1.7.26 · D5Thermodynamics
Question bank — Thermodynamic potentials — U, H, F, G (preview)
Before we start, three words we will lean on constantly, built from zero:
Reminders of the four differentials we will keep quoting (from the parent):
Two pictures to hold in your head as you work through the traps:
True or false — justify
holds for any process, reversible or not.
False — as an equality of state-function changes it always holds (all terms are exact differentials), but the identifications and only hold for a reversible path. So the equation is universal; its interpretation term-by-term is not.
The natural variables of are and .
False — they are and , because is expressed in terms of and . You get as an output (), not as an input knob.
is always the potential that is minimized at equilibrium.
False — is minimized only at constant and . Change the constraints (say constant ) and becomes the boss instead. The right potential is the one whose natural variables you are holding fixed.
"Free energy" means energy you can extract at no cost.
False — it is the maximum useful work extractable at fixed (and for ). The term is energy locked away by the second law as a "tax" to the surroundings; "free" means available for work, not free of charge.
For a spontaneous reaction at constant , of the system must decrease.
True — at constant the total entropy of the universe increasing is exactly equivalent to . That translation is why chemists watch .
Enthalpy equals the heat content of a body in general.
False — only at constant pressure (and along a reversible path) does . Call "heat content" only in that setting; at constant volume it is whose change equals the heat instead.
can be larger than .
False under normal conditions — with the usual positive absolute temperature () and non-negative entropy (), the product , so is always less than or equal to . The entropy tax only ever subtracts; it never adds.
A Maxwell relation is an extra experimental law of nature.
False — it is pure mathematics: equality of mixed second partial derivatives of a state function (order of differentiation doesn't matter). No new physics, just the fact that are exact differentials. See Maxwell Relations.
Spot the error
", by analogy with ."
Wrong sign on the term. Adding gives ; the cancels the in and what survives is . Correct: .
", therefore ."
Missed the minus sign. Matching the coefficient gives , so . The extra negative is what makes entropy come out positive.
"Since , at constant minimizing is the same as minimizing ."
No — the term matters. A reaction can be endothermic () yet spontaneous because a large positive makes dominate and . Entropy, not just heat, decides.
", so temperature is a derivative of volume."
Confused which variable is fixed. is the derivative of with respect to holding constant; volume is the frozen variable, not the thing being differentiated.
"The Legendre transform changes the physics of the system."
It changes only the description, swapping which variable is the independent knob (e.g. ) by subtracting off the product . Same system, same physics, more convenient bookkeeping. See Legendre Transform.
"."
Mixes potentials. Heat capacity at constant pressure uses enthalpy: , because at constant it is (not ) whose change equals the heat. . See Heat Capacities Cp and Cv.
Why questions
Why do we need four potentials instead of just ?
Because different experiments fix different variables. You rarely control entropy, but you can control (thermostat) and (open beaker). Each potential is engineered so the variables you actually hold fixed become its natural variables.
Why does subtracting "promote" to a natural variable?
Because , and the piece exactly cancels the inside . What's left is an term, so (hence ) now appears as the differential — the knob.
Why is the "chemistry" potential?
Reactions in open flasks run at atmospheric pressure and room temperature — constant and — which are precisely the natural variables of . Its minimum is the equilibrium of that exact lab setup. See Gibbs Free Energy and Chemical Equilibrium.
Why does "entropy of the universe maximized" turn into "a system potential minimized"?
For a system exchanging heat/work with surroundings at fixed , the surroundings' entropy change can be written as . Combining with the system's own shows: total entropy up (i.e. if and only if) down. The constraint just re-expresses the second law.
Why does have a minus on the but potentials for have a plus on ?
The minus encodes that the system loses energy when it does expansion work ( of work leaving). When we Legendre-transform to make natural, the added flips it to : a bookkeeping sign, not a physics change.
Why can a Maxwell relation be useful in practice?
It swaps a hard-to-measure quantity for an easy one. E.g. replaces an entropy slope (unmeasurable directly) with a pressure–temperature slope you read off a gauge. See Maxwell Relations.
Edge cases
At constant volume, does still equal the heat absorbed?
No — at constant (and along a reversible path) it is that equals the heat ( when ). requires constant pressure. Mixing these up is the classic trap.
What happens to the entropy tax in as ?
As , , so : at absolute zero the free energy and internal energy coincide. All of the internal energy becomes "available," because there is no thermal disorder to pay for.
If a process holds and fixed, which potential decides equilibrium?
— because are the natural variables of , and at fixed the second law forces to a minimum. This is the isolated, rigid box (rare in practice, but the conceptual anchor).
For an incompressible solid (), how do and , and and , relate?
With nearly constant and modest, changes are tiny, so and . The distinction between the "" pairs collapses when volume can't change — which is why solid-state work often ignores it.
Can ever describe a real observed reaction?
Yes, but only driven — coupled to another process with a larger (e.g. biochemistry coupling to ATP hydrolysis). Alone, at constant , a positive reaction does not proceed spontaneously in the forward direction.
Is meaningful if is held fixed everywhere (adiabatic)?
The derivative is a local slope of the -vs- surface; it's defined whether or not any given process changes . Even in an adiabatic step where stays constant, that slope still equals the system's temperature.
What if two potentials share the same value — does that fix the state?
Not necessarily. A single number never pins down the state; you need the natural-variable pair. Equilibrium is about a potential being at its minimum over the allowed states, not about numerical coincidences between potentials.
Recall One-line self-test
Name the constraint that makes each potential the boss. ::: constant ::: constant ::: constant ::: constant