Ideal Gas PV = nRT, and Each Process Traces Differently
In an ideal gas, pressure, volume, and temperature are tied together by the single relation PV = nRT. Even between the same two points, the trace on the P-V plane differs depending on whether the process is isothermal, isobaric, isochoric, or adiabatic.
Move the volume and temperature. The pressure follows as P = nRTV. The three always satisfy PV = nRT. When a tire heats up under the summer sun or after highway driving, the volume stays about the same while the temperature T rises, so the pressure P climbs with it, which is exactly this relation.
Hold the temperature fixed and PV stays constant. Drag the volume and the point moves along an inverse curve, a hyperbola. This is the trace of an isothermal process. Just as blocking a syringe's tip and slowly pushing to halve the volume doubles the pressure, when the temperature is held constant P and V move in opposite directions.
See two processes with the buttons. Isobaric fixes the pressure, so it is a horizontal line; isochoric fixes the volume, so it is a vertical line. The two simplest traces. Throw a sealed can into a fire and the volume stays fixed while only the temperature rises, so the pressure shoots up, which is isochoric; warm the gas under a freely moving piston and it swells while holding atmospheric pressure, which is isobaric.
An adiabatic process exchanges no heat, so PVγ stays constant. Starting from the same point, the adiabatic curve is steeper than the isotherm, because the gas also cools as it expands. Air rising over a mountain, expanding and cooling to make clouds and rain, and a spray can's nozzle turning cold as you spray, are both adiabatic: expanding too fast for heat to enter or leave, so the temperature drops.
Finally, overlay the four processes at one spot. Switch isothermal, isobaric, isochoric, and adiabatic with the buttons and you see at a glance how differently each trace leaves the same start. Since the area under the curve was the work, a different process means different work. In the next chapter we move to heat engines built from these cycles, and the second law. Real car engines and refrigerators, too, run on a cycle that strings these processes into a loop, and the area that closed loop encloses on the P-V plane is the net work exchanged over one turn.