Problem 16 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The Henry's law constant for the solubility of \(\mathrm{N}_{2}\) gas in water at \(298 \mathrm{~K}\) is \(1.0 \times 10^{5}\) atm. The mole fraction of \(\mathrm{N}_{2}\) in air is \(0.8\). The number of moles of \(\mathrm{N}_{2}\) from air dissolved in 10 moles of water at \(298 \mathrm{~K}\) and 5 atm pressure is (a) \(4.0 \times 10^{-4}\) (b) \(4.0 \times 10^{5}\) (c) \(5.0 \times 10^{-4}\) (d) \(4.0 \times 10^{-6}\)

Step-by-Step Solution

Answer

(a) \(4.0 \times 10^{-4}\)

Step 1: Understanding Henry's Law

Henry's law relates the solubility of a gas in a liquid to the partial pressure of the gas above the liquid. It's given by the formula: \[ C = k \times P \], where \( C \) is the concentration of the gas (in terms of mole fraction), \( k \) is the Henry's law constant, and \( P \) is the partial pressure of the gas.

Step 2: Calculate Partial Pressure of N2

First, to find the partial pressure of \( \mathrm{N}_2 \) in air, use the given mole fraction, \( 0.8 \), and the total pressure, \( 5 \) atm. The formula is: \[ P_{\mathrm{N}_2} = \text{mole fraction of } \mathrm{N}_2 \times \text{total pressure} = 0.8 \times 5 = 4 \text{ atm} \].

Step 3: Use Henry's Law to Find Mole Fraction of Dissolved N2

Now, use Henry's Law to find the concentration of \( \mathrm{N}_2 \) in water. Using \( k = 1.0 \times 10^{5} \) atm, \( P = 4 \) atm, apply the formula \[ C = \frac{P}{k} = \frac{4}{1.0 \times 10^5} = 4.0 \times 10^{-5} \].

Step 4: Calculate Moles of Dissolved N2

Given you have 10 moles of water, the number of moles of \( \mathrm{N}_2 \) dissolved is found by multiplying the mole fraction by the number of moles of water. Therefore, \[ \text{Moles of } \mathrm{N}_2 = 4.0 \times 10^{-5} \times 10 = 4.0 \times 10^{-4} \text{ moles} \].

Step 5: Finalize the Answer

Compare the calculated moles of dissolved \( \mathrm{N}_2 \) to the options provided. It matches option (a) \( 4.0 \times 10^{-4} \).

Key Concepts

Gas Solubility

The solubility of a gas in a liquid refers to how much of the gas can dissolve at a specific temperature and pressure. It's crucial in various scientific fields, including chemistry and environmental science. According to Henry's Law, the gas solubility depends on its partial pressure above the liquid and the Henry's Law constant specific to that gas. Understanding solubility helps us predict and control how gases like oxygen dissolve in water, crucial for aquatic life.
For example, in this exercise, we are determining how much nitrogen gas ( 2) can dissolve in water at room temperature and a certain pressure. Knowing solubility principles helps us ensure we can calculate such scenarios accurately.

Partial Pressure

Partial pressure is a term used to describe the pressure of a single gas in a mixture of gases. It's especially important in situations involving gas solubility and chemical reactions. The partial pressure of a gas depends on its concentration in the mixture and the total pressure.

• In our example, we calculated the partial pressure of nitrogen ( 2) in air to solve the problem by using the mole fraction provided: 0.8.
• The total atmospheric pressure given as 5 atm, combined with this mole fraction, results in a partial pressure of 4 atm for nitrogen gas.

This concept is fundamental for predicting how gases will behave when they dissolve in liquids.

Mole Fraction

A mole fraction is a way of expressing the concentration of a component in a mixture. It's the ratio of the number of moles of a component to the total number of moles in the mixture. This is an important concept in chemistry as it allows us to calculate properties like partial pressures and concentrations.
In our problem, the mole fraction of nitrogen ( 2) in air is 0.8, meaning nitrogen makes up 80% of the total gas present. By applying this concept, we can find how much nitrogen will dissolve in water when a total pressure is given, as seen in the exercise.

N2 Solubility

Nitrogen gas ( 2) is one of the major components of Earth's atmosphere, and understanding its solubility in water is important for various applications ranging from nature to industrial processes. Using Henry's Law, we can determine how much nitrogen gas will dissolve in water at a specific temperature and pressure.

• In the given exercise, after calculating the partial pressure of nitrogen, we used the Henry's Law constant to find out the solubility in terms of mole fraction.
• The calculation showed that under the given conditions, 4.0 × 10^{-4} moles of nitrogen would dissolve in 10 moles of water.

This exercise helps illustrate how nitrogen's behavior in water can be predicted using fundamental principles.