Problem 106 - A Complete Resource Book in Chemistry for JEE Main

Question

Equal weights of ethane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction to total pressure exerted by hydrogen is (a) \(1: 2\) (b) \(1: 1\) (c) \(1: 16\) (d) \(15: 16\)

Step-by-Step Solution

Answer

The fraction of total pressure exerted by hydrogen in the mixture is 15:16.

Step 1: Identify Components' Molecular Weights

We have hydrogen (H_2) and ethane (C_2H_6). The molecular weight of hydrogen is 2 g/mol, and the molecular weight of ethane is 30 g/mol.

Step 2: Determine Moles in Equal-Weight Mixture

Since the weights of H_2 and C_2H_6 are equal, let's assume each gas is 1 gram. The number of moles of H_2 is \( \frac{1}{2} \) moles, and the number of moles of C_2H_6 is \( \frac{1}{30} \) moles.

Step 3: Calculate Total Moles in Mixture

Total moles = Moles of H_2 + Moles of C_2H_6 = \( \frac{1}{2} + \frac{1}{30} = \frac{15}{30} + \frac{1}{30} = \frac{16}{30} \).

Step 4: Determine Mole Fraction of Hydrogen

The mole fraction of H_2 is \( \frac{\text{moles of } H_2}{\text{total moles}} = \frac{\frac{1}{2}}{\frac{16}{30}} = \frac{15}{16} \).

Step 5: Comparison to Options

The fraction of total pressure that hydrogen exerts, proportional to its mole fraction, is \( \frac{15}{16} \). When comparing this fraction against the options, (d) \( 15:16 \) is correct.

Key Concepts

Mole fraction

Mole fraction is a crucial concept when working with mixtures of gases. Essentially, it tells us the ratio of the number of moles of a particular gas to the total number of moles of all gases in a mixture. You can think of it as the gas's share in the mixture.

To compute the mole fraction, use the formula:

• \[ ext{Mole Fraction} = \frac{ ext{Moles of the Component Gas}}{ ext{Total Moles in the Mixture}} \]

In our scenario, we're interested in the mole fraction of hydrogen. By computing the total moles in the mixture and dividing the moles of hydrogen by this total, we found that hydrogen's mole fraction is \( \frac{15}{16} \). Understanding this concept is essential, as it directly influences the gas's behavior in mixtures and its contribution to overall pressure.

Partial pressure

Partial pressure is the pressure exerted by a single gas in a mixture. Each gas in a container contributes to the total pressure in proportion to its mole fraction. The sum of partial pressures of each gas in the mixture is equal to the total pressure.

The formula for determining the partial pressure of a gas is:

• \[ P_i = ext{Mole Fraction of Gas} imes ext{Total Pressure} \]

In our case, the mole fraction of hydrogen is \( \frac{15}{16} \), meaning it contributes \( \frac{15}{16} \) of the total pressure. This helps us understand why hydrogen's presence is a significant part of the total pressure in the mixture.

Molecular weight

Knowing the molecular weight of each component in a gas mixture is essential for calculating moles, and consequently, mole fractions and partial pressures. Molecular weight refers to the sum of the atomic weights of all atoms in a molecule. It gives us a way to convert between grams and moles, as molecular weight is expressed in grams per mole.

For this exercise:

• Hydrogen (H\(_2\)) has a molecular weight of 2 g/mol.
• Ethane (C\(_2\)H\(_6\)) has a molecular weight of 30 g/mol.

By knowing these weights, we derived that from an equal mass of each gas, hydrogen comprises more moles than ethane, leading to its significant mole fraction. Understanding these dynamics can help explain why gases with lower molecular weights often exert a larger influence in a mix.

Ideal gas law

The ideal gas law is a fundamental equation that connects various gas properties: pressure (P), volume (V), temperature (T), and the number of moles (n). Represented by the formula:

• \[ PV = nRT \]

Where \( R \) is the ideal gas constant. Although our specific exercise does not require direct application of this law, it's critical in understanding gas behavior at standard conditions.

When mixing gases such as ethane and hydrogen, ideal gas law principles imply that the gases will behave predictably based on their individual and combined moles, temperature, and volume. This law provides the foundation for calculating related properties such as partial pressures and reinforces why gas mixtures can predictably use mole fractions to determine each component's contribution to the total pressure.