In an adiabatic expansion, a gas does 25J of work while in an adiabatic compression 100J of work is dione on a gas. The chagne of internal energy in the two processes respectively are

Options

A. 25J and -100J
B. `-25J` and 100J
C. `25J` and `-100J`
D. 25J and 100J

Explanation

To solve the problem of finding the change in internal energy for both adiabatic processes, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. This means that the heat transfer (ΔQ) is zero.

### Step 2: Use the First Law of Thermodynamics
The first law of thermodynamics states:
\[ \Delta U = \Delta Q + \Delta W \]
Since ΔQ = 0 for an adiabatic process, the equation simplifies to:
\[ \Delta U = \Delta W \]

### Step 3: Analyze the Adiabatic Expansion
In the adiabatic expansion, the gas does 25 J of work. According to the sign convention, when the gas does work on the surroundings, the work done (ΔW) is positive. Therefore:
\[ \Delta W_1 = -25 \text{ J} \]
Thus, the change in internal energy for the adiabatic expansion (ΔU1) is:
\[ \Delta U_1 = -\Delta W_1 = -(-25) = -25 \text{ J} \]

### Step 4: Analyze the Adiabatic Compression
In the adiabatic compression, 100 J of work is done on the gas. Here, the work done on the gas is considered negative:
\[ \Delta W_2 = +100 \text{ J} \]
Thus, the change in internal energy for the adiabatic compression (ΔU2) is:
\[ \Delta U_2 = -\Delta W_2 = -100 \text{ J} \]

### Step 5: Summarize the Results
- For the adiabatic expansion, the change in internal energy (ΔU1) is:
\[ \Delta U_1 = -25 \text{ J} \]
- For the adiabatic compression, the change in internal energy (ΔU2) is:
\[ \Delta U_2 = +100 \text{ J} \]

### Final Answer:
The change in internal energy in the two processes respectively are:
- Adiabatic Expansion: **-25 J**
- Adiabatic Compression: **+100 J**

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