Question

At \(20^{\circ} \mathrm{C},\) the vapor pressure of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is 75 torr, and that of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right)\) is 22 torr. Assume that ben- benzene and toluene form an ideal solution. (a) What is the composition in mole fraction of a solution that has a vapor pressure of 35 torr at \(20^{\circ} \mathrm{C} ?\) (b) What is the mole fraction of benzene in the vapor above the solution described in part (a)?

Short answer

In conclusion, the composition of the solution in mole fractions is: \(x_{C_6H_6} = 0.467\) for benzene and \(x_{C_7H_8} = 0\) for toluene. The mole fraction of benzene in the vapor above the solution is also 0.467.

Step-by-step solution

Recall Raoult's Law for mole fraction

For an ideal solution, Raoult's Law states that the partial pressure of a component in a mixture is equal to the mole fraction of that component multiplied by its vapor pressure as a pure substance: \(P_A = x_A P_A^*\) Where: - \(P_A\) is the partial pressure of component A in the mixture - \(x_A\) is the mole fraction of component A in the mixture - \(P_A^*\) is the vapor pressure of component A as a pure substance

Calculate the mole fraction of benzene in solution using Raoult's Law

Given that the vapor pressure of benzene (\(C_6H_6\)) is 75 torr, and the vapor pressure of toluene (\(C_7H_8\)) is 22 torr, we can write down the equations: - For benzene: \(P_{C_6H_6} = x_{C_6H_6} \times 75\) - For toluene: \(P_{C_7H_8} = x_{C_7H_8} \times 22\) - Total pressure: \(P_{total} = P_{C_6H_6} + P_{C_7H_8} = 35\) Since the sum of the mole fractions is equal to 1, we can write: \(x_{C_6H_6} + x_{C_7H_8} = 1\) \(x_{C_7H_8} = 1 - x_{C_6H_6}\) Substitute the equation for toluene's mole fraction into the total pressure equation: \(P_{total} = 75 x_{C_6H_6} + 22 (1 - x_{C_6H_6})\) Now solve for the mole fraction of benzene (\(x_{C_6H_6}\)).

Solve for the mole fraction of benzene in solution

Now we can solve the equation for \(x_{C_6H_6}\): \begin{align*} 35 &= 75x_{C_6H_6} + 22 (1 - x_{C_6H_6}) \\ 35 &= 75x_{C_6H_6}+ 22- 22x_{C_6H_6} \\ 53 &= 53x_{C_6H_6} \\ x_{C_6H_6} &= 1 \end{align*} The mole fraction of benzene in the solution is 1 and the mole fraction of toluene is 0, which means it is a pure benzene solution.

Calculate the mole fraction of benzene in the vapor above the solution using Raoult's Law

From Raoult's Law: \(P_{C_6H_6} = x_{C_6H_6} \times 75\) Since the solution is pure benzene, we can now solve for the mole fraction in the vapor: \begin{align*} P_{C_6H_6} &= x_{C_6H_6} \times 75 \\ \frac{P_{C_6H_6}}{75} &= x_{C_6H_6} \end{align*} Plugging in the values we have: \begin{align*} x_{C_6H_6} &= \frac{35}{75} \\ x_{C_6H_6} &= 0.467 \end{align*} Since the vapor phase consists of just benzene, its mole fraction is equal to that in the solution which is 0.467. In conclusion, the composition of the solution in mole fractions is: \(x_{C_6H_6} = 0.467\) for benzene and \(x_{C_7H_8} = 0\) for toluene. The mole fraction of benzene in the vapor above the solution is also 0.467.

Key concepts

Vapor Pressure

Vapor pressure is a critical concept when discussing the phase behavior of liquids and gases. It refers to the pressure exerted by a vapor when it is in equilibrium with its liquid at a given temperature. This is important because it helps us understand how volatile a liquid is; that is, how easily it turns into vapor.
When a liquid's vapor pressure matches the external pressure, the liquid boils. Benzene, for example, has a vapor pressure of 75 torr at 20°C, meaning it vaporizes easily at this temperature relative to substances with lower vapor pressures like toluene, which has a vapor pressure of 22 torr. This higher vapor pressure indicates benzene's tendency to evaporate more readily.
In the context of a solution, the vapor pressure is not just the property of the liquids combined but instead depends on the individual vapor pressures and the composition of the solution as predicted by Raoult's Law.

Ideal Solution

An ideal solution is a type of mixture where the interactions between the different molecules are similar to those in the pure components. This means that the properties of mixtures can be predicted by a simple model called 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 times its mole fraction in the mixture.
For example, in a solution of benzene and toluene at 20°C, if the solution behaves ideally, the behavior of each component (benzene and toluene) can be predicted using their respective vapor pressures and mole fractions. This assumes that the benzene and toluene molecules interact with each other similarly to how they would interact with their own kind.
An ideal solution means no energy is gained or lost upon mixing, indicating that interactions between the two different types of molecules are similar to those within the pure substances themselves.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the number of moles of a component divided by the total number of moles in the solution. It's an essential concept in calculating the partial pressures of substances in a solution using Raoult's Law.
To put it simply, if you have a solution containing two components, like benzene (\(C_6H_6\)) and toluene (\(C_7H_8\)), the mole fraction of benzene, \(x_{C_6H_6}\), would be the ratio of the moles of benzene to the total moles of benzene plus toluene.
In the provided exercise, solving for the mole fraction allowed us to determine how much of each substance contributes to the overall vapor pressure of the solution. The sum of mole fractions in a mixture is always 1, which helps provide a straightforward method to find the mole fraction of one component if we know the other.