Boundary Work Is the Area Under the P-V Curve
Let us push out the piston of a cylinder full of gas. Drag the volume up and the gas pushes the piston outward, doing work. The force on the piston is the gas's pressure times the piston's area, and when that force moves the piston through a distance, that is work. Add up the little bits of work as it pushes out, and you get pressure times change in volume, summed all the way along. Draw this on a P-V diagram and, remarkably, it becomes the area under the curve. The work the gas does shows up as an area in the picture. So even with the same start and end, the area, and therefore the work, depends on which path you took. This is the first step of the chapter.
First, get a feel for how work arises. The gas in the cylinder is pushing on the piston. Drag the volume to expand the gas, and the piston is pushed outward, doing work. The force on the piston is pressure times area, and as that force moves the piston a little, that much work piles up. Watch the work bar fill on screen. The key is that when a gas grows in volume, it does work on the outside world. A car engine pushing the piston with exploded gas to turn the wheels, and a balloon swelling to shove the surrounding air aside, are both this expansion work. Conversely, if you clamp the piston so it cannot move, then no matter how hard the gas presses, the distance it pushes through is zero, so no work arises at all.
Now count that work exactly. Even if the pressure is not constant and changes as the gas expands, over a very short stretch the pressure is nearly constant. The work over that short stretch is pressure times the change in volume there, that is, the area of one thin rectangle. Use the slider to grow the volume from V1 to V2. The thin bars add one by one and the work piles up. Adding up all these finely sliced bars is integration, which is why boundary work is written W = ∫P dV. Whether the pressure changes or not, multiply the pressure at each instant by the change in volume at that instant and add it all up, and that is the work the gas does. The finer you slice the bars, the more the staircase hugs the true curve, so in the limit of infinitely thin slices this sum closes in exactly on the real area under the curve.
Adding up all the sliced bars means, in the end, finding the area under the curve. So on a P-V diagram the boundary work is just the area between the curve and the horizontal axis. Take the simplest case and expand at constant pressure. As you adjust the pressure with the slider, the work is simply the area of a rectangle, W = P times (V2 − V1). The higher the pressure, the larger the area for the same change in volume, so the more work. For reference, heating an ideal gas at constant pressure so it expands is exactly this case. Seen as an area, the size of the work is clear at a glance. Because it is an area, a tall narrow rectangle and a short wide one can do the same work, so what sets the work is not the pressure or the volume change alone but the area that multiplies the two.
Now the key point. Even going from the same start to the same end, there can be several paths. Use the buttons to switch between two routes. One path expands while holding a high pressure first, the other lowers the pressure first and then expands. The areas under the two curves are noticeably different. The start and end are the same, yet the work done differs. So work is not a property of state. You cannot tell it from the current state alone; you have to know what process was taken. For example, expanding an ideal gas at constant temperature gives a smooth path with PV held constant, whose area is different again. Different path, different area; different area, different work. A heat engine exploits exactly this path dependence, expanding along a high-pressure path to draw out a lot of work and returning along a low-pressure path that costs only a little, pocketing the difference as net work every cycle.
The last piece is the sign convention. Use the slider to move the volume to the left and right of the starting point. In an expansion, where the volume grows, the gas does work on the outside, so we agree to call the work positive. In a compression, where the volume shrinks, the outside does work on the gas, so the work is negative. The sign flips exactly at the reference line in the middle. This convention pairs with the first law of thermodynamics you will meet next lesson: work done by the gas is positive, work done on the gas is negative. So a device like an engine, where gas expands and sends out work, deals in positive work, while a compressor, which crams gas in, deals in negative work. In the next lesson, we bind this work together with heat into energy conservation, the first law. Press a bicycle pump quickly to compress the air, and in this compression, where the work counts as negative, the energy the outside hands to the gas piles up inside, so you can feel the pump grow warm in your hand.