Problem 84 - A Complete Resource Book in Chemistry for JEE Main

Question

Equal masses of methane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by hydrogen is (a) \(1 / 2\) (b) \(8 / 9\) (c) \(1 / 9\) (d) \(16 / 17\)

Step-by-Step Solution

Answer

The fraction of total pressure exerted by hydrogen is \(\frac{8}{9}\).

Step 1: Understand the Problem

We are given equal masses of methane (CH₄) and hydrogen (H₂) mixed in a container. We need to find the fraction of the total pressure exerted by hydrogen.

Step 2: Define Molar Masses

Calculate the molar masses of methane and hydrogen. Methane (CH₄) has a molar mass of 16 g/mol, and hydrogen (H₂) has a molar mass of 2 g/mol.

Step 3: Calculate Moles

Let the mass of methane and hydrogen in the mixture be the same, say 'm' grams. Then, moles of methane = \(\frac{m}{16}\) and moles of hydrogen = \(\frac{m}{2}\).

Step 4: Calculate Total Moles

The total moles in the container = moles of methane + moles of hydrogen = \(\frac{m}{16} + \frac{m}{2} = \frac{m}{16} + \frac{8m}{16} = \frac{9m}{16}\).

Step 5: Calculate Mole Fraction of Hydrogen

Mole fraction of hydrogen (X_H2) = \(\frac{\text{moles of hydrogen}}{\text{total moles}} = \frac{\frac{m}{2}}{\frac{9m}{16}} = \frac{m \times 16}{2 \times 9m} = \frac{16}{18} = \frac{8}{9}\).

Step 6: Determine Hydrogen's Fraction of Total Pressure

The fraction of total pressure exerted by hydrogen is equal to the mole fraction of hydrogen, as the pressure ratio is proportional to the mole ratio.

Key Concepts

Gas Laws

Understanding gas laws is essential when dealing with gases and their mixtures. Gas laws are mathematical relationships that describe the behavior of gases under different conditions:

• The **Boyle's Law** states that for a fixed amount of gas at constant temperature, the volume of the gas is inversely proportional to the pressure. That means if you decrease the volume, the pressure increases.
• The **Charles's Law** describes how gases tend to expand when heated. It states that the volume of a gas is directly proportional to its temperature, provided the pressure remains constant.
• The **Avogadro's Law** states that the volume of a gas at a given temperature and pressure is directly proportional to the number of moles of gas present.

These laws help us understand how gases behave individually and in mixtures, like in our exercise with methane and hydrogen.

Mole Fraction

The concept of mole fraction is fundamental in chemistry, particularly when dealing with mixtures of gases. It's a way to express the concentration of a component in a mixture. The mole fraction is simply the ratio of moles of a component to the total moles in the mixture.
To calculate the mole fraction ( X_i ):

• Find the number of moles of the component you're interested in.
• Find the total number of moles in the mixture.
• Divide the number of moles of the component by the total number of moles.

In our exercise problem, the mole fraction of hydrogen is computed by dividing the moles of hydrogen by the total moles in the container.
The importance of the mole fraction is that it directly correlates with the partial pressure of that gas in the mixture, making it a prevalent measure when using gas laws for calculations.

Ideal Gas Equation

The Ideal Gas Equation is another crucial concept for understanding how gases behave under various conditions. It's an equation of state for a hypothetical "ideal" gas. The formula is given by:\[ PV = nRT \]where:

• P represents the pressure.
• V is the volume.
• n is the number of moles.
• R is the universal gas constant.
• T is the temperature in Kelvin.

This formula lets us relate the conditions under which a gas is stored. It assumes that gas molecules don't attract each other, and these interactions can often predict real gases' behavior under common conditions. By understanding and applying the Ideal Gas Equation, you can solve for any of the variables if all others are known, and thus predict how the gas will react when changing conditions like pressure or temperature.