To tackle this question, we need to consider the properties of gases and how they respond to heat. In your scenario, we have two cylinders containing the same ideal diatomic gas, both initially at 300 K. The key difference is that the piston in cylinder A is free to move, while the piston in cylinder B is fixed. This distinction is crucial in understanding how the gas behaves when heat is added.
Understanding the Basics of Heat Transfer in Gases
When heat is added to a gas, the temperature change depends on the amount of heat supplied and the specific heat capacity of the gas. For an ideal diatomic gas, the specific heat at constant volume (Cv) and constant pressure (Cp) are important. In cylinder A, the gas can expand freely, allowing it to do work on the piston. In contrast, the gas in cylinder B cannot expand because its piston is fixed, meaning all the heat energy goes into increasing the internal energy of the gas, which directly raises its 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):
In cylinder A, since the piston is free to move, the gas does work as it expands. Therefore, some of the heat added (Q) is used to do work (W), resulting in a smaller increase in internal energy (ΔU) and consequently a smaller rise in temperature.
In cylinder B, the piston is fixed, so the gas cannot do any work (W = 0). Thus, all the heat added contributes to increasing the internal energy of the gas, leading to a larger increase in temperature.
Calculating the Temperature Rise
Given that the rise in temperature of the gas in cylinder A is 30 K, we can analyze the situation further. For an ideal diatomic gas, the relationship between heat added, temperature change, and the specific heat capacities can be expressed as:
- For constant volume: Q = n * Cv * ΔT
- For constant pressure: Q = n * Cp * ΔT
In cylinder A, the heat added results in a temperature rise of 30 K, but some of that energy is used for work. In cylinder B, since the piston is fixed, the entire amount of heat contributes to the temperature increase. The specific heat at constant volume (Cv) for a diatomic gas is approximately 5R/2, while at constant pressure (Cp) it is approximately 7R/2, where R is the gas constant.
Since the same amount of heat is added to both cylinders, we can infer that the temperature rise in cylinder B will be greater than in cylinder A. Specifically, if we denote the rise in temperature in cylinder B as ΔT_B, we can set up a proportion based on the heat capacities:
Substituting the values, we have:
- ΔT_A = 30 K
- Cv = 5R/2, Cp = 7R/2
Thus, we can express the ratio:
Solving for ΔT_B gives us:
- ΔT_B = (30 K * 7) / 5 = 42 K
Final Thoughts
Therefore, the rise in temperature of the gas in cylinder B, where the piston is fixed, will be 42 K. This illustrates how the ability of a gas to do work (by expanding) affects its temperature change when heat is added. In summary, the constraints on the system significantly influence how energy is distributed within the gas, leading to different temperature changes in the two cylinders.
