Pick out the correct option which represents for the work done by the system on the surroundings

Options

A. `p_("in") gt p_("ex")`
B. `p_("in") = p_("ex")`
C. `p_("in") lt p_("ex")`
D. `p_("ex") ge 1` atm

Answer

To solve the question regarding the work done by the system on the surroundings, we can follow these steps:

### Step 1: Understand the Concept of Work Done
Work done by a system on its surroundings occurs during expansion. When a gas expands against an external pressure, it does work on the surroundings. The relationship can be expressed as:
\[ W = -P_{\text{external}} \Delta V \]
where \( W \) is the work done, \( P_{\text{external}} \) is the external pressure, and \( \Delta V \) is the change in volume.

### Step 2: Identify the Conditions for Work Done
For work to be done by the system on the surroundings, the internal pressure (\( P_{\text{internal}} \)) of the system must be greater than the external pressure (\( P_{\text{external}} \)):
\[ P_{\text{internal}} > P_{\text{external}} \]
This indicates that the system is expanding against the external pressure.

### Step 3: Analyze the Given Options
Now, let’s analyze the options provided in the question:
1. **Option A**: \( P_{\text{internal}} > P_{\text{external}} \) - This is correct as it indicates that the system can do work on the surroundings.
2. **Option B**: \( P_{\text{internal}} = P_{\text{external}} \) - This is incorrect; no work is done when the pressures are equal.
3. **Option C**: \( P_{\text{internal}} < P_{\text{external}} \) - This is also incorrect; the system cannot expand against a greater external pressure.
4. **Option D**: \( P_{\text{external}} > 1 \, \text{atm} \) - This does not directly relate to the work done by the system.

### Step 4: Conclusion
Based on the analysis, the correct option that represents the work done by the system on the surroundings is:
**Option A: \( P_{\text{internal}} > P_{\text{external}} \)**