Calculating Work for a Variable External Pressure

[ronaldor9]

Homework Statement

Consider 5.5L of a gas at a pressure of 3 atm in a cylinder with a movable piston. The external pressure is changed so that the volume changes to 10.5L. Calculate the work done.

Calculate the work if the process is done in two steps. At the end of the first step the volume is 7L and after the second step the volume is 10.5L

Homework Equations

W=PV?

The Attempt at a Solution

Not sure since the external pressure is not constant I don't think I can use W=PV to calculate work done.

 

[anathema]

I'll need to use the formula for calculating work done in a non-constant pressure system, which is W=∫PdV. To calculate the work done in the first step, I'll use the initial and final volumes (5.5L and 7L) and integrate the external pressure over that range. Then for the second step, I'll use the final and initial volumes (7L and 10.5L) and integrate the external pressure again. Finally, I'll add the two values together to get the total work done in both steps.

Alternatively, I could use the formula W=∫V1V2PdV, where V1 and V2 are the initial and final volumes, respectively, and integrate the external pressure over the entire range of volume change (from 5.5L to 10.5L). This would give me the same result as the previous method.

In either case, the work done would depend on the external pressure changing during the process, so I cannot simply use W=PV. It's important to consider the entire range of volume change and take into account the changing pressure in order to accurately calculate the work done in this system.

 

[kerimek]

Instead, I would need to use the formula W = ∫PdV, where P is the external pressure and dV is the change in volume. So for the first step, the work done would be W = ∫3dV, with the limits of integration being from 5.5L to 7L. For the second step, the work done would be W = ∫PdV, with the limits of integration being from 7L to 10.5L. Then, the total work done would be the sum of the work done in each step.