A pressure volume diagram, or PV diagram, shows how the pressure and volume of a gas change during a thermodynamic process. It is one of the most useful tools for connecting graphs to physical quantities like work, heat, and internal energy. For an ideal gas, the shape of the path tells you what is being held constant and how energy is transferred.
Engineers and scientists use PV diagrams to analyze engines, refrigerators, pumps, and natural gas expansion.
On a PV diagram, the work done by the gas is the area under the curve, W = ∫P dV. Expansion gives positive work by the gas because volume increases, while compression gives negative work by the gas because volume decreases. The first law of thermodynamics, ΔU = Q - W, connects the graph to energy changes.
For an ideal gas, internal energy depends only on temperature, so processes with no temperature change have ΔU = 0.
Understanding Physics: PV Diagrams and Thermodynamic Processes
Each named process places a different restriction on the gas. In an isothermal process, temperature remains fixed. If the gas expands, its particles would normally slow down as they push outward.
Heat must enter from the surroundings to prevent that cooling. During compression, heat leaves so the gas does not warm up. For a fixed amount of ideal gas at fixed temperature, pressure times volume stays constant.
This produces a downward curving path. A slow expansion in a cylinder touching a large temperature controlled water bath is a useful model. Slow transfer matters because the gas needs time to exchange heat.
An adiabatic process permits no heat transfer between the gas and its surroundings. When the gas expands, it spends energy doing work. Its internal energy falls, so its temperature falls too.
During adiabatic compression, work done on the gas raises its internal energy and temperature. On a graph, an adiabatic expansion curve drops more steeply than an isothermal curve beginning at the same state. Pressure falls faster because there is no incoming heat to replace the energy used in expansion.
A process can be approximately adiabatic when it happens very quickly, such as air being compressed in a bicycle pump. Good insulation can produce a similar result. No heat transfer does not mean no energy transfer, since work can still transfer energy.
In an isobaric process, pressure stays fixed. A gas under a movable piston with a constant load is a common example. Heating it makes the piston rise, increasing volume while the pressure remains near the same value.
The path is horizontal when pressure is plotted vertically and volume horizontally. In an isochoric process, volume stays fixed because the container is rigid. Heating a sealed metal tank raises the temperature and pressure, but the gas cannot push the container boundary outward.
Therefore it does no mechanical work. This is why the pressure in a nearly fixed volume tyre rises after driving. Energy supplied as heat mainly increases the internal energy of the gas.
Many real machines use a sequence of processes that forms a closed loop. The gas returns to its starting pressure, volume, and temperature after one cycle. Its total internal energy change for the complete cycle is zero.
The enclosed area represents the net work for that cycle. A clockwise loop usually means the gas delivers net work, as in an engine. An anticlockwise loop requires net work input, as in a refrigerator or heat pump.
When studying these graphs, always follow the direction arrows. Check whether the volume rises or falls, then decide the sign of the work.
Identify horizontal paths, vertical paths, and curved paths before doing calculations. Different paths between the same two states can involve different amounts of heat and work, even though the overall internal energy change is the same.
Key Facts
- Ideal gas law: PV = nRT.
- Work done by a gas on a PV diagram: W = ∫P dV.
- For constant pressure: W = PΔV.
- First law of thermodynamics: ΔU = Q - W, where W is work done by the gas.
- For an ideal gas, internal energy depends only on temperature: ΔU = nCvΔT.
- Isothermal process: ΔT = 0 and ΔU = 0, so Q = W for an ideal gas.
Vocabulary
- PV Diagram
- A graph of pressure versus volume that shows the path of a gas during a thermodynamic process.
- Isothermal Process
- A thermodynamic process that occurs at constant temperature.
- Adiabatic Process
- A thermodynamic process in which no heat is transferred between the system and its surroundings.
- Isochoric Process
- A thermodynamic process that occurs at constant volume, so no work is done by the gas.
- Isobaric Process
- A thermodynamic process that occurs at constant pressure.
Common Mistakes to Avoid
- Confusing work done by the gas with work done on the gas. If the gas expands, W by the gas is positive, but work done on the gas is negative.
- Using W = PΔV for every process. This formula only works when pressure is constant, while curved paths require area under the curve or integration.
- Assuming heat and temperature change are the same thing. Heat is energy transferred across a boundary, while temperature is related to the average kinetic energy of particles.
- Thinking ΔU depends on the shape of the path for an ideal gas. Internal energy depends only on temperature change, so different paths with the same initial and final temperatures have the same ΔU.
Practice Questions
- 1 A gas expands at constant pressure 2.0 x 10^5 Pa from 0.030 m^3 to 0.080 m^3. How much work is done by the gas?
- 2 An ideal gas absorbs 600 J of heat while doing 250 J of work on its surroundings. What is the change in internal energy of the gas?
- 3 Two different paths connect the same initial and final states on a PV diagram. One path has a larger area under the curve than the other. Explain which path has more work done by the gas and whether the change in internal energy must be different.