PV diagrams show how a gas changes pressure and volume during a thermodynamic process. This cheat sheet helps students connect graph shape, process type, and work done by or on a gas. It is useful for solving problems involving engines, compression, expansion, and the first law of thermodynamics.
Students need these relationships to interpret diagrams quickly and avoid sign errors.
Key Facts
- Thermodynamic work done by a gas is , which equals the signed area under a curve on a PV diagram.
- For an isobaric process with constant pressure, the work is .
- For an isochoric process with constant volume, the work is because .
- For an ideal gas undergoing an isothermal process, the work is .
- For an adiabatic ideal gas process, and .
- Expansion gives and usually , meaning the gas does work on the surroundings.
- Compression gives and usually , meaning work is done on the gas.
- For a complete cycle on a PV diagram, the net work is the enclosed area, with clockwise cycles giving and counterclockwise cycles giving .
Vocabulary
- PV Diagram
- A graph of pressure versus volume that represents the thermodynamic state and process of a gas.
- Thermodynamic Work
- Energy transferred when a gas changes volume under pressure, calculated by .
- Isobaric Process
- A thermodynamic process that occurs at constant pressure, so work is .
- Isochoric Process
- A thermodynamic process that occurs at constant volume, so the gas does no work and .
- Isothermal Process
- A thermodynamic process that occurs at constant temperature, with ideal gas work .
- Adiabatic Process
- A thermodynamic process with no heat transfer, so and for an ideal gas .
Common Mistakes to Avoid
- Using for every process is wrong because that formula only applies when pressure is constant.
- Ignoring the sign of work is wrong because expansion has for work done by the gas, while compression has .
- Calculating the area under an isochoric line as nonzero is wrong because a vertical line has , so .
- Confusing work done by the gas with work done on the gas is wrong because these values have opposite signs, so .
- Assuming all curved PV paths have the same work between two endpoints is wrong because work depends on the path, not only the initial and final states.
Practice Questions
- 1 A gas expands isobarically at from to . Find the work done by the gas.
- 2 An ideal gas undergoes an isothermal expansion with , , , and . Calculate using .
- 3 A thermodynamic cycle encloses an area of on a PV diagram and runs counterclockwise. What is for work done by the gas?
- 4 Two processes connect the same initial and final states on a PV diagram, but one path stays at higher pressure for most of the expansion. Explain which process does more work and why.
Understanding PV Diagrams and Thermodynamic Work Reference
Pressure, volume, and temperature describe the state of a gas, but work depends on the route between two states. A gas can go from one starting point to one ending point by many different processes. Each route can produce a different amount of work because the pressure can be different at each volume.
For an ideal gas, internal energy depends only on temperature. This means the change in internal energy is the same for any route with the same beginning and ending temperatures.
The heat transferred can still differ. The first law states that change in internal energy equals heat added to the gas minus work done by the gas.
The shape of a path reflects what is happening physically. A constant pressure expansion can occur when a piston pushes against a fixed load. A constant volume change can occur in a rigid sealed container, where heating raises pressure but cannot move a boundary.
An isothermal change requires heat to enter or leave at the right rate to keep temperature steady. An adiabatic change has no heat transfer. It can happen in a well insulated system or during a process that occurs too quickly for much heat to flow.
Starting from the same state, an adiabatic expansion drops in pressure more sharply than an isothermal expansion. The gas uses its own internal energy to do work, so its temperature falls.
A moving piston makes the graph easier to picture. Gas molecules collide with the piston and push it outward. If the piston moves slowly, the gas stays close to equilibrium, so each point on the graph has a clear pressure and volume.
This is the condition assumed in most school calculations. The units reveal why the area on the graph represents energy. Pressure is measured in pascals and volume in cubic metres.
One pascal multiplied by one cubic metre equals one joule. Real engines use repeated expansions and compressions to turn a crankshaft. Refrigerators and air conditioners use cycles too, though work is supplied to move heat from a cooler place to a warmer one.
When reading a diagram, identify the direction of every path before calculating anything. Arrows matter because reversing a path reverses the sign of the work. Check whether the vertical axis is pressure and the horizontal axis is volume, then read the scale carefully.
Curved paths usually need a given formula or a stated relationship between pressure and volume. For a closed cycle, the gas returns to its original state. Its total change in internal energy is therefore zero.
Under the convention where work means work done by the gas, the net heat added during the cycle equals the net work done. Keep this convention visible throughout a solution, since some books define work as work done on the gas instead.