Problem 115 - A Complete Resource Book in Chemistry for JEE Main

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) \(168 \mathrm{~mm}\)

Step-by-Step Solution

Answer

(c) 148 mm

Step 1: Calculate the partial pressures

The partial pressure of each gas when they occupy the total volume (500 mL) can be calculated using the relationship: \( P_1 V_1 = P_2 V_2 \). First, for oxygen (\( O_2\)). The initial partial pressure is \( P_{O_2} = 220 \text{ mmHg} \) and its volume \( V_{O_2} = 200 \text{ mL} \). Using the formula: \[220 \text{ mmHg} \times 200 \text{ mL} = P_{O_2}' \times 500 \text{ mL}\]Solve for \( P_{O_2}' \) (Pressure in the mixed state):\[P_{O_2}' = \frac{220 \times 200}{500} = 88 \text{ mmHg}\]

Step 2: Calculate the partial pressure of Nitrogen

Next, calculate the pressure of nitrogen (\( N_2\)). The initial pressure is \( P_{N_2} = 100 \text{ mmHg} \) and its volume \( V_{N_2} = 300 \text{ mL} \). Using the formula:\[100 \text{ mmHg} \times 300 \text{ mL} = P_{N_2}' \times 500 \text{ mL}\]Solve for \( P_{N_2}' \):\[P_{N_2}' = \frac{100 \times 300}{500} = 60 \text{ mmHg}\]

Step 3: Calculate Total Pressure of the Mix

Total pressure (\( P_{Total} \)) in the gaseous mixture is simply the sum of the partial pressures:\[ P_{Total} = P_{O_2}' + P_{N_2}' = 88 \text{ mmHg} + 60 \text{ mmHg} = 148 \text{ mmHg} \]

Key Concepts

Partial Pressure

Partial pressure refers to the pressure that a gas would exert if it were alone in the container. When dealing with gas mixtures, it's important to consider the contribution of each individual gas to the total pressure.
For instance, when oxygen or nitrogen is confined in separate containers, each exerts its own partial pressure. This pressure arises due to the collisions of the gas molecules with the container walls.
In this exercise, oxygen has a partial pressure of 220 mmHg in a 200 mL flask, while nitrogen has a partial pressure of 100 mmHg in a 300 mL flask. When combining gases, we use their partial pressures to calculate their effect when they share the same volume.

To find the new partial pressures after gases mix, use the formula:

• For oxygen: \( P_{O_2}' = \frac{220 \times 200}{500} = 88 \text{ mmHg} \)
• For nitrogen: \( P_{N_2}' = \frac{100 \times 300}{500} = 60 \text{ mmHg} \)

These calculations help measure how each gas contributes to the total pressure in a mixed system.

Dalton's Law of Partial Pressures

Dalton’s Law of Partial Pressures is a fundamental principle that assists in calculating the total pressure of a gas mixture. This law posits that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of each individual gas in the mixture.
This concept is incredibly useful when dealing with air or any gas system consisting of different gases that don't chemically react with each other.

According to Dalton's Law, if you know the individual pressures of the components, you can easily calculate the total pressure:
\[ P_{Total} = P_{O_2}' + P_{N_2}' \]
From our exercises:

• The partial pressure of oxygen in the mixed volume is 88 mmHg.
• The partial pressure of nitrogen in the mixed volume is 60 mmHg.

Adding these gives a total pressure of 148 mmHg.

Using Dalton’s Law simplifies the calculation of total pressure in a multi-gas system and helps understand how each component contributes.

Gas Mixtures

When gases are mixed, they occupy the same volume and their individual properties contribute to the characteristics of the mixture. Unlike liquids, gases will completely mix to form a uniform composition. This property is due to the high kinetic energy and low molecular interaction in gaseous state.
In our example, flasks of oxygen and nitrogen are opened to a shared environment, leading them to mix evenly.

Let's break down the process of gas mixing in terms of pressure:

• Each gas retains its identity and exerts only its own pressure.
• The total pressure in the container can be predicted using Dalton’s Law.

Understanding gas mixtures and how they work allows chemists and scientists to predict behavior in numerous applications.
From clinical gas solutions to industrial applications, gases are frequently mixed to leverage property benefits like different rates of diffusion or reactivity, making this knowledge foundational in fields involving gas dynamics.