Problem 59 - A Complete Resource Book in Chemistry for JEE Main

Question

If increase in temperature and volume of an ideal gas is two times, then the initial pressure \(\mathrm{P}\) changes to (a) \(4 \mathrm{P}\) (b) \(2 \mathrm{P}\) (c) \(\mathrm{P}\) (d) \(3 \mathrm{P}\)

Step-by-Step Solution

Answer

The initial pressure \( P \) remains unchanged: \( P \).

Step 1: Understand the Relationship

According to the ideal gas law, the relationship between pressure \( P \), volume \( V \), and temperature \( T \) for a given amount of gas is expressed as \( PV = nRT \), where \( n \) is the number of moles and \( R \) is the ideal gas constant. In this scenario, both temperature and volume are increased by a factor of 2.

Step 2: Adjust the Equation for Changes

Since both the temperature and volume are doubled, modify the initial condition. The new condition would be \( P' \cdot 2V = nR \cdot 2T \). Simplifying this gives \( P' \cdot 2V = 2nRT \).

Step 3: Solve for New Pressure

Divide both sides by \( 2V \): \( P' = \frac{2nRT}{2V} \). Simplify the equation to \( P' = \frac{nRT}{V} = P \). This shows that the pressure remains unchanged under the given conditions.

Key Concepts

Pressure Changes

When considering pressure changes in an ideal gas, it's crucial to understand how pressure is influenced by other factors such as volume and temperature. In the ideal gas equation, pressure (\( P \)) directly relates to the volume (\( V \)) and temperature (\( T \)) when the quantity of the gas and the gas constant remain the same.

If we keep the amount of gas and the gas constant (\( R \)) unchanged, any change in volume or temperature will affect the pressure. In a closed system, when the volume increases, pressure tends to decrease if the temperature remains constant. Conversely, increasing temperature raises the pressure if the volume is constant.

In the given exercise, because both temperature and volume double, the changes seem to counterbalance each other, leading to an unchanged pressure, as derived from simplifying the modified gas law equation.

Ideal Gas Relationship

The ideal gas law is an essential equation in thermodynamics. It establishes a relationship between pressure (\( P \)), volume (\( V \)), and temperature (\( T \)) for an ideal gas, using the equation:

• \( PV = nRT \)

where \( n \) is the number of moles and \( R \) is the gas constant.

An ideal gas is a hypothetical construct that assumes no interactions between gas molecules, which only applies perfectly in certain conditions. However, this model often serves to approximate real gas behavior under a range of conditions.

Adjusting parameters in this equation allows us to explore how changes in temperature or volume affect the pressure. When both the temperature and volume double, as in the exercise, the ideal gas law helps explain why the overall pressure stays constant, despite these significant changes.

Temperature and Volume Changes

Understanding how temperature and volume changes affect an ideal gas is critical. Temperature is a measure of the average kinetic energy of gas molecules. When temperature rises, molecules move faster, potentially increasing the pressure or requiring more volume to accommodate the increased activity.

Volume affects pressure when it changes, assuming the amount of gas is constant. Greater volume typically leads to reduced pressure if temperature stays the same, as molecules collide with walls less frequently.

In this exercise, both the temperature and volume of the gas double. This simultaneous change means:

• Temperature increase tries to raise pressure.
• Volume increase tries to lower pressure.

The two effects cancel each other out, resulting in an unchanged pressure, consistent with the derived equation \( P' = P \). This demonstrates a robust understanding of the interplay of gas laws in this situation.