P-V diagram boundary work

First feel how the work arises. The gas inside the cylinder is pushing the piston, and when you expand the gas the piston is pushed outward and work is done. The force on the piston is pressure times area, and when that force moves the piston a little, that much work piles up, which is the work bar filling on the screen. The point is that a gas does work on the outside world when it increases its volume. A car engine pushing a piston with exploded gas to turn the wheels, and a balloon swelling and pushing the air around it, are both this expansion work. Hold the piston so it cannot move and however hard the gas presses, the distance pushed is 0 and 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 can be taken as nearly constant. The work of that short stretch is the pressure times the volume change of that stretch, the area of one thin rectangle. Increase the volume with the slider from V1 to V2 and thin bars add one by one and the work piles up. The sum of all these finely split bars is the integral, so the boundary work is written W = ∫P dV. Whether the pressure changes or not, multiply the pressure of each instant by the volume change of that instant and add along, and that is the work the gas did. The finer you split the bars, the closer the staircase clings to the real curve, and in the limit of infinitely thin slices this sum approaches the real area under the curve exactly.

Adding every split bar is in the end finding the area under the curve, so on a P-V diagram the boundary work is the area between the curve and the horizontal axis. In the simplest case, hold the pressure constant and expand, and adjust the pressure with the slider, and the work is just the rectangle area W = P (V2 − V1). The higher the pressure, the larger the area for the same volume change, so the work grows, and heating an ideal gas and expanding it at constant pressure is exactly this case. Seen as an area, the size of the work is plain at a glance, and just as a narrow tall rectangle and a wide low rectangle can give the same work, it is not pressure alone or volume change alone but the area of the two multiplied that sets the work.

Now the heart of it. Even going from the same start to the same end, the path can branch several ways, and switching the two paths with the buttons shows one path expanding first while holding a high pressure and the other lowering the pressure first and then expanding. The areas under the two curves differ plainly, so the work done differs even though the start and the end are the same. So work is not a property of the state. You cannot know it from the state alone, and you have to know which process was taken. Expand an ideal gas at constant temperature and the path is a smooth curve on which the product of pressure and volume stays constant, and that area is different again. A different path means a different area, and a different area means different work. A heat engine uses exactly this path dependence, expanding at high pressure to draw out a large work and returning at low pressure while giving back only a small work, and it keeps the difference as net work each turn.

Last is the sign agreement. Move the volume to the left and the right of the start with the slider. In an expansion where the volume grows, the gas does work on the outside, so the work is agreed to be 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 center line. This agreement pairs with the 1st law of thermodynamics that the next lesson meets. Work the gas does is positive and work the gas receives is negative, so a device that sends work out as the gas expands, like an engine, handles positive work, and a device that squeezes gas in, like a compressor, handles negative work. The next lesson ties this work together with heat into conservation of energy, the law 1. Press a bicycle pump fast and compress the air, and in this compression where the work is negative the energy the outside hands the gas piles up inside and you can feel the pump grow warm in your hand.