One equation to seed them all
Everything in this rung flows from a single compact statement, the fundamental thermodynamic relation: dU = T·dS − P·dV. In words, the tiny change in a system's internal energy equals the heat it soaks up (temperature times the change in entropy) minus the work it does pushing out its boundary (pressure times the change in volume). It is just the first law and the second law, fused into one line and written for the smallest of steps.
This little equation says something profound: the *natural* variables of internal energy are entropy S and volume V. If you know U as a function of S and V, you know everything about the system — its temperature, its pressure, all of it — just by looking at how U slopes. The whole rest of this guide is a series of clever rearrangements that swap those awkward natural variables for ones you can actually control on a bench.
Four potentials for four sets of conditions
Starting from U(S,V), a simple algebraic move (adding or subtracting a product like P·V or T·S) trades one natural variable for its partner. Do this systematically and you generate exactly four thermodynamic potentials, each tailored to a different pair of variables you might choose to hold fixed.
- Internal energy U(S,V) — natural for fixed entropy and volume.
- Enthalpy H = U + P·V, with natural variables S and P — natural for fixed entropy and pressure.
- Helmholtz energy A = U − T·S, natural variables T and V — the everyday choice for a sealed, fixed-volume vessel.
- Gibbs energy G = H − T·S, natural variables T and P — the everyday choice for an open beaker at fixed pressure.
Each potential carries its own short relation, and Gibbs energy's is the most used of all: dG = −S·dT + V·dP (for a fixed amount of substance). Read off its slopes and two facts pop out for free. Raise the temperature and G falls (the slope is −S, and entropy is always positive). Raise the pressure and G rises (the slope is +V). Those two innocent-looking statements quietly govern why substances boil, why high pressure favours the denser phase, and much more.
Maxwell relations: free measurements
Now comes a piece of pure mathematical magic. For any smooth function of two variables, the order in which you take two successive slopes does not matter. Apply that humble fact to each potential and out drop the Maxwell relations — a set of surprising equalities between rates of change that, on the surface, have nothing to do with each other.
Their value is intensely practical. One Maxwell relation says that how entropy changes when you squeeze a substance equals how its volume changes when you heat it. The second quantity — thermal expansion — is trivial to measure with a ruler and a thermometer. The first — an entropy change under pressure — is nearly impossible to measure directly. The relation hands you the impossible one for free, in exchange for the easy one. That is the everyday work these identities do throughout physical chemistry.
Gibbs–Helmholtz: how free energy bends with temperature
We know G tilts with temperature, but a tabulated ΔG° is usually quoted at just one temperature, 25 °C. How do you find it somewhere else? The Gibbs–Helmholtz equation answers exactly this. It says, in clean form, that the way ΔG/T changes with temperature is governed entirely by ΔH. In other words: if you know the enthalpy of a reaction, you can predict how its free energy — and therefore its equilibrium constant — drifts as you warm or cool the system.
This is enormously useful. It is why you can predict that warming an endothermic reaction (positive ΔH) pushes its equilibrium toward products, while warming an exothermic one pushes it back toward reactants — the rule of thumb you may have met as Le Chatelier's response to temperature. Gibbs–Helmholtz is the rigorous engine underneath that rule, and it is the direct ancestor of the equation used in the equilibrium chapter to chart K against temperature.
Fugacity: keeping the tidy formulas for messy gases
There is one last loose thread. The clean formulas that relate free energy to pressure assume a perfectly ideal gas — molecules that ignore each other. Real gases, especially when squeezed or cooled, attract and jostle, and the tidy equations start to lie. Rather than throw the equations away, chemists patch them. They invent an effective pressure that makes a real gas obey the ideal formulas exactly. That repaired pressure is the fugacity.
Fugacity is the pressure-world cousin of activity, the effective-concentration idea from the equilibrium guide. The recipe is simple: wherever an ideal-gas formula calls for pressure, slip in the fugacity instead, and every clean relation — including the link to chemical potential — works for a real gas with no rewriting. At low pressure, where gases behave ideally, fugacity equals pressure and the patch vanishes. At high pressure it quietly carries the real-gas corrections, so the elegant machinery of free energy never has to be abandoned.