Drop a cold pack's contents into water and it gets colder, even though nothing forced it to. Roll a single ice cube out of the freezer and it melts, but only above 0 °C. Hold a carrot up to the light and it is orange, while the chain of carbon atoms inside it is nothing but carbon and hydrogen. Three ordinary moments, three questions: which way does it go, why does it stop where it stops, and why that colour?
Chemistry has a branch that answers these with numbers rather than recipes. It is called physical chemistry, and its graduates keep a small set of formulas in their heads as other people keep a few phone numbers: the Gibbs energy, the Boltzmann factor, the energy levels of a particle in a box, the energy of the hydrogen atom, and the photon relation E = hν.
The second law (see Thermodynamics, 34/14) says total entropy must not fall. A chemist rarely wants to track the surroundings, so the book-keeping is rolled into one number for the reaction itself, measured at a fixed temperature and pressure. Two things compete. Enthalpy ΔH: does the reaction release heat (negative, favourable) or absorb it? Entropy ΔS: does it spread energy and matter over more arrangements (positive, favourable)? The temperature decides how much the entropy side is worth:
That minus sign is the whole story of melting. Ice melting takes in 6.01 kJ per mole (ΔH > 0) and creates a lot of disorder (ΔS = +22.0 J/mol·K). At low temperature the heat term wins and ice stays. When T·ΔS grows to match ΔH, at T = ΔH ÷ ΔS = 6010 ÷ 22.0 = 273 K, the two balance, and above it the liquid wins. Limestone behaves the same way, only hotter: CaCO₃ → CaO + CO₂ has ΔH about +178 kJ/mol and ΔS about +159 J/mol·K, so ΔG changes sign near 1,120 K, around 850 °C, which is why lime kilns run hot. The ammonia reaction has both terms negative: heat released but order created, so the sign flips the other way, favourable below roughly 460 K and unfavourable above.
ΔH and ΔS are held constant with temperature, which is a good approximation over a modest range only. Preset values are rounded textbook table values for standard conditions (ammonia per mole of reaction, 2 NH₃). R = 8.314 J/mol·K (CODATA).
The Gibbs energy does more than say yes or no. Under standard conditions, its value tells you exactly where the reaction stops, as an equilibrium constant K:
A worked number: for A ⇌ B with ΔG° = −10 kJ/mol at 25 °C, K = e10000 ÷ (8.314 × 298.15) = 56.5, so at equilibrium B outnumbers A 56 to 1, 98.3% B. The relation is gentle in one direction and ruthless in the other. A reaction with ΔG° = +10 is not "forbidden", merely 1.7% converted. One with ΔG° = +40 kJ/mol has K near 10⁻⁷ and is, for practical purposes, stuck.
A ⇌ B with equal-sized standard states, so K = [B]/[A] = e−ΔG°/RT and the fraction of B is K/(1+K). ΔG° is held fixed as T changes (a simplification).
Why should a reaction stop where it does? Think about a molecule that can sit in one low-energy state or in any of several higher ones. At temperature T each molecule borrows energy from jostling neighbours, and the chance that it is in a state with extra energy ΔE is smaller by the Boltzmann factor e−ΔE/kT. But there can be many such states. Ludwig Boltzmann's formula, which is engraved on his gravestone in Vienna, counts them: S = k ln W, where W is the number of arrangements. Put the two together:
So a high-energy form wins at high temperature even though it is higher in energy, simply because it has more ways to be. This is ΔG = ΔH − TΔS in disguise: the ΔH is the energy gap and the ΔS is R ln g. At room temperature RT is only 2.48 kJ/mol (kT = 25.7 meV), which is why the vibrations and rotations of molecules are active at room temperature but the electrons stay in their ground state.
Quantum mechanics (see Quantum Mechanics, 34/83) says a trapped particle can only have certain energies. The simplest case is a particle in a one-dimensional box of length L, where the allowed energies are
The energies climb as n², and a longer box has lower and more closely spaced levels. Now the leap that earned this model its place in the course: treat the π electrons of a chain of alternating single and double bonds as free particles in a box along the chain. N electrons fill the levels two at a time, up to level N/2. The smallest jump a photon can drive is from level N/2 up to N/2 + 1, with gap (N + 1)h² ÷ (8mL²). A longer chain makes a longer box, a smaller gap, and a photon of lower energy, so absorption moves from the ultraviolet into the visible. With L = N × 140 pm (about the carbon–carbon bond length) butadiene (N = 4) comes out at 207 nm; the measured value is about 217 nm.
Be honest about its limits. For longer chains the model keeps falling behind: octatetraene it puts at 460 nm against a measured value near 290 nm, and β-carotene, the pigment of carrots, around 1,360 nm against about 450. Real chains have alternating bond lengths that open a larger gap than a uniform box allows. The lesson is the trend, not the digits: this crude picture explains why chain length steers colour, and a later course refines it.
Box length L = N × (length per atom). Measured absorption maxima, approximate textbook values: ethene 165 nm, butadiene 217 nm, hexatriene 258 nm, octatetraene 290 nm, β-carotene about 450 nm. The swatch colours are rough, from a simple wavelength-to-RGB formula.
Put one electron near one proton and Schrödinger's equation (1926) can be solved completely. The energies are
The levels crowd together as n grows and reach zero at n = ∞, where the electron is free. The solution also hands you the orbitals, labelled by the shape number l: s (l = 0, spherical), p (l = 1, two lobes), d (l = 2). The size is set by the Bohr radius a₀ = 52.9 pm. The 1s electron is most likely found at exactly one a₀. A 2p electron is most likely at four, a 3d at nine. Every orbital has n − l − 1 radial nodes, spots where the chance of finding the electron is zero. Pick one and see.
Every other atom is harder, and chemistry borrows these hydrogen-like orbitals as the vocabulary for them: an s, p or d label in an electron configuration is a hydrogen answer lent out.
Radial probability P(r) = r²R(r)², with R the hydrogen radial functions; each was normalised numerically to 1.0000. Distances in Bohr radii (a₀ = 52.918 pm).
Matter is a ladder of allowed energies, and light is the way to measure the rungs. A molecule absorbs a photon only if the photon's energy matches the gap between two levels exactly. Johann Balmer fitted the visible lines of hydrogen in 1885, and Niels Bohr explained them in 1913 as jumps between the n levels above. A jump from n = 3 to n = 2 releases 1.89 eV, a red photon of 656.5 nm in vacuum (the famous H-alpha line, 656.3 nm in air). The unifying relation:
What kind of rung you measure decides what kind of light you need. Electrons moving between orbitals take UV and visible photons. Molecules stretching and bending like springs, a C–H bond near 3,000 cm⁻¹, take infrared photons of about 0.37 eV. Molecules rotating take microwaves: a 2.45 GHz photon has an energy near 10 µeV, tens of thousands of times smaller. Core electrons take X-rays. Ahmed Zewail, who received the 1999 Nobel Prize in Chemistry for femtosecond spectroscopy, used flashes of light lasting only 10⁻¹⁵ s to watch the bonds in a reaction break and form. The detail of the instruments belongs to Analytical Chemistry & Spectroscopy (35/42).
Region boundaries are conventional and approximate. 1 eV = 96.485 kJ/mol; thermal energy kT at 298 K is 25.7 meV (CODATA constants).
A reaction goes where Gibbs energy falls, ΔG = ΔH − TΔS, and stops at an equilibrium whose constant is e−ΔG°/RT. Under that is counting: states with more arrangements are favoured as temperature rises. Confinement quantises energy, light measures the gaps, and the colour of a molecule is the gap in its electrons.
Try this: watch ice melt in a glass at room temperature, then in a bowl of salty water. The salt lowers the freezing point. Ask which way the T·ΔS term is pulling. Also notice that anything coloured, whether a leaf, a juice or a dye, is absorbing the opposite colour.
Sources. Gibbs energy, K = e−ΔG°/RT, the Boltzmann factor, particle in a box, hydrogen-atom energies and radial functions, and photon relations are standard textbook physical chemistry (Atkins, de Paula and Keeler). Thermodynamic data (ice fusion 6.01 kJ/mol; NH₃, CaCO₃, CaO and CO₂ formation enthalpies and standard entropies) are standard table values (CRC Handbook-style); the ΔS for limestone and ammonia were summed from them. Constants: CODATA 2018. Nobel Prize in Chemistry 1999 (Zewail): nobelprize.org via the Wikipedia list of laureates. Balmer 1885 and Bohr 1913 are standard history-of-science dates. The free-electron model of conjugated dyes was worked out in the late 1940s; Hans Kuhn is the name usually attached. Measured absorption maxima are approximate textbook values. All interactive numbers were computed and checked in node.