PV Diagrams for AP Physics B
Practice problems for these concepts can be found at:
Imagine a gas in a cylinder with a moveable piston on top, as shown in Figure 18.1.
When the gas expands, the piston moves up, and work is done by the gas. When the piston compresses, work is done on the gas. The point of a heat engine is to add heat to a gas, making the gas expand—in other words, using heat to do some work by moving the piston up.
To visualize this process mathematically, we use a graph of pressure vs. volume, called a PV diagram. On a PV diagram, the x-axis shows the volume of the gas, and the y-axis shows the pressure of the gas at a particular volume. For the AP exam, you're expected to be familiar with the PV diagrams for four types of processes:
- Isothermal: The gas is held at a constant temperature.
- Adiabatic: No heat flows into or out of the gas.
- Isobaric: The gas is held at a constant pressure.
- Isochoric: The gas is held at a constant volume.
Not all processes on a PV diagram fall into one of these four categories, but many do. We look at each of these processes individually.
Look at the ideal gas law: PV = NkBT. In virtually all of the processes you'll deal with, no gas will be allowed to escape, so N will be constant. Boltzmann's constant is, of course, always constant. And, in an isothermal process, T must be constant. So the entire right side of the equation remains constant during an isothermal process. What this means is that the product of P and V must also be constant. A PV diagram for an isothermal process is shown in Figure 18.2.
What this diagram shows is the set of all P–V values taken on by a gas as it goes from point "a" to point "b." At point "a," the gas is held at high pressure but low volume. We can imagine that there's a lot of gas squished into a tiny cylinder, and the gas is trying its hardest to push up on the piston—it is exerting a lot of pressure.
As the volume of the gas increases—that is, as the piston is pushed up—the pressure of the gas decreases rapidly at first, and then slower as the volume of the gas continues to increase. Eventually, after pushing up the piston enough, the gas reaches point "b," where it is contained in a large volume and at a low pressure.
Since the temperature of the gas does not change during this process, the internal energy of the gas remains the same, because U = (3/2)NkbT. For an isothermal process, ΔU = 0.
In an adiabatic process, no heat is added to or removed from the system. In other words, Q = 0. By applying the first law of thermodynamics, we conclude that ΔU = W. This means that the change in internal energy of the gas equals the work done ON the system. In an adiabatic process, you can increase the temperature of the gas by pushing down on the piston, or the temperature of the gas can decrease if the gas itself pushes up on the piston. The PV diagram for an adiabatic process looks similar, but not identical, to the diagram for an isothermal process—note that when a gas is expanded adiabatically, as from "a" to "b" in Figure 18.3, the final state is on a lower isotherm. (An isotherm is any curve on a PV diagram for which the temperature is constant.) In other words, in an adiabatic expansion, the temperature of the gas decreases.
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