Question

Dry nitrogen gas is bubbled through liquid benzene (C. ${\mathrm{H}}_6)$ at ${20.0}^{\circ }\mathrm{C}.$ From 100.0 $\mathrm{L}$ of the gaseous mixture of nitrogen and benzene, 24.7 g benzene is condensed by passing the mixture through a trap at a temperature where nitrogen is gaseous and the vapor pressure of benzene is negligible. What is the vapor pressure of benzene at ${20.0}^{\circ }\mathrm{C}$ ?

Short answer

The vapor pressure of benzene at 20°C is 0.0768 atm.

Step-by-step solution

Calculate the number of moles of condensed benzene

To calculate the number of moles of condensed benzene, we will use its molar mass, which is 78.11 g/mol. n_Benzene = mass of benzene / molar mass n_Benzene = $\frac{24.7g}{78.11g/mol}$ n_Benzene = 0.316 mol

Compute the mole fraction of benzene

To compute the mole fraction of benzene in the gaseous mixture, we can use the Ideal Gas Law: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. Here, we know that V = 100 L T = 20°C + 273.15 = 293.15 K R = 0.0821 L atm/mol K (Universal Gas Constant) n_Benzene = 0.316 mol Now, rearrange the formula to get pressure (P): P_Benzene = $\frac{n_{B}enzene\ast R\ast T}{V}$ P_Benzene = $\frac{0.316mol\ast 0.0821Latm/molK\ast 293.15K}{100L}$ P_Benzene = 0.0768 atm

Calculate the vapor pressure of Benzene

Now that we have found the partial pressure of benzene in the gaseous mixture, we can calculate the vapor pressure of benzene at 20°C. Since all the benzene has been condensed from the gaseous mixture, the partial pressure of benzene in the mixture equals its vapor pressure. Vapor pressure of benzene = P_Benzene Vapor pressure of benzene = 0.0768 atm Thus, the vapor pressure of benzene at 20°C is 0.0768 atm.

Key concepts

ideal gas law

The Ideal Gas Law is a useful equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as $PV=nRT$, where $P$ stands for pressure, $V$ for volume, $n$ for the number of moles, $R$ is the ideal gas constant, and $T$ represents the temperature in Kelvin. This equation helps us understand how gases behave under different conditions.

In the given exercise, nitrogen gas and benzene are considered ideal gases. We used the Ideal Gas Law to calculate the partial pressure of benzene when 24.7 g of it is condensed and trapped. By knowing the molar mass of benzene, we can determine the moles of benzene. From there, we apply the Ideal Gas Law to find the pressure contributions from benzene in the gaseous mixture before condensation.

The ideal gas constant $R$ is typically given as $0.0821\text{ L atm/mol K}$, and this plays a key role in solving the problem, as seen when calculating $P_{B}enzene=\frac{n_{B}enzene\times R\times T}{V}$. This relates temperature, pressure, and volume to quantify the amount and pressure exerted by benzene vapor in the mixture.

mole fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. 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. Mathematically, it is expressed as ${\chi }_i=\frac{n_i}{n_{total}}$.

In our exercise, as we wanted to find the vapor pressure of benzene, we were not directly calculating the mole fraction but rather focused on deriving the partial pressure using another method. However, understanding mole fractions can be beneficial, as they often help in calculating pressures of individual components in gaseous systems via Dalton's Law of Partial Pressures.

When applying the Ideal Gas Law, mole fractions can help to understand how much each species in a mixture contributes to the overall behavior of the gas, especially when dealing with vapor-liquid equilibria and systems at constant temperature and pressure. While the solution does not strictly require calculating the mole fraction, it's worth understanding that the concentration of gaseous benzene relative to nitrogen would be impactful in a scenario comparing both gases or looking for the solution in a vapor-liquid equilibrium context.

partial pressure

Partial pressure is a concept that refers to the pressure exerted by a single gas in a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. Mathematically, this is expressed as each gas's partial pressure fulfilling $P_i={\chi }_i\times P_{total}$.

In the context of the exercise, we found the partial pressure of benzene in a nitrogen and benzene gas mixture before the benzenes condensed. It required calculating the actual pressure benzene exerts without being affected by the other gas present.

By applying Dalton's Law and recognizing that, given the exercise condition, the partial pressure equates to vapor pressure under the set temperature, we conclude the vapor pressure is 0.0768 atm. The measured pressure corresponds effectively to its partial pressure since the system's setup has minimized benzene exposure in the gas phase. This approach effectively highlights how single gas pressures can help in determining properties specific to one component, like the benzene in question.