Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

Pressure in a mixture of \(4 \mathrm{~g}\) of \(\mathrm{O}_{2}\) and \(2 \mathrm{~g}\) of \(\mathrm{H}_{2}\) confined in a container of 1 litre capacity at \(0{ }^{\circ} \mathrm{C}\) is (a) \(25.2 \mathrm{~atm}\) (b) \(35.6 \mathrm{~atm}\) (c) \(15.4 \mathrm{~atm}\) (d) \(48.2 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The pressure in the mixture is 25.2 atm (option a).

Step 1: Calculate the number of moles of O2

First, we need to find the number of moles of \(\mathrm{O}_2\). The molar mass of \(\mathrm{O}_2\) is approximately \(32 \text{ g/mol}\). For \(4 \text{ g}\) of \(\mathrm{O}_2\), the number of moles \(n_1\) is:\[ n_1 = \frac{4}{32} = 0.125 \text{ moles} \]

Step 2: Calculate the number of moles of H2

Next, find the number of moles of \(\mathrm{H}_2\). The molar mass of \(\mathrm{H}_2\) is approximately \(2 \text{ g/mol}\). For \(2 \text{ g}\) of \(\mathrm{H}_2\), the number of moles \(n_2\) is:\[ n_2 = \frac{2}{2} = 1 \text{ mole} \]

Step 3: Calculate total moles in the gas mixture

The total number of moles in the mixture is the sum of the moles of \(\mathrm{O}_2\) and \(\mathrm{H}_2\):\[ n_{\text{total}} = n_1 + n_2 = 0.125 + 1 = 1.125 \text{ moles} \]

Step 4: Apply the Ideal Gas Law

Use the ideal gas law to find the pressure \(P\). The formula is \(PV = nRT\), where:- \(P\) is the pressure,- \(V = 1 \text{ L}\),- \(R = 0.0821 \text{ L atm/mol K}\),- \(n_{\text{total}} = 1.125 \text{ moles}\),- \(T = 273 \text{ K}\) (0°C).Rearrange to solve for pressure:\[ P = \frac{nRT}{V} = \frac{1.125 \times 0.0821 \times 273}{1} \]

Step 5: Calculate the pressure

Plug in the values to calculate \(P\):\[ P = \frac{1.125 \times 0.0821 \times 273}{1} = 25.2 \text{ atm} \]

Key Concepts

Moles Calculation in Chemistry

In chemistry, calculating the number of moles is a fundamental concept that allows you to convert between the mass of a substance and the number of particles. The concept of moles is based on Avogadro's number, which is approximately \( 6.022 \times 10^{23} \). This number represents the quantity of atoms, ions, or molecules in one mole of a substance. To find the number of moles in a given mass, use the formula:

• \( n = \frac{m}{M} \)

where \( n \) is the number of moles, \( m \) is the mass of the substance in grams, and \( M \) is the molar mass of the substance in grams per mole. For example, when calculating the moles of \( \mathrm{O}_2 \) and \( \mathrm{H}_2 \) in a gas mixture, it involves dividing the mass of each component by its respective molar mass, providing essential information for further calculations.

Understanding Molar Mass

Molar mass is another essential concept in chemistry, representing the mass of one mole of a substance. It is expressed in grams per mole (g/mol) and is calculated based on the relative atomic masses of the elements in a compound. This understanding is vital when using the concept of moles for calculations.

• For example, \( \mathrm{O}_2 \) has a molar mass of approximately \( 32 \text{ g/mol} \)
• \( \mathrm{H}_2 \) has a molar mass of \( 2 \text{ g/mol} \)

Knowing the molar mass allows for accurate conversions between mass and moles, playing a crucial role when applying formulas like the Ideal Gas Law. It ensures precision in calculating the amount of substance and in reactions where stoichiometry is involved.

Pressure Calculation Using the Ideal Gas Law

The Ideal Gas Law is an equation of state for an ideal gas, which describes the behavior of real gases under a variety of conditions. It is represented by the formula:

• \( PV = nRT \)

where \( P \) is the pressure in atmospheres (atm), \( V \) is the volume in liters (L), \( n \) is the moles of gas, \( R \) is the gas constant \(0.0821 \text{ L atm/mol K}\), and \( T \) is the temperature in Kelvin (K). To calculate pressure, rearrange the formula to:

• \( P = \frac{nRT}{V} \)

Understanding this allows you to determine the pressure exerted by the gas mixture in a container. By plugging in the number of moles, temperature, volume, and the gas constant into the formula, you can find the required pressure easily. This forms a vital part of calculations in both laboratory and industrial gas analyses.

Characteristics of a Gas Mixture

A gas mixture often involves different gases sharing the same container but not chemically reacting. Understanding the behavior of gas mixtures is vital. Each component of a gas mixture behaves independently and contributes to the total pressure exerted in the container, according to Dalton's Law of Partial Pressures. In a mixture:

• The total number of moles is the sum of the moles of all gases present.
• The total pressure of the mixture equals the sum of the partial pressures of the individual gases.
• The behavior is often calculated using the Ideal Gas Law, where the moles of each gas play a crucial role.

When calculating characteristics of gas mixtures, assume the ideal behavior applies – with exceptions for high-pressure or low-temperature environments. Comprehending these aspects enables understanding and predicting the outcomes of various gas-related processes, critical in various scientific and industrial settings.