr and propanol at 44.6 torr, both at 40°C. These values indicate that methanol is more volatile than propanol as it has a higher vapor pressure at the same temperature. Understanding vapor pressure helps us use Raoult's Law to calculate the behavior of solutions, especially in calculating the mole fractions in vapor composition.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the moles of one component to the total moles of all components in the solution. It gives us a proportion, helping to understand the composition without units.

In our example, to find the mole fraction of methanol and propanol in the solution, we divide the number of moles of each component by the total moles (1.000 for methanol and 3.18 for propanol). For methanol, the mole fraction is calculated as follows:

• Mole fraction of methanol = \( \frac{1.000}{4.18} \).
• Mole fraction of propanol = \( \frac{3.18}{4.18} \).

Each mole fraction provides insight into the proportion of each component, vital for tasks like identifying the behavior of the solution under different conditions.

Partial Pressure

Partial pressure refers to the pressure that a single component of a gas mixture would exert if it occupied the whole volume alone at the same temperature. Raoult's Law relies heavily on this concept when applied to liquid solutions.

Using Raoult's Law, we calculate each component's partial pressure in the mixture by multiplying its mole fraction by the vapor pressure of the pure component. For methanol, the partial pressure is calculated as \( P_{CH_3OH} = \left(\frac{1.000}{4.18}\right) \times 303 \text{ torr} \). Similarly, propanol's partial pressure is \( P_{CH_3CH_2CH_2OH} = \left(\frac{3.18}{4.18}\right) \times 44.6 \text{ torr} \).

The sum of these individual partial pressures gives the total pressure exerted by the solution's vapor, essential for finding the mole fractions in the vapor phase.

Methanol

Methanol, known chemically as \( \text{CH}_3\text{OH} \), is a simple alcohol often found in chemical laboratories and industrial processes. With a higher vapor pressure of 303 torr at 40°C compared to propanol, methanol is more prone to evaporating.

In this exercise, methanol's behavior in forming part of the vapor is critical. The exercise shows us that methanol will contribute a higher portion to the pressure and, thus, hold a larger mole fraction in vapor due to its high volatility. Such characteristics make methanol a significant player in determining vapor compositions alongside other components like propanol.

Propanol

Propanol, represented as \( \text{CH}_3\text{CH}_2\text{CH}_2\text{OH} \), is another alcohol frequently used in chemical applications, albeit less volatile than methanol. It has a much lower vapor pressure of 44.6 torr at 40°C.

In our solution, even though there is a higher initial mole content of propanol (3.18 moles), its relative contribution to the vapor is diminished due to its lower volatility compared to methanol. This indicates that while the liquid solution contains more moles of propanol initially, methanol's greater vapor pressure gives it a more substantial presence in the vapor phase composition than its liquid phase concentration might suggest. Understanding such a role highlights the importance of each component's properties in predicting real-world liquid and vapor behaviors.