**Problem Statement:**
A 0.01 mole sample of a gas is compressed from 4L to 1L at a constant temperature of 25°C. The external pressure during the compression process is 4 bar. We need to calculate the work done for this process.

**Solution:**

To calculate the work done during the compression process, we can use the formula:

**Work Done = -Pext * ΔV**

where,
Pext is the external pressure,
ΔV is the change in volume.

In this case, the initial volume (Vi) is 4L and the final volume (Vf) is 1L. Therefore, the change in volume (ΔV) can be calculated as:

ΔV = Vf - Vi
= 1L - 4L
= -3L

The negative sign indicates that the volume has decreased during the compression process.

Now, let's convert the external pressure from bar to Pascal (Pa). 1 bar is equal to 100,000 Pa. Therefore, the external pressure (Pext) can be calculated as:

Pext = 4 bar * 100,000 Pa/bar
= 400,000 Pa

Now, we can calculate the work done using the formula:

Work Done = -Pext * ΔV
= -400,000 Pa * -3L
= 1,200,000 Pa.L

Since 1 L is equal to 0.001 m^3, we can convert the work done to the SI unit Joules (J) as follows:

Work Done = 1,200,000 Pa.L * 0.001 m^3/L
= 1200 J

Therefore, the work done for this process is 1200 Joules.

In summary,
- The initial volume (Vi) is 4L and the final volume (Vf) is 1L.
- The change in volume (ΔV) is -3L.
- The external pressure (Pext) is 400,000 Pa.
- The work done for this process is 1200 Joules.

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