Two cylinders A and B fitted with pistons contain equal amounts of an ideal diatomic gas at 300 K.Teh piston of A is free to move,while that of B is held fixed.Same amount of heat is given to the gas in each cylinder.If the rise in temperature of the gas in A is 30 K,then what is the rise in temperature of the gas in B?

To tackle this problem, we need to consider 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 an initial temperature of 300 K. Cylinder A has a movable piston, while cylinder B has a fixed piston. When the same amount of heat is added to both cylinders, the behavior of the gas will differ due to the constraints imposed by the pistons.

Understanding the System

In cylinder A, the piston can move freely. This means that when heat is added, the gas can expand, doing work on the piston. The energy added as heat will not only increase the internal energy of the gas but also allow it to perform work, which means that the temperature increase will be significant.

In contrast, cylinder B has a fixed piston. When heat is added to the gas in this cylinder, the gas cannot expand; instead, all the added energy goes into increasing the internal energy of the gas, which directly translates into an increase in temperature.

Applying the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W):

ΔU = Q - W

• For cylinder A (movable piston): The gas does work on the piston as it expands, so some of the heat added (Q) is used for work (W). Therefore, the change in internal energy (ΔU) is less than the heat added.
• For cylinder B (fixed piston): The gas cannot do any work since the piston is fixed. Thus, all the heat added (Q) goes into increasing the internal energy, leading to a greater temperature rise.

Calculating Temperature Changes

For an ideal diatomic gas, the relationship between the change in temperature (ΔT) and the heat added can be expressed using the specific heat capacity at constant volume (C_v) and constant pressure (C_p). The specific heat at constant volume for a diatomic gas is:

C_v = (5/2)R

And at constant pressure:

C_p = (7/2)R

In cylinder A, since the piston is free to move, the heat added results in a temperature increase of 30 K. This means:

Q = nC_vΔT_A

Where ΔT_A = 30 K.

For cylinder B, since all the heat goes into increasing the internal energy, we can express the temperature change as:

Q = nC_vΔT_B

Relating the Two Cylinders

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

nC_vΔT_A = nC_vΔT_B

From this, we can see that:

ΔT_B = ΔT_A

However, we must remember that the work done in cylinder A means that not all the heat contributes to the temperature rise. The actual temperature rise in cylinder B will be greater than that in cylinder A. Since we know that the temperature rise in A is 30 K, we can conclude that:

Final Temperature Rise in Cylinder B

Given that the gas in cylinder B experiences no work done against a piston, the temperature increase will be higher than 30 K. In fact, for an ideal diatomic gas, the temperature rise in cylinder B can be calculated as:

ΔT_B = 30 K + additional temperature rise due to fixed volume

In practical terms, if we assume that the heat added is sufficient to raise the temperature significantly, we can estimate that the rise in temperature in cylinder B could be around 60 K, as it would absorb all the heat without performing work. Thus, the final answer is:

ΔT_B is approximately 60 K.