Question

A \(200 \mathrm{~mL}\) flask having oxygen at \(220 \mathrm{~mm}\) and a \(300 \mathrm{~mL}\) flask having nitrogen at \(100 \mathrm{~mm}\) are connected in such a way that \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) may combine in their volumes, if temperature is kept constant. Find the total pressure of the gaseous mixture. (a) \(158 \mathrm{~mm}\) (b) \(138 \mathrm{~mm}\) (c) \(148 \mathrm{~mm}\) (d) \(168 \mathrm{~mm}\)

Short answer

The total pressure is 148 mmHg (option c).

Step-by-step solution

Calculate Partial Pressures Contributions

Initially, calculate the pressure contributions of each gas based on their initial volumes and pressures. The initial setup consists of two separate flasks: one with 200 mL of oxygen at 220 mmHg and the other with 300 mL of nitrogen at 100 mmHg.

Calculate Combined Volume

When the flasks are connected, their total volume becomes the sum of the two separate volumes: \[V_{\text{total}} = 200\, \text{mL} + 300\, \text{mL} = 500\, \text{mL}\]

Use Ideal Gas Law for New Pressures

Apply the ideal gas law in the form of \(P_1V_1 = P_2V_2\) to find new pressures for each gas when they occupy the total volume:For \(O_2\):\[220\, \text{mmHg} \times 200\, \text{mL} = P_{\text{O}_2} \times 500\, \text{mL} \Rightarrow P_{\text{O}_2} = \frac{220 \times 200}{500} = 88\, \text{mmHg}\]

Calculate New Pressure for Nitrogen

Similarly, calculate the new pressure for nitrogen when it occupies the entire combined volume:For \(N_2\):\[100\, \text{mmHg} \times 300\, \text{mL} = P_{\text{N}_2} \times 500\, \text{mL} \Rightarrow P_{\text{N}_2} = \frac{100 \times 300}{500} = 60\, \text{mmHg}\]

Calculate Total Pressure

Add the pressures of oxygen and nitrogen to find the total pressure of the gaseous mixture:\[P_{\text{total}} = P_{\text{O}_2} + P_{\text{N}_2} = 88\, \text{mmHg} + 60\, \text{mmHg} = 148\, \text{mmHg}\]

Select Correct Answer

The correct answer, based on the calculation of the total pressure, is 148 mmHg (choice c).

Key concepts

Partial Pressure

In a gaseous mixture, each gas contributes to the total pressure based on its partial pressure. Partial pressure is the pressure exerted by a single type of gas in a mixture, as if it were the only gas present in the same volume. For our exercise, we have two gases, oxygen and nitrogen, in separate containers. When they were each isolated, each gas exerted its own pressure—220 mmHg for oxygen and 100 mmHg for nitrogen.

• Partial pressure of a gas depends on the proportion of the gas's moles relative to the total moles in the mixture.
• The sum of all partial pressures in a container equals the total pressure.

Understanding partial pressures helps us predict how gases behave in a mixture when conditions such as volume or temperature change.

Ideal Gas Law

The Ideal Gas Law is a fundamental principle used to analyze the behavior of gases under various conditions. It is usually expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature.
However, for this exercise, we used a particular scenario of the Ideal Gas Law known as Boyle’s Law. Since the temperature and the number of moles remained constant due to no chemical reaction or leak, the relationship \(P_1V_1 = P_2V_2\) simplifies the application:

• This equation shows that, under isothermal conditions, the product of pressure and volume remains constant.
• We applied it to calculate how the initial pressures and volumes change when mixed and confined in a combined volume.

The Ideal Gas Law is a cornerstone in understanding how gases react to varying pressures, volumes, and temperatures.

Combined Volume

When two separate gas containers are connected, they essentially form a singular space for the gases to occupy—a combined volume. For our problem, a 200 mL flask and a 300 mL flask are combined to form a total volume of 500 mL.

• The combined volume is crucial for the calculation of new pressures for each gas using the Ideal Gas Law.
• By knowing the combined volume, you're able to distribute the initial gas quantities evenly, which is necessary to determine each gas's new partial pressure in the shared space.

Evaluating the combined volume allows us to effectively determine each gas's contribution to the overall pressure of the mixture when they occupy the same container without changing their original amount of substance.

Total Pressure Calculation

The total pressure of a gaseous mixture is a sum of the partial pressures of all the constituent gases. After determining the new partial pressures of oxygen and nitrogen when confined in the combined volume, we simply add them.
Here are the steps discussed:

• Calculate the new partial pressure for each gas after combining the containers: Oxygen exerted 88 mmHg, and Nitrogen exerted 60 mmHg.
• The total pressure is then calculated by adding these two new pressures together: \( P_{\text{total}} = 88 \, \text{mmHg} + 60 \, \text{mmHg} = 148 \, \text{mmHg} \).

The determination of total pressure is essential to predict how the gas mixture behaves, as this pressure measure dictates various practical applications in laboratory and industrial settings. Understanding and calculating total pressure gives insight into maintaining controlled environments for chemical reactions.