Problem 39 - A Complete Resource Book in Chemistry for JEE Main

Question

If pressure of 2 mol of an ideal gas at \(546 \mathrm{~K}\) having volume \(44.8\) litre is (a) \(2 \mathrm{~atm}\) (b) \(3 \mathrm{~atm}\) (c) \(4 \mathrm{~atm}\) (d) \(1 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The correct pressure is \(2\, \text{atm}\).

Step 1: Identify the Ideal Gas Equation

The equation to use for this problem is the Ideal Gas Law, which is \( PV = nRT \). Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Insert Known Values into the Equation

Given: \( n = 2 \) moles, \( T = 546 \) K, \( V = 44.8 \) liters. The ideal gas constant \( R \) is often used as \( 0.0821 \; \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}\). Substituting the values, the equation becomes: \( P \times 44.8 = 2 \times 0.0821 \times 546 \).

Step 3: Calculate Right Hand Side of the Equation

Perform the multiplication first: \( 2 \times 0.0821 \times 546 = 89.6832 \).

Step 4: Solve for Pressure \( P \) in Atmospheres

Rearrange the equation to solve for \( P \): \( P = \frac{89.6832}{44.8} \).

Step 5: Compute the Final Value of Pressure

Perform the division: \( P \approx 2 \) atm.

Step 6: Determine the Correct Answer Option

Match the calculated pressure to the given options: the correct answer is option (a) \(2 \; \text{atm}\).

Key Concepts

Pressure Calculation

In the ideal gas law, calculating pressure involves solving the equation \( PV = nRT \). You need to rearrange the equation to calculate for pressure \( P \). The formula becomes \( P = \frac{nRT}{V} \). By inserting the known values for moles \( n \), the ideal gas constant \( R \), temperature \( T \), and volume \( V \), you can solve for the pressure inside the container. Ensuring units are consistent (like using liters for volume) is crucial to getting the right answer.

Ideal Gas Constant

The ideal gas constant \( R \) is a crucial component in the ideal gas law equation \( PV = nRT \). It bridges the relationship between physical quantities like pressure, volume, temperature, and amount of gas. The commonly used value for \( R \) is \( 0.0821 \; \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K} \).
This constant allows the equation to work seamlessly across different quantities, ensuring unity and consistency when conducting gas calculations.
Remember, the choice of \( R \) ensures your units match up correctly, which is key to getting the correct results.

Temperature and Volume Relationship

In the context of the ideal gas law, temperature and volume share a direct relationship when pressure and the number of moles are held constant. As you adjust the temperature of a gas sample, the volume will change accordingly, assuming the pressure is constant. This means that as temperature increases, the volume of gas expands.
If you find the volume not matching the temperature change, it could mean other variables, like pressure, are changing too.
Understanding this relationship helps in predicting how gases will behave under different conditions of temperature.

Moles in Gas Calculations

Understanding the mole concept is vital in gas calculations because moles represent the amount of substance. In the ideal gas law \( PV = nRT \), \( n \) represents the number of moles. It gives you a way to quantify the amount of gas in a reaction or process.
In scenarios where other variables like pressure, volume, or temperature change, keeping track of the moles of gas helps maintain consistency using the ideal gas law.
For instance, knowing that one mole of any ideal gas occupies 22.4 L at standard temperature and pressure (STP) helps in estimating calculations across various conditions.