Equal amount of an ideal diatomic gas is taken in two cylinders A and B at 300 K. The piston of A is free to move while that of B is held fixed. Same quantity of heat is given to both the cylinders. If the rise in temperature of gas in A is 30 K, then the rise in temp of the gas in B is (A) 84 K (B) 21 K (C) 30 K (D) 42

To solve this problem, we need to understand how heat transfer affects the temperature of gases in different conditions. In this scenario, we have two cylinders containing the same ideal diatomic gas at the same initial temperature of 300 K. The key difference is that the piston in cylinder A is free to move, while the piston in cylinder B is fixed. Let's break down the implications of these conditions and find the rise in temperature for cylinder B.

Understanding the Concepts

When heat is added to a gas, the temperature change depends on the amount of heat added, the number of moles of gas, and the specific heat capacity of the gas. For an ideal diatomic gas, the specific heat capacities at constant volume (C_v) and constant pressure (C_p) are important.

Heat Transfer in Cylinder A

In cylinder A, since the piston is free to move, the gas can expand. The heat added to the gas does work on the piston, and the temperature change can be described using the formula:

• Q = nC_vΔT + PΔV

Here, Q is the heat added, n is the number of moles, C_v is the specific heat at constant volume, ΔT is the change in temperature, P is the pressure, and ΔV is the change in volume. Since we know that the rise in temperature (ΔT) for cylinder A is 30 K, we can express the heat added in terms of this temperature change.

Heat Transfer in Cylinder B

In cylinder B, the piston is fixed, meaning the gas cannot expand. Therefore, the heat added only increases the internal energy of the gas, and we can use the formula:

• Q = nC_vΔT

Since the same amount of heat is added to both cylinders, we can set the heat added in both cases equal to each other:

• Q_A = Q_B

Relating the Temperature Changes

For cylinder A, we have:

• Q = nC_v(30 K) + PΔV

For cylinder B, since the volume does not change, we have:

• Q = nC_vΔT_B

Equating the two expressions for Q, we can simplify the relationship:

• nC_v(30 K) + PΔV = nC_vB

Since the gas in cylinder B does not do any work (because the volume is constant), the term PΔV becomes zero. Thus, we can simplify further:

• nC_v(30 K) = nC_vB

Finding the Temperature Change in Cylinder B

Now, we can cancel out nC_v from both sides (assuming they are not zero), leading us to:

• 30 K = ΔT_B

This means that the rise in temperature of the gas in cylinder B is also 30 K. Therefore, the correct answer is:

Final Answer

(C) 30 K

This result illustrates how the constraints on the system (fixed vs. movable piston) affect the distribution of heat and the resulting temperature changes in the gas. In both cases, the same amount of heat leads to the same temperature increase due to the nature of the ideal gas and the specific heat capacities involved.