The change in internal energy of the gas and workdone by the gas in an adiabatic process are

Options

A. Unequal
B. Equal
C. Only numerically equal
D. No relation.

Answer

To solve the question regarding the change in internal energy of the gas and the work done by the gas in an adiabatic process, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- An adiabatic process is defined as a thermodynamic process in which there is no heat exchange with the surroundings. Hence, the heat transfer (ΔQ) is zero.

2. **Apply the First Law of Thermodynamics**:
- The First Law of Thermodynamics states that:
\[ \Delta Q = \Delta U + W \]
- Where:
- ΔQ = Heat added to the system
- ΔU = Change in internal energy
- W = Work done by the system

3. **Substitute for an Adiabatic Process**:
- Since we know ΔQ = 0 for an adiabatic process, we can substitute this into the equation:
\[ 0 = \Delta U + W \]

4. **Rearrange the Equation**:
- Rearranging the equation gives us:
\[ W = -\Delta U \]

5. **Interpret the Result**:
- This means that the work done by the gas (W) is equal to the negative of the change in internal energy (ΔU). Therefore, the magnitude of work done by the gas is equal to the magnitude of the change in internal energy, but they have opposite signs.

6. **Conclusion**:
- In an adiabatic process, the work done by the gas is numerically equal to the change in internal energy, but they have opposite signs.

### Final Answer:
The change in internal energy of the gas and the work done by the gas in an adiabatic process are numerically equal but opposite in sign.