Problem 96 - Chemistry The Central Science

Question

The vapor pressure of a volatile liquid can be determined by slowly bubbling a known volume of gas through it at a known temperature and pressure. In an experiment, 5.00 L of \(\mathrm{N}_{2}\) gas is passed through 7.2146 \(\mathrm{g}\) of liquid benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) at \(26.0^{\circ} \mathrm{C} .\) The liquid remaining after the experiment weighs 5.1493 g. Assuming that the gas becomes saturated with benzene vapor and that the total gas volume and temperature remain constant, what is the vapor pressure of the benzene in torr?

Step-by-Step Solution

Answer

The vapor pressure of benzene is approximately 150.8 torr.

Step 1: Calculate the mass of benzene vaporized

Firstly, we need to find how much benzene was vaporized in the experiment. We can do this by subtracting the final mass of benzene from the initial mass. Mass of vaporized benzene = Initial mass - Final mass Mass of vaporized benzene = 7.2146 g - 5.1493 g = 2.0653 g

Step 2: Calculate the moles of vaporized benzene

Now convert the mass of vaporized benzene to moles using the molar mass of benzene, which is 78.11 g/mol for C6H6. Moles of vaporized benzene = (Mass of vaporized benzene)/(Molar mass of benzene) Moles of vaporized benzene = (2.0653 g)/(78.11 g/mol) = 0.02645 mol

Step 3: Calculate the initial pressure of nitrogen gas

Use the ideal gas law equation (PV = nRT) to find the initial pressure of the nitrogen gas. We are given the volume of gas (5.00 L) and temperature (\(26^{\circ} \mathrm{C}\); converted to Kelvin as \(26 + 273.15 = 299.15K\)). We know that R, the ideal gas constant, is 0.0821 L atm/mol K. First, we need to find the moles of nitrogen gas. To do this, we can use nitrogen's molar mass (28.02 g/mol) and the fact that nitrogen makes up 79% of the atmosphere by volume (assuming other gases do not contribute to the pressure). Then, we can find the atmospheric pressure P in atm by using the fact that the atmospheric pressure is 760 torr. Because the nitrogen gas is 79% by volume, we can calculate the initial pressure of nitrogen gas as follows. Atmospheric pressure = 760 torr = 1 atm (approximately) Initial pressure of nitrogen = 0.79 * Atmospheric pressure Initial pressure of nitrogen = 0.79 * 1 atm = 0.79 atm Now we can find the moles of nitrogen gas (n). PV = nRT n = PV/(RT) n = (0.79 atm * 5.00 L)/ (0.0821 L atm/mol K * 299.15 K) = 0.1617 mol

Step 4: Calculate the total moles of gas

Next, we need to find the total moles of gas present after the nitrogen gas is saturated with benzene vapor. We can do this by adding the moles of nitrogen gas and the moles of vaporized benzene. Total moles of gas = Moles of nitrogen gas + Moles of vaporized benzene Total moles of gas = 0.1617 mol + 0.02645 mol = 0.18815 mol

Step 5: Find the new pressure

Now we need to find the new pressure experienced by the gas mixture using the ideal gas law equation. PV = nRT P = nRT/V P = (0.18815 mol * 0.0821 L atm/mol K * 299.15 K)/(5.00 L) = 0.9887 atm

Step 6: Calculate the partial pressure of benzene vapor

Finally, let's find the vapor pressure of benzene, which can be determined by finding the partial pressure of benzene in the gas mixture. We can do this by subtracting the initial pressure of nitrogen gas from the new pressure. Partial pressure of benzene vapor = New pressure - Initial pressure of nitrogen Partial pressure of benzene vapor = 0.9887 atm - 0.79 atm = 0.1987 atm Now we need to convert this pressure to torr. Vapor pressure of benzene = 0.1987 atm * (760 torr/1 atm) = 150.8 torr The vapor pressure of benzene is approximately 150.8 torr.

Key Concepts

Ideal Gas Law

The ideal gas law is a useful equation for understanding the behavior of gases under various conditions. It's given by the formula \( PV = nRT \), where \( P \) represents the pressure of the gas, \( V \) is its volume, \( n \) stands for the number of moles of gas, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is the temperature in Kelvin.

By using this equation, we can calculate different properties of a gas when the others are known. For example, if we know the pressure, volume, and temperature of a gas, we can find the number of moles. Similarly, knowing the moles, volume, and temperature allows us to solve for pressure.

• Temperature Conversion: Always remember to convert the temperature to Kelvin by adding 273.15 to the Celsius temperature, as the ideal gas law uses Kelvin.
• Unit Consistency: Ensure all units match the ideal gas constant's units for consistent results.

In our exercise, we used the ideal gas law to calculate the initial pressure of the nitrogen and later determine the pressure of the gas mixture after the benzene had vaporized.

Partial Pressure

Partial pressure refers to the pressure that a single component in a mixture of gases would exert if it alone occupied the entire volume of the original mixture at the same temperature. It’s a crucial concept when dealing with gas mixtures, as it allows us to understand how each gas contributes to the total pressure.

In Dalton’s law of partial pressures, the total pressure in a gas mixture is the sum of the partial pressures of each individual gas. Therefore, the partial pressure of gas can be calculated by subtracting the partial pressures of the other gases present.

• Dalton's Law: \( P_{total} = P_1 + P_2 + ext{...} + P_n \)
• Adjustments in Calculations: For the benzene vapor, the partial pressure is found by subtracting the initial nitrogen pressure from the total pressure after vaporization.

This principle was used in the exercise to find the vapor pressure of benzene by determining its partial pressure in the mixture.

Benzene Vaporization

Benzene vaporization is an example of a phase change, where benzene translates from a liquid to a gas. When a liquid vaporizes, molecules gain enough energy to break free from the liquid's surface to become vapor. This process is influenced by factors like temperature and the nature of the liquid itself.

In our context, we pass nitrogen gas through benzene. As nitrogen bubbles through, it carries away benzene molecules, and this results in benzene vaporization until the gas is saturated.

• Mass Difference Calculation: By measuring the mass before and after vaporization, we can determine how much benzene vaporized.
• Molar Calculations: We then convert this mass into moles to understand the amount of benzene that entered the gaseous phase.

The key to solving the initial exercise problem was calculating the vapor pressure of benzene, achieved by observing its partial pressure after a known portion vaporized.