Problem 61 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The volumes of the two vessels are in the ratio of \(2: 1\). One contains nitrogen and the other oxygen at \(800 \mathrm{~mm}\) and \(680 \mathrm{~mm}\) pressure, respectively. Determine the resulting pressure when they are connected together. (a) \(760 \mathrm{~mm}\) (b) \(670 \mathrm{~mm}\) (c) \(1140 \mathrm{~mm}\) (d) \(1480 \mathrm{~mm}\)

Step-by-Step Solution

Answer

1480 mmHg

Step 1: Understanding the Law Governing the Mixture

When two gases mix, their partial pressures add up to the total pressure of the mixture. The partial pressure of each gas is determined by its mole fraction in the mixture and the total pressure. This is rooted in Dalton's Law of Partial Pressures.

Step 2: Determining the Mole Fraction for Each Gas

The volumes of the gases are in the ratio of 2:1, which means the volume of nitrogen is twice that of oxygen. Since both gases are at the same temperature, the ratio of their volumes is also the ratio of their moles (by Avogadro's Law). Therefore, nitrogen has twice as many moles as oxygen.

Step 3: Calculating the Partial Pressures

To find the total final pressure, calculate partial pressures using the mole ratio. Nitrogen’s partial pressure is \((2/3) \times P_{total}\) since it accounts for two moles out of three total moles. Oxygen’s partial pressure is \((1/3) \times P_{total}\).

Step 4: Expressing the Total Pressure

The total pressure is the sum of the individual partial pressures: \((2/3) \times P_{total} = 800 \, \text{mm}\) for nitrogen and \((1/3) \times P_{total} = 680 \, \text{mm}\) for oxygen. Therefore, \((2/3) \times P_{total} + (1/3) \times P_{total} = 800 \, \text{mm} + 680 \, \text{mm}\).

Step 5: Solving for Total Pressure

Solve the equation from step 4: \((2/3) \times P_{total} + (1/3) \times P_{total} = 1480 \, \text{mm}\). Combine the terms to yield \((3/3) \times P_{total} = 1480 \, \text{mm}\), which simplifies to \P_{total} = 1480 \, \text{mm}\.

Key Concepts

Partial Pressure

Imagine carrying a basket full of different fruits, each kind of fruit representing a different gas in a container. Just as each fruit contributes to the total weight of the basket, each gas contributes to the total pressure in the container. This contribution is what we call the partial pressure of a gas.

According to Dalton's Law of Partial Pressures, the total pressure exerted by a mixture of non-reacting gases in a container is equal to the sum of the partial pressures of individual gases. Mathematically, if container A has gases X, Y, and Z with partial pressures P_X, P_Y, and P_Z, then the total pressure P_total is:

• P_total = P_X + P_Y + P_Z

This rule helps explain why, when our textbook problem's vessels containing nitrogen and oxygen are connected, the total pressure is a simple sum of the individual gases’ partial pressures. Understanding this concept is crucial in predicting the behavior of gases in practical applications, such as in diving, medical gas administration, and aeronautics.

Mole Fraction

To further dissect the textbook problem, we need to comprehend the concept of mole fraction, an integral element in calculating partial pressures. Suppose that in a group of people, we are interested in finding out the fraction of people who are left-handed. Similarly, the mole fraction tells us the proportion of a specific gas mixed with other gases.

The mole fraction, often denoted by the Greek letter Chi \(\chi\), is a way of expressing the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture. It is calculated as follows:

• \(\chi_\text{component} = \frac{n_{\text{component}}}{n_{\text{total}}}\)

Where \(n_{\text{component}}\) is the number of moles of the component and \(n_{\text{total}}\) is the total number of moles of all components. In our textbook solution, this concept plays a pivotal role in determining the partial pressures of nitrogen and oxygen by considering their mole ratios.

Avogadro's Law

Avogadro's law is like a democratic principle applied to gases: Each gas mole is granted the same amount of space when the temperature and pressure are consistent. This law states that equal volumes of all gases, under the same conditions of temperature and pressure, contain the same number of molecules. The formula for this is:

• V ∝ n (at constant temperature and pressure)

,
where V stands for volume and n for the number of moles. This relationship implies that the volume of a gas is directly proportional to the number of moles of the gas when the temperature and pressure are held constant.

In the context of our textbook problem, using Avogadro’s law allows us to conclude that because the volume ratio of nitrogen to oxygen is 2:1, at the same temperature and pressure the mole ratio is also 2:1. This direct ratio is key to calculating mole fraction and eventually the partial pressures of the gases.