two cylinders A and B filled with pistons contain equal amount of an ideal diatomic gas at 300 k ,the piston of A is free to more while that of B is held fixed. The same amount of heat is given to the gas in each cylinder .It was found that the rise in temperature of gas A=30k Obtain the rise in temperature

To tackle this problem, we need to consider the properties of the ideal gas and how heat transfer affects temperature in different scenarios. We have two cylinders, A and B, filled with the same amount of an ideal diatomic gas at an 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 situation step by step.

Understanding the System

In cylinder A, the gas can expand freely when heat is added. This means that the gas does work on the piston as it expands, which affects how the temperature changes. In contrast, cylinder B has a fixed piston, meaning the gas cannot expand and does not do any work on the piston. Therefore, all the heat added to the gas in cylinder B goes into increasing its internal energy, which directly raises its temperature.

Applying the Ideal Gas Law

The ideal gas law states that:

• PV = nRT

Where:

• P = pressure
• V = volume
• n = number of moles of gas
• R = ideal gas constant
• T = temperature in Kelvin

Heat Transfer and Temperature Change

When heat (Q) is added to a gas, the change in internal energy (ΔU) can be expressed as:

• ΔU = nC_vΔT

Where:

• C_v = molar heat capacity at constant volume
• ΔT = change in temperature

Analyzing Cylinder A

For cylinder A, since the piston is free to move, the gas does work (W) as it expands. The first law of thermodynamics states:

• ΔU = Q - W

In this case, some of the heat added (Q) is used to do work (W), which means the change in internal energy (ΔU) and thus the temperature change (ΔT) will be less than what it would be if the gas were at constant volume.

Analyzing Cylinder B

In cylinder B, the piston is fixed, so no work is done (W = 0). Therefore, all the heat added goes into increasing the internal energy:

• ΔU = Q

Thus, the temperature change in cylinder B can be calculated as:

• ΔT_B = Q / (nC_v)

Relating the Two Cylinders

Given that the rise in temperature for gas A is 30 K, we can denote this as:

• ΔT_A = 30 K

Since the same amount of heat is added to both cylinders, we can express the heat added in terms of the temperature changes:

• Q = nC_vΔT_A for cylinder A
• Q = nC_vΔT_B for cylinder B

Since both expressions equal Q, we can set them equal to each other:

• nC_vΔT_A = nC_vΔT_B

From this, we can see that:

• ΔT_B = ΔT_A

However, since cylinder A does work, the effective temperature change in cylinder B will be greater than that in cylinder A. We can express this relationship as:

• ΔT_B = ΔT_A + ΔT_{work}

Calculating the Temperature Rise in Cylinder B

To find the rise in temperature for cylinder B, we need to consider that the work done by the gas in cylinder A reduces the temperature rise. For an ideal diatomic gas, the ratio of heat capacities (γ) is approximately 1.4. Thus, we can relate the temperature changes as follows:

• ΔT_B = \frac{ΔT_A}{1 - \frac{1}{γ}} = \frac{30 K}{1 - \frac{1}{1.4}} = \frac{30 K}{0.2857} \approx 105 K

Therefore, the rise in temperature of the gas in cylinder B is approximately 105 K. This illustrates how the constraints on the system (fixed vs. free piston) significantly affect the temperature change when heat is added.