Question

A 2.00 L container is filled with \(\operatorname{Ar}(g)\) at 752 mm Hg and \(35^{\circ} \mathrm{C} .\) A \(0.728 \mathrm{g}\) sample of \(\mathrm{C}_{6} \mathrm{H}_{6}\) vapor is then added. (a) What is the total pressure in the container? (b) What is the partial pressure of \(\mathrm{Ar}\) and of \(\mathrm{C}_{6} \mathrm{H}_{6} ?\)

Short answer

(a) The total pressure in the vessel is the sum of the partial pressures of $\mathrm{Ar}(g)$ and $\mathrm{C}_{6} \mathrm{H}_{6}$. (b) The partial pressure of each gas in the mixture can be determined using their respective mole fractions along with the total pressure.

Step-by-step solution

Convert the given values to appropriate units

The temperature will be converted from Celsius to Kelvin by using the formula \[T(K) = T(°C) + 273.15\]. The volume is already in litres and the initial pressure of Ar(g) is in mm Hg. Convert this to atmospheres by using \(1 \, \text{atm} = 760 \, \text{mm Hg}\).

Applying the ideal gas law to Ar(g)

The ideal gas law, \(PV = nRT\), is used to calculate the number of moles of Ar(g), since the volume, pressure and temperature are known. Here, R is the ideal gas constant and its value is \(0.0821 \, \text{L atm/mol K}\).

Calculate the number of moles of C6H6

The number of moles of \(C_6H_6\) can be calculated by using the molar mass of \(C_6H_6\), which is approximately 78.11 g/mol. Here, \(n = \text{mass} / \text{molar mass}\).

Calculate total pressure using Dalton's Law

Dalton's Law states that the total pressure exerted by a mixture of gases is the sum of the partial pressures of each individual gas. The total pressure is given by the equation \(P_{\text{total}} = n_{\text{total}}RT / V\), where \(n_{\text{total}}\) is the sum of the number of moles of Ar(g) and \(C_6H_6\).

Calculate the partial pressure for Ar(g) and C6H6

The partial pressure of each gas can be found using the formula \(P_{i} = X_{i}P_{\text{total}}\), where \(X_{i}\) is the mole fraction of the gas in the mixture and is given by the number of moles of the gas divided by the total number of moles in the mixture.

Key concepts

Dalton's Law of Partial Pressures

When dealing with mixtures of gases, Dalton's Law of Partial Pressures becomes very handy. It states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas in the container. This means if you have a container with multiple gases, you can simply add up the pressures that each gas would exert if it were the only one present in the container.
To calculate the total pressure using Dalton's Law, you first need to know the individual pressures each gas exerts, known as partial pressures. You can get the partial pressure of a gas if you know the number of moles of that gas and apply the ideal gas law, adjusting for the conditions of the container. Once you have that, summing them gives you the total pressure.
In practice, like in the original exercise, you'll often first find the moles of each gas, use the ideal gas law to find individual pressures, and then sum them according to Dalton's law. This principle is key in understanding how gases behave in a mixture.

Mole Fraction

The concept of mole fraction is crucial when calculating partial pressures. The mole fraction is simply the ratio of the number of moles of a particular gas to the total number of moles of all gases present in the mixture.

• For example, if there are 3 moles of gas A and 2 moles of gas B, the mole fraction of gas A ( X_A ) is 3/(3+2) = 0.6, and for gas B ( X_B ) is 2/(3+2) = 0.4.

The mole fraction has no units because it's a ratio. It's a simple yet powerful component when you're calculating partial pressures. Once you know the mole fraction, you can find the partial pressure of that gas by multiplying its mole fraction by the total pressure of the mixture:
\( P_{i} = X_{i} \times P_{\text{total}} \).
Using the mole fraction streamlines the process of finding each gas's contribution to the overall pressure in a gas mixture setting.

Converting Units in Gas Calculations

Being adept in converting units is essential for successful gas calculations, as gases can be measured using various units. The most common unit of pressure is atmospheres (atm), but pressure can also be in millimeters of mercury (mmHg), Pascal (Pa), or others.

• Each unit has a conversion factor: 1 atm equals 760 mm Hg, so converting between the two requires division or multiplication by 760.
• Temperature in gas laws must use the Kelvin scale, so conversion from Celsius is often necessary: \( T(K) = T(°C) + 273.15 \).

Volume is usually measured in liters (L) in gas law equations like the ideal gas law. Paying attention to these conversions is crucial because incorrect units can lead to incorrect results.
In the provided exercise, the pressure was initially given in mm Hg and needed to be converted to atm for the calculations, and the temperature in Celsius had to be converted to Kelvin. Mastering these conversions helps streamline calculations and ensure accuracy.