Problem 65 - A Complete Resource Book in Chemistry for JEE Main

Question

If two moles of an ideal gas at a temperature \(546 \mathrm{~K}\), occupy a volume of \(44.8\) litres its pressure must be (a) \(4 \mathrm{~atm}\) (b) \(3 \mathrm{~atm}\) (c) \(2 \mathrm{~atm}\) (d) \(1 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The pressure is 2 atm, corresponding to option (c).

Step 1: Review the Ideal Gas Law

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (approximately \(0.0821 \frac{L \, atm}{mol \, K}\)), and \( T \) is the temperature in Kelvin.

Step 2: Plug in Known Values

We know from the problem that \( n = 2 \) moles, \( T = 546 \mathrm{~K} \), and \( V = 44.8 \) litres. These values will be substituted into the ideal gas law equation to find the pressure \( P \).

Step 3: Calculate using the Ideal Gas Law

Substitute the values into the ideal gas law equation. We have: \[P \times 44.8 = 2 \times 0.0821 \times 546\] Solving for \( P \), we calculate: \[P \times 44.8 = 89.5812\] To find \( P \), divide both sides by 44.8:\[P = \frac{89.5812}{44.8} \approx 2\]

Step 4: Determine Best Answer Choice

The calculated pressure \( P \) is \( 2 \) atm. This matches option (c) from the given answer choices.

Key Concepts

Ideal Gas Constant

The ideal gas constant, represented as \( R \), plays a vital role in the ideal gas law equation, \( PV = nRT \). This constant is a key factor that links the physical properties of gases: pressure (\( P \)), volume (\( V \)), temperature (\( T \)), and the amount of gas in moles (\( n \)). The value of \( R \) is approximately \( 0.0821 \frac{L \cdot atm}{mol \cdot K} \), and it is consistent across all ideal gases. This means that regardless of the type of gas, as long as it behaves ideally, \( R \) remains the same.
- The constancy of \( R \) simplifies calculations significantly. It allows for straightforward plugging in of values to solve for unknowns in gas law calculations.
- When using the ideal gas law, make sure that all units are consistent with those used for \( R \). This keeps calculations accurate and avoids conversion errors.
The ideal gas constant reminds us how gas particles move and interact under ideal conditions, and it's an anchor for understanding other thermodynamic concepts.

Pressure Calculation

Pressure is a measure of force applied over an area. In gases, we often calculate pressure using the ideal gas law, \( PV = nRT \), rearranged to solve for pressure: \( P = \frac{nRT}{V} \). In this specific problem, we are given the number of moles \( n \), temperature \( T \), and volume \( V \), which allows us to compute \( P \).
To calculate:

- Use the known values: \( n = 2 \) moles, \( T = 546 \mathrm{~K} \), and \( V = 44.8 \text{ liters} \).- Substitute these into the rearranged equation: \( P = \frac{2 \times 0.0821 \times 546}{44.8} \).

The computed result for \( P \) is approximately \( 2 \text{ atm} \). This pressure aligns with real-world expectations when observing the behavior of gases under controlled conditions.

Volume of Gas

Gas volume is the space that gas particles occupy, and this can change with pressure and temperature variations. In the ideal gas equation, \( V \) stands for volume, and understanding its role helps us predict how gases expand or compress.
- Given a constant amount of gas and temperature, increasing the volume will lead to a decrease in pressure, and vice versa, according to Boyle's Law.
- At a fixed pressure and amount of gas, if the temperature rises, the gas volume will increase. This relationship is described by Charles's Law: \( V \propto T \).
The volume influences how generous gases are with space. Larger volumes mean lower pressures when holding the amount of gas and temperature constant, providing a clear picture of the interplay between pressure and volume.