Gibbs-Helmholtz equation: Foundations — Physics deep dive

This page assumes nothing. Every letter that appears in the parent note gets built here, brick by brick, before you are allowed to use it. If you can count and read a graph, you can follow line one.

0. The three things we can dial: $T$ , $P$ , $V$

Before any energy, there are three state variables — numbers that describe what condition the system is in right now, not how it got there.

Figure s01 — the three dials. A gas box (black outline) holds particles (teal dots) each carrying a jiggle arrow (orange) whose length is temperature

T

. The plum arrows on the right wall are the outward push

P

; the double arrow beneath is the room

V

. Why we need this picture: the phrase "at constant

P

" in Gibbs–Helmholtz only makes sense once you can see

P

as one independent dial you can freeze while turning the

T

dial.

Why the topic needs these: Gibbs–Helmholtz is stated "at constant $P$ ". That phrase is meaningless unless you know $P$ is a dial we can hold fixed while we change $T$ . The little subscript $P$ in $(\frac{\partial \cdot}{\partial T})_{P}$ literally means "keep $P$ frozen."

1. Enthalpy $H$ — the "heat content"

At constant pressure, the change $Δ H$ (the symbol $Δ$ , "delta," just means final minus initial, i.e. how much something changed) equals the heat that flows in or out. That is why chemists love $H$ : it is measurable with a thermometer.

Why the topic needs it: $H$ is the prize. Gibbs–Helmholtz exists to extract $Δ H$ . See Enthalpy.

2. Entropy $S$ — the "spread-out-ness"

The combination $T S$ has units of energy. Picture it as a "disorder tax": the hotter it is ( $T$ large), the more each unit of disorder $S$ costs you in energy. This product $T S$ is the villain we will make disappear.

Why the topic needs it: $S$ is the term we want to cancel. The whole "divide by $T$ " trick exists to make the $T S$ part vanish. See Entropy.

3. Gibbs free energy $G$ — the referee

Figure s02 — $G$ is what's left after the tax. Each orange bar is the full heat energy

H

; the hatched plum cap on top is the disorder tax

T S

, and it grows taller as

T

climbs left→right. The teal line marks what survives,

G = H - T S

. Why we need this picture: it shows in one glance why merely watching

G

shrink as

T

rises tells you about the tax, not the heat

H

— the exact confusion Gibbs–Helmholtz is built to dodge.

Why the topic needs it: $G$ is the quantity we measure (via equilibrium — see below) and the quantity we differentiate. See Gibbs free energy.

4. Helmholtz free energy $A$ — the constant-volume twin

Why the topic needs it: the parent note's twin equation $(\frac{\partial ( A / T )}{\partial ( 1/ T )})_{V} = U$ is the exact mirror of the $G$ version. Learn one, get the other free. See Helmholtz free energy.

5. The differential $d$ and where $d G = - S d T + V d P$ comes from

The parent note asserts $d G = - S d T + V d P$ . Let us actually build it, so no step is a magic trick.

Step A — differentiate $H = U + P V$ . Using the product rule on $P V$ : $d H = d U + P d V + V d P .$ Substitute $d U = T d S - P d V$ ; the $- P d V$ and $+ P d V$ cancel: $d H = T d S + V d P .$

Step B — differentiate $G = H - T S$ . Again the product rule on $T S$ : $d G = d H - T d S - S d T .$ Substitute $d H = T d S + V d P$ ; the $+ T d S$ and $- T d S$ cancel: $d G = - S d T + V d P$

Why this step? Now we have earned the parent's fundamental relation instead of quoting it. Reading off the coefficient of $d T$ at constant $P$ ( $d P = 0$ ) gives the identity the whole derivation hinges on: $(\frac{\partial G}{\partial T})_{P} = - S .$

Why the topic needs it: the derivative is about rates of change, and $d$ is the alphabet of change.

6. The partial derivative $(\frac{\partial G}{\partial T})_{P}$ — "slope while everything else is frozen"

This is the single most-used piece of notation in the parent note, so we build it fully.

Figure s03 — a partial derivative is one directional slope. The teal curve is

G

as a function of

T

with

P

held fixed; the dashed orange line is its tangent (its steepness) at the plum dot. That steepness is

(\frac{\partial G}{\partial T})_{P} = - S

, and it is negative because

G

slopes downward as

T

rises. Why we need this picture: it converts the abstract symbol

\partial G / \partial T

into something you can literally see — the tilt of a tangent line.

Why the topic needs it: the entire equation is a statement about slopes, e.g. $(\frac{\partial G}{\partial T})_{P} = - S$ . Without partials there is no Gibbs–Helmholtz.

7. The quotient rule — worked through for $G / T$

The parent note differentiates $G / T$ , not $G$ . To do that you need one calculus tool, and we will run it all the way to the punchline.

Substitute $f = G$ and $g = T$ . Since $\frac{\partial g}{\partial T} = \frac{\partial T}{\partial T} = 1$ : $(\frac{\partial}{\partial T} \frac{G}{T})_{P} = \frac{T ( \frac{\partial G}{\partial T} ) _{P} - G \cdot 1}{T ^{2}} = \frac{T ( \frac{\partial G}{\partial T} ) _{P} - G}{T ^{2}} .$

Now show the numerator is $- H$ . Start from the definition $G = H - T S$ and the slope we earned in §5: $(\frac{\partial G}{\partial T})_{P} = - S ⟹ S = - (\frac{\partial G}{\partial T})_{P} .$ Put that $S$ back into $G = H - T S$ : $G = H - T [- (\frac{\partial G}{\partial T})_{P}] = H + T (\frac{\partial G}{\partial T})_{P} .$ Rearrange for $H$ : $H = G - T\left(\frac{\partial G}{\partial T}\right)_P \tag{$\star$}$ Multiply $(⋆)$ by $- 1$ : the numerator of our quotient is exactly $T (\frac{\partial G}{\partial T})_{P} - G = - [G - T \partial_{T} G] = - H .$

Assemble the result. $(\frac{\partial ( G / T )}{\partial T})_{P} = \frac{- H}{T ^{2}} = - \frac{H}{T ^{2}}$

8. Changing the variable to $1/ T$ — the explicit chain rule

The chain rule step, done explicitly. The chain rule says a slope in one variable converts to a slope in another by multiplying by the conversion factor: $(\frac{\partial ( G / T )}{\partial ( 1/ T )})_{P} = (\frac{\partial ( G / T )}{\partial T})_{P} \cdot \frac{\partial T}{\partial ( 1/ T )} .$ We need $\frac{\partial T}{\partial ( 1/ T )}$ . Let $u = 1/ T$ , so $T = 1/ u$ and $\frac{\partial T}{\partial u} = \frac{\partial}{\partial u} (\frac{1}{u}) = - \frac{1}{u ^{2}} = - T^{2} .$ Substitute both pieces: $(\frac{\partial ( G / T )}{\partial ( 1/ T )})_{P} = (- \frac{H}{T ^{2}}) \cdot (- T^{2}) = H .$ The two minus signs and the two $T^{2}$ 's cancel exactly: $(\frac{\partial ( G / T )}{\partial ( 1/ T )})_{P} = H$

Equivalently, in differential language $d (1/ T) = - \frac{1}{T ^{2}} d T$ , i.e. $d T = - T^{2} d (1/ T)$ — the same $- T^{2}$ factor that just cancelled the $- 1/ T^{2}$ .

Why the topic needs it: plotting $Δ G / T$ against $1/ T$ gives a straight line whose slope is $Δ H$ . That is the practical payoff.

9. The equilibrium constant $K$ , the gas constant $R$ , and the standard state $^{\circ}$

The bridge $Δ G^{\circ} = - R T ln K$ is how we actually measure $Δ G$ : measure $K$ in the lab, take a log, multiply by $- R T$ (with $R = 8.314 J mol^{- 1} K^{- 1}$ ).

Why the topic needs it: this is why $Δ G$ is "the thing we can measure," and feeding it into Gibbs–Helmholtz produces the van 't Hoff equation.

10. When is Gibbs–Helmholtz allowed? (validity conditions)

Common mistake The hidden assumptions you must not forget

The clean result $(\partial (G / T) / \partial T)_{P} = - H / T^{2}$ silently assumes several things. State them or get burned:

Constant pressure — the whole derivation used $d P = 0$ ; that is why the subscript $P$ is there.
Only pressure–volume work — we used $d U = T d S - P d V$ , which has no extra work terms. If the system also does electrical, surface, or other non- $P V$ work, an extra term appears and the simple equation no longer holds as written.
Fixed composition — no matter is added or removed and the reaction extent is held for the derivative; a changing amount of substance adds chemical-potential terms.
For the integrated two-point formula: $Δ H$ roughly constant over the temperature range (it actually drifts via $Δ C_{p}$ — see Kirchhoff's law (thermochemistry)).

Why this matters: Gibbs–Helmholtz is a theorem, not a slogan — it is only true inside these fences.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer is a surprise, reread that section.

What does the subscript

P

(\frac{\partial G}{\partial T})_{P}

mean?

Hold pressure fixed while nudging only

T

Define enthalpy

H

in one formula.

H = U + P V

(internal energy plus room-making energy).

What does

Δ

in front of a quantity mean?

Final value minus initial value (the change).

What is entropy

S

, in a picture?

How spread-out / disordered the energy is (ink dispersing in water).

Write the definition of Gibbs energy

G

G = H - T S

Which identity do we start the

d G

derivation from?

d U = T d S - P d V

(the fundamental thermodynamic identity).

Derive

d G

in terms of

d T

and

d P

d G = - S d T + V d P

(via

d H = T d S + V d P

, then

G = H - T S

Why is

G

called "free" energy?

It's the part of

H

left free for work after paying the

T S

disorder tax.

What is the Helmholtz twin

A

A = U - T S

(constant-volume analogue of

G

State the quotient rule for

f / g

\frac{g \partial _{T} f - f \partial _{T} g}{g ^{2}}

After substituting

f = G, g = T

, what is the numerator, and what does it equal?

T \partial_{T} G - G

, which equals

- H

(⋆)

Compute

\partial T / \partial (1/ T)

- T^{2}

(let

u = 1/ T

T = 1/ u

d T / d u = - 1/ u^{2} = - T^{2}

Why does the sign flip between the

T

form and the

1/ T

form?

The

- T^{2}

chain-rule factor cancels the

- 1/ T^{2}

, leaving

+ H

Give the value and units of the gas constant

R

R = 8.314 J mol^{- 1} K^{- 1}

What does the superscript

^{\circ}

Δ G^{\circ}

mean?

Standard state: reference conditions of

1 bar

(and

1 mol/L

for solutes) at the stated

T

How do we actually measure

Δ G

in the lab?

From equilibrium:

Δ G^{\circ} = - R T ln K

Name three assumptions Gibbs–Helmholtz needs.

Constant

P

, only

P V

work, fixed composition (plus constant

Δ H

for the integrated form).

Connections

Parent topic — everything here feeds it.
Gibbs free energy — the quantity we differentiate.
Enthalpy — the prize $H$ we extract.
Entropy — the term that cancels.
Helmholtz free energy — the constant-volume twin $A$ .
Maxwell relations — the same partial-derivative machinery.
van 't Hoff equation — what you get after plugging in $K$ .
Kirchhoff's law (thermochemistry) — how $Δ H$ drifts with $T$ .