Problem 96 - Master Resource Book in JEE Main Physics

Question

An ideal gas is found to obey an additional law \(p V^{2}=\) constant. The gas is initially at temperature \(T\) and volume \(V\), Then it expands to a volume \(2 \mathrm{~V}\), the temperature becomes (a) \(T / \sqrt{2}\) (b) \(2 \mathrm{~T}\) (c) \(2 T / 2\) (d) \(4 \underline{T}\)

Step-by-Step Solution

Answer

The final temperature is \(T/2\), not matching listed options.

Step 1: Understand the Given Ideal Gas Law

The given problem states that the ideal gas follows a modified relationship: \(pV^2 = \text{constant}\). This differs from the standard ideal gas law \(pV = nRT\). Here, the product of pressure and the square of volume remains constant during changes in state.

Step 2: Use Relationship for Initial and Final State

For initial state, the condition is \(p_i V_i^2 = \text{constant}\). Similarly, for the final state, \(p_f V_f^2 = \text{constant}\). Since the two constants are equal, we equate them: \(p_i V_i^2 = p_f (2V_i)^2\) since the final volume is given as \(2V\).

Step 3: Simplify and Relate Initial and Final Pressures

From \(p_i V_i^2 = 4p_f V_i^2\), we can simplify to find the relationship between the initial and final pressures: \(p_i = 4p_f\).

Step 4: Apply Ideal Gas Law and Relate Temperatures

Using the ideal gas law for both states, for the initial state: \(p_i V_i = nRT_i\), and for the final state: \(p_f (2V_i) = nRT_f\). Since \(p_i = 4p_f\), substitute: \(4p_f V_i = nRT_i\). Also \(p_f (2V_i) = nRT_f\), which simplifies to \(2p_f V_i = nRT_f\).

Step 5: Solve for Final Temperature

From equations \(4p_f V_i = nRT_i\) and \(2p_f V_i = nRT_f\), divide the first by the second: \((4p_f V_i) / (2p_f V_i) = T_i / T_f\). This simplifies to \(2 = T_i / T_f\), hence \(T_f = T_i / 2\). The temperature becomes \(T/2\), which matches option (c), though the listed option should be revisited for misprint.

Key Concepts

Boyle's Law

Boyle's Law is one of the foundational principles in understanding gas behavior. This law states that at a constant temperature, the pressure of a gas is inversely proportional to its volume. In simpler terms, as the volume of a gas decreases, its pressure increases, provided the temperature remains unchanged. Boyle's Law can be mathematically represented as:

• \( PV = ext{constant} \)

This relationship can help us predict how a gas will respond when we change the space it occupies. For instance, if you compress a gas within a piston by reducing its volume, the pressure inside will rise. Conversely, if you expand the gas by increasing the volume, the pressure drops. This balance maintains the product \( PV \) as a constant, illustrating the principle of inversely related pressure and volume.

Charles's Law

Delving into the realm of gas laws, Charles's Law is another fundamental rule that highlights the relationship between volume and temperature. According to Charles's Law, the volume of a gas is directly proportional to its temperature when pressure is kept constant. To express this mathematically:

• \( \frac{V}{T} = ext{constant} \)

This law implies that if the temperature of a gas increases, its volume will expand as long as the pressure doesn't change. Conversely, if the temperature drops, the gas will contract. This is why hot air balloons rise: as the air inside the balloon heats up, it expands, causing the balloon to float. Charles's Law is fundamentally about the control and predictability of gas volume with temperature changes, making it incredibly useful in numerous practical applications.

Pressure-Volume Relationship

Understanding the pressure-volume relationship is essential in studying gases because it forms the basis for much of ideal gas behavior. The relationship involves how pressure changes with volume given certain constraints like constant temperature, aligning with Boyle's Law. However, when we introduce other conditions such as variable temperature or altered gas contents, this relationship can change.In the context of the given problem, we see a modified pressure-volume relationship: \( pV^2 = ext{constant} \). This implies that the pressure times the square of the volume remains a constant. This is not typical of ideal gas laws but provides an interesting perspective on how gases could behave under specific conditions, where the usual directly or inversely proportional relationships might be altered. These kinds of expressions are handy in complex systems or theoretical models where traditional laws need adjustments to fit empirical observations. Knowing how to manipulate and understand these expressions is key in physics and chemistry, especially when typical conditions are not guaranteed.