The internal energy of a system remains constant when it undergoes

Options

A. cyclic process
B. an isothermal process
C. any process in which the heat given out by the system is equal to work done on the system.
D. all of the above.

Answer

To determine when the internal energy of a system remains constant, we can analyze the problem step by step using the first law of thermodynamics.

### Step-by-Step Solution:

1. **Understanding the First Law of Thermodynamics**:
The first law of thermodynamics states that the change in internal energy (dU) of a system is equal to the heat added to the system (dQ) minus the work done by the system (dW):
\[ dQ = dU + dW \]

2. **Condition for Constant Internal Energy**:
If the internal energy of a system remains constant, it implies that the change in internal energy is zero:
\[ dU = 0 \]
Substituting this into the first law gives:
\[ dQ = 0 + dW \implies dQ = dW \]
This means that the heat added to the system is equal to the work done by the system.

3. **Identifying Processes**:
- **Isothermal Process**: In an isothermal process, the temperature of the system remains constant. For an ideal gas, the internal energy is a function of temperature alone. Therefore, if the temperature is constant, the internal energy does not change:
\[ dU = 0 \]
- **Cyclic Process**: In a cyclic process, the system returns to its initial state after completing a cycle. Since internal energy is a state function, the change in internal energy over a complete cycle is also zero:
\[ dU = 0 \]

4. **Conclusion**:
From the analysis, we conclude that the internal energy of a system remains constant during:
- An **isothermal process** (constant temperature).
- A **cyclic process** (the system returns to its initial state).

Thus, the answer to the question is that the internal energy remains constant during both isothermal and cyclic processes.