Let’s dive into the Carnot cycle and tackle each part of your question step by step. The Carnot cycle is a theoretical thermodynamic cycle that serves as a standard for the maximum efficiency of heat engines. It consists of two isothermal processes and two adiabatic processes. In your problem, we are given the temperatures for the isothermal expansion and compression as well as the heat absorbed during expansion. Let’s break it down.
Work Done During Isothermal Expansion
During isothermal expansion, the work done by the gas can be calculated using the formula:
W = Q
where W is the work done by the gas, and Q is the heat absorbed. In this case, during the isothermal expansion, the gas absorbs 2090 J of heat.
Thus:
W = 2090 J
So, the work performed by the gas during isothermal expansion is 2090 J.
Heat Rejected During Isothermal Compression
To find the heat rejected during isothermal compression, we can use the relationship between the heat absorbed and the heat rejected in a Carnot cycle:
Q_H / Q_C = T_H / T_C
Where:
- Q_H is the heat absorbed during the isothermal expansion (2090 J).
- Q_C is the heat rejected during the isothermal compression.
- T_H is the absolute temperature during the isothermal expansion (412 K).
- T_C is the absolute temperature during the isothermal compression (297 K).
Now, rearranging the equation to solve for Q_C gives us:
Q_C = Q_H × (T_C / T_H)
Plugging in the values:
Q_C = 2090 J × (297 K / 412 K)
Calculating this gives:
Q_C ≈ 2090 J × 0.7204 ≈ 1505.4 J
Therefore, the heat rejected from the gas during isothermal compression is approximately 1505.4 J.
Work Done on the Gas During Isothermal Compression
The work done on the gas during isothermal compression is the negative of the work done by the gas. Since work done by the gas is associated with energy leaving the system, the work done on the gas can be derived from the heat rejected during isothermal compression:
W_C = -Q_C
Thus, substituting the value we found:
W_C = -1505.4 J
This indicates that 1505.4 J of work is done on the gas during isothermal compression.
Summary of Results
- Work performed by the gas during isothermal expansion: 2090 J
- Heat rejected from the gas during isothermal compression: 1505.4 J
- Work done on the gas during isothermal compression: -1505.4 J
By understanding these relationships and equations, you can see how energy transfers and work interactions occur in a Carnot cycle. If you have any further questions or need clarification on any part of this, feel free to ask!
