Work is a path function

Take two hikers who start in the same car park and end at the same summit. Any map will tell you they gained the same height.Strictly, the same change in gravitational potential energy, which is what the map is quietly measuring. Nothing on the map will tell you how far they walked, and that number is the one their legs remember. Height gained depends only on where you started and where you ended up. Distance walked depends on the route. The first kind of quantity is called a state function; the second is a path function.

Thermodynamics is largely an exercise in keeping these two kinds of quantities apart. Internal energy, entropy, pressure and temperature are state functions: a system in a given state has one value of each, regardless of its history. Heat and work are not. They are not things a system has; they are things that happen to it on the way from one state to another.This is why textbooks write $dU$ for a change in energy but $\delta Q$ and $\delta W$ , with a different d, for heat and work. The bar is a warning: these are not differences of anything. See Fermi [1]. The distinction is the subject of the first law, and it is the reason the law is worth stating at all.

The diagram everyone draws

The standard picture is a gas in a cylinder with a piston. Push the piston in and you do work on the gas; let it push out and the gas does work on you. The work done in a small step is $\delta W = P\,dV$ , and the total is $W = \int P\,dV$ along whatever route the gas takes through pressure–volume space. That integral is the area under the curve, and the area under a curve depends on the curve.

W = ∫P dV825 JΔU113 JQ = ΔU + W938 J

Figure 1. Drag the control point to change the path from A to B. The shaded area is the work done by the gas; the change in internal energy is fixed by the endpoints alone. Ideal monatomic gas, arbitrary units.

Drag the control point in the figure above. States A and B are fixed, so the change in internal energy is fixed too; that readout does not move. The work does. Take the high road, expanding at high pressure before the pressure falls, and the gas does a great deal of work. Take the low road and it does very little. The difference has to come from somewhere, and it comes from heat: every route has its own heat as well. $Q = \Delta U + W$ in the convention where $W$ is work done by the gas. Chemists usually flip the sign [2]. Either way, $\Delta U$ is the same on every path and $Q$ and $W$ are not.

Why this matters more than it seems

Once you notice the distinction it turns up everywhere. The level of a lake is a state function; the water that has flowed through it is not. The balance of a bank account is a state function; the total that has ever passed through it is not. The first kind of number is what you can look up. The second kind is what actually happened.

State functions are what you can look up. Path functions are what actually happened.

**Figure 2.** An indicator diagram traced by a steam engine, the instrument that first drew pressure against volume. The area inside the loop is the work per stroke.

Careers are path functions

This site is named after the distinction because it is the most useful frame I know for thinking about work in the other sense. A résumé is a list of states: a degree, a title, a company, another title. Every one of them is a state function. Two people can hold the same title at the same company and have taken wildly different routes to get there, done wildly different amounts of work, and learned wildly different things.The interviewer's problem, in these terms, is that they can only observe states and are trying to infer the path.

What you actually accumulate over a career, the skills, the judgement, the people who will pick up the phone, is the integral along the path. It is not a function of the endpoints. This is obvious once said, and yet almost every tool we have for making decisions about work, from job boards to org charts, is a table of states.

Three consequences follow:

The same destination can cost very different amounts. Choose the route, not just the summit.
Nobody can read your path from your position. If it matters, you have to say it.
Reversibility is a property of the path. Some routes can be retraced; most cannot.

Quantity	Kind	Depends on
Elevation	State	Where you are
Distance walked	Path	The route
Internal energy	State	P, V, T
Work, heat	Path	The process
Job title	State	Current position
What you learned	Path	Everything in between

Appendix: computing the work

The figure is a hand-written SVG; there is no charting library. The path is a quadratic Bézier from A to B, and the work is the area under it, which is easiest to get by sampling:

function work(c) {
  let w = 0, prev = A;
  for (let i = 1; i <= 240; i++) {
    const t = i / 240;
    const pt = {
      v: quad(t, A.v, c.v, B.v),
      p: quad(t, A.p, c.p, B.p),
    };
    w += (pt.p + prev.p) / 2 * (pt.v - prev.v);  // trapezium
    prev = pt;
  }
  return w;  // bar·L; ×100 for joules
}

That is the whole computation. The figure re-renders on every pointer move because the arithmetic is cheap enough that there is no reason not to.

**Figure 3.** A wide figure. It keeps the left edge of the text and extends into the note column, which is empty beside a figure anyway.

Next time: entropy, and why the second law is a statement about which paths exist at all.

Bibliography

Fermi, E. Thermodynamics. Dover, 1956. Chapter 1, on the distinction between exact and inexact differentials.
Atkins, P. and de Paula, J. Physical Chemistry, 8th ed. Oxford, 2006. Section 2.3 on the sign convention for work.
Watt, J. and Southern, J. The indicator diagram, c. 1796. See Hills, R. L. Power from Steam. Cambridge, 1989.