Problem 36 - General Chemistry: Principles and Modern Applications

Question

A 7.53 I. sample of \(\mathrm{N}_{2}(\mathrm{g})\) at \(742 \mathrm{mmHg}\) and \(45.0^{\circ} \mathrm{C}\) is bubbled through \(\mathrm{CCl}_{4}(1)\) at \(45.0^{\circ} \mathrm{C} .\) Assuming the gas becomes saturated with \(\mathrm{CCl}_{4}(\mathrm{g}),\) what is the volume of the resulting gaseous mixture, if the total pressure remains at \(742 \mathrm{mm} \mathrm{Hg}\) and the temperature remains at \(45^{\circ} \mathrm{C} ?\) The vapor pressure of \(\mathrm{CCl}_{4}\) at \(45^{\circ} \mathrm{C}\) is \(261 \mathrm{mmHg}\)

Step-by-Step Solution

Answer

The volume of the resulting gaseous mixture is 62.8 L

Step 1: Evaluate the Partial Pressure of Nitrogen

First, compute the partial pressure of the nitrogen gas. Because \(N_{2}\) and \(CCl_{4}\) are in equilibrium, the partial pressure of \(N_{2(g)}\) is also 742 mmHg. This is obtained by subtracting the vapor pressure of the \(CCl_{4}(g)\) from the total pressure. \(P_{N2} = P_{total} - P_{CCl4} = 742 mmHg - 261 mmHg = 481 mmHg\)

Step 2: Convert Temperatures to Kelvin

Now convert the temperature from Celsius to Kelvin. Kelvin is the SI unit for temperature and is necessary to use the ideal gas law equation. \(T = 45^{\circ}C + 273 = 318 K\)

Step 3: Calculate the Volume of Nitrogen Gas

Utilize the ideal gas law to calculate the original volume of nitrogen gas. Rearrange the formula \(P_{N2}V=N_{N2}RT\) for volume: \(V_{N2} = (N_{N2}RT) / P_{N2}\). Using \(R = 0.0821 L·atm/mol·K\), convert the pressure of \(N_{2(g)}\) to atmospheres (1 atm = 760 mmHg), which gives us \(P_{N2} = 481 mmHg * (1 atm / 760 mmHg) = 0.633 atm\). The number of moles of nitrogen, \(N_{N2}\), is equal to 7.53I. So, volume of \(N_{2(g)}\) becomes \(V_{N2} = (7.53 mol * 0.0821 L·atm/mol·K * 318K) / 0.633 atm = 31.4 L\)

Step 4: Calculate Volume of Carbon Tetrachloride Gas

The volume of \(CCl_{4}(g)\) is the same as that of \(N_{2(g)}\) because they share the same conditions of pressure and temperature. Hence, \(V_{CCl4} = V_{N2} = 31.4 L\)

Step 5: Calculate Total Volume of the Resulting Mixture

The total volume of the mixture, \(V_{total}\), can be found by adding the individual volumes of the gases. \(V_{total} = V_{N2} + V_{CCl4} = 62.8 L\)

Key Concepts

Partial Pressure

Understanding partial pressure is essential when dealing with gas mixtures. It refers to the pressure that a gas would exert if it alone occupied the entire volume of the mixture at a constant temperature. In the given exercise, we start by computing the partial pressure of nitrogen gas

• The total pressure of the mixture is given as 742 mmHg.
• Since the nitrogen gas ( $N_2$ ) mixes with carbon tetrachloride ( $CCl_4$ ), we must subtract the vapor pressure of $CCl_4$ , which is 261 mmHg, from the total pressure.
• This calculation leads to a partial pressure for $N_2 (g)$ of 481 mmHg.

It's this understanding of partial pressures that allows us to isolate the contribution of each gas within the mixture.

Vapor Pressure

Vapor pressure is the pressure exerted by the vapor when it is in thermodynamic equilibrium with its liquid at a given temperature. This concept is crucial when gases become saturated with vapor. In our exercise:

• We know that the temperature is maintained at 45°C.
• The vapor pressure of carbon tetrachloride ( $CCl_4$ ) is given as 261 mmHg at this temperature.
• This information allows us to define the pressure contribution of $CCl_4 (g)$ in the mixture.

The vapor pressure informs us how much carbon tetrachloride gas is mixed with the nitrogen, affecting overall calculations.

Volume Calculation

Volume calculation involves the application of the ideal gas law, which is pivotal for understanding the behavior of gases under various conditions. The ideal gas law is expressed as $PV = nRT$ where:

• $P$ is the pressure,
• $V$ is the volume,
• $n$ is the number of moles,
• $R$ is the ideal gas constant (0.0821 L·atm/mol·K),
• $T$ is the temperature in Kelvin.

In the exercise, we determined:

• The temperature is converted to Kelvin: 318 K.
• The pressure of nitrogen is converted to atm: approximately 0.633 atm.
• The nitrogen gas volume calculated using its partial pressure becomes 31.4 L.

Moreover, since $N_2$ and $CCl_4$ are at equilibrium, their volumes are additive, leading to a total volume calculation of 62.8 L for the mixture. Volume calculation is crucial to quantify the outcome when gases mix.

Nitrogen Gas

Nitrogen gas is a colorless, odorless gas that is abundant in Earth's atmosphere. When dealing with gas mixtures, it typically acts inertly due to its non-reactive nature under standard conditions. In the context of the exercise:

• Nitrogen is represented as $N_2$ due to its diatomic molecular form.
• It begins with a sample size of 7.53 L, which is processed through the calculations.
• Its partial pressure helps determine its behavior in the mixture with $CCl_4$ .

Being familiar with nitrogen's properties and handling is useful when analyzing its behavior and calculating changes in ideal gas scenarios.