5.9Modeling Using Variation
Section · module m51281
What leans on this
Chemistry that uses this algebra — the answer to why you are learning it.
The end-of-chapter exercise asks why density = mass/volume is intensive when mass and volume are each extensive — the answer is that both are directly proportional to the amount of matter, so their ratio behaves like the constant k in y = kx and is scale-invariant; a student unfamiliar with what direct proportionality formally guarantees can state the fact but not explain it.
- Alongside3.2Electromagnetic Energy
c = λν is used throughout this section as an inverse relationship — as λ increases, ν decreases for fixed c — to answer qualitative ranking questions (which region of the spectrum has the higher frequency) without solving an equation; naming this explicitly as inverse variation sharpens the chemistry reasoning, and the chemistry gives the algebra concept a concrete home, but a student can still get the rankings right from plain proportional reasoning.
- Alongside3.4Development of Quantum Theory
The de Broglie wavelength λ = h/(mv) is explained explicitly through inverse proportionality — 'de Broglie wavelength is inversely proportional to both particle mass and velocity' — and explaining why a thrown softball has no detectable wavelength depends on recognizing that inverse variation drives λ toward zero as mv grows, a reasoning move that naming as formal variation sharpens without strictly requiring it.
- Applies5.2Valence Bond Theory
The Coulomb force F = Q₁Q₂/d² is walked through as an explicit inverse-square comparison — doubling the charges while halving nothing versus doubling the separation — the same joint/inverse structure as a variation problem, just never named as one.
- Applies6.4Molarity
Because moles of solute are held constant while diluting, M (or C) is inversely proportional to V — C₁V₁=C₂V₂ is the same inverse-variation pattern as any 'holding one product constant' relationship, just introduced by name as 'the dilution equation' rather than as variation.
Amontons's law (P∝T), Charles's law (V∝T), and Boyle's law (P∝1/V) are each stated in the text as explicit proportionalities before being converted to the P₁/T₁=P₂/T₂ or P₁V₁=P₂V₂ working forms — the same direct- and inverse-variation vocabulary as a variation-modeling problem, just introduced under the historical names of the laws.
Lattice energy ΔHlattice = C(Z⁺)(Z⁻)/Ro is a joint-variation relationship — directly proportional to the product of the ionic charges and inversely proportional to interionic distance — which is why doubling both charges while holding Ro fixed quadruples the lattice energy rather than doubling it.
- Applies11.4Solubility
Henry's law Cg = kPg is direct variation by name; the worked examples find k = Cg/Pg from one data point and reuse that same constant to predict solubility at a new pressure, the canonical define-k-then-reuse-it pattern.
- Applies11.5Colligative Properties
ΔTb = Kb·m, ΔTf = Kf·m, Π = MRT (at fixed T), and Raoult's law PA = XA·PA* are four separate colligative properties built from the identical y = kx direct-variation pattern with a different proportionality constant each time.
- Applies18.2Periodicity
Recovering the mass of fish that delivers a fatal 0.20 g HgCl₂ dose from a 30 ppm mercury concentration treats ppm as a direct-variation constant (mass Hg = k × mass fish) and solves for the fish mass at a fixed dose — the same y = kx inversion Modeling Using Variation trains, and mixing up which quantity is the constant of variation misplaces a decimal by orders of magnitude.
Computing Δ_oct from an absorbed wavelength chains ν = c/λ into E = hν — two variation substitutions in a row — and the classic wrong intuition (longer wavelength means more energy) is exactly the error variation problems are built to catch, since E is inversely proportional to λ, not directly.