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Boundary Work and its Process Path

V1
V2
V1 < V2

P1
P2
P1 > P2

Initial State:
Final State:
Where:

We are going to investigate the relationship between

work

and our

process path

Piston-and-cylinder device that illustrates pressure.

Remember that

work

is the product of the force driving the expansion and the displacement in the direction of the force.

If the piston never accelerates, then:

Our process path can best be visualized on a P-V diagram.

Lesson
4A Blog

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Lesson 4A: page 9 of 31

The first type of work that we will consider is called boundary work or PV work.
This form of work occurs when the system expands or is compressed. The boundary moves against a restraining force, so work is done.
This type of work is very important in thermodynamics, so let’s see what we can learn about it.
In the diagram shown here, let’s define our system to be the gas inside the cylinder.
As the gas expands, it does work on the surroundings.
If the piston rises slowly, without accelerating, then all of the forces acting on the piston must be balanced.
That means that the force that the system is overcoming in order to expand is always equal to the force exerted on the piston by the gas inside the cylinder.
This seems like a subtle point, but you’ll see why it is important later in this lesson.
The force exerted on the piston by the gas is equal to the pressure inside the system multiplied by the cross-sectional area of the piston, PR2.
This works for a piston of any shape as long as the area doesn’t change. If the piston and cylinder were tapered, the force applied to the piston by the gas would change as the piston moved upward. That would definitely be a more difficult problem to solve. But let’s not think about that too much at this point. We will just assume that A is not a function of x.
Here we substitute dV for A dx, because the cross-sectional area multiplied by the differential distance that the piston moves is equal to the differential change in the volume of the system, dV.
Now, we need to evaluate this integral.
It would be very easy if P were constant, but as the gas expands, we know that the pressure in the system is going to drop.
Since P is not a constant, we can’t just pull it outside of the integral.
Nope. Instead we must solve the integral in order to determine the area under a curve.
This can be visualized on a P-V Diagram.
Flip the page to exactly what area I am talking about.

Boundary Work and its Process Path

work

process path

work

Ch 4, Lesson A, Page 9 - Boundary Work and its Process Path