4.2Linear Functions
Section · module m51270
What leans on this
Chemistry that uses this algebra — the answer to why you are learning it.
The Celsius–Fahrenheit formula is built in front of the student as slope-intercept form — m = Δy/Δx from the anchor points (0°C, 32°F) and (100°C, 212°F), then b = y − mx — so seeing y = mx + b assembled from two real data points here reinforces, and is reinforced by, the same construction taught for linear functions, without either one strictly gating the other.
- Applies10.4Phase Transitions
The text names ln P = −(ΔHvap/R)(1/T) + ln A ‘the linear equation’ outright, with 1/T standing in for x and ln P for y — a slope-intercept form where the slope is literally a measurable enthalpy of vaporization, which makes ‘what is slope, really’ far more concrete than an abstract y = mx + b example.
- Applies12.5Free Energy
ΔG = ΔH − TΔS is linear in T with slope −ΔS and intercept ΔH, which is why its sign — and hence spontaneity — can flip exactly once as T increases; reading ΔG this way alongside linear functions makes the sign-table in this section (ΔG<0 vs >0) feel like evaluating one line rather than memorizing three cases.
- Applies14.7Buffers
pH = pKa + log([A⁻]/[HA]) is a linear function of the quantity log([A⁻]/[HA]) with slope 1 and intercept pKa — recognizing this is what makes it obvious that equal concentrations (ratio = 1, log = 0) give pH = pKa exactly, the fact used repeatedly at titration half-equivalence points.
The Nernst equation Ecell = E°cell − (0.0592 V/n) log Q is a linear function of log Q with slope −0.0592/n and intercept E°cell — recognizing this linear structure is what lets a student predict how Ecell moves as Q changes without recomputing from scratch each time.
The text writes ln[A]ₜ = −kt + ln[A]₀ directly over y = mx + b (and does the same for 1/[A]ₜ and for [A]ₜ) and asks students to read k off as the negative of a plotted slope and the initial concentration off as the y-intercept; a student who cannot recognize a rearranged equation as a disguised line has no way to know which two quantities on a data plot correspond to slope and intercept, making the graphical-determination examples unworkable rather than just harder.
- Needs17.6Collision Theory
The Arrhenius equation's linear form ln k = (−Eₐ/R)(1/T) + ln A is set beside y = mx + b in the text with the input relabeled 1/T instead of x; a student who has not learned to read a rearranged equation for its slope and intercept will not realize that plotting ln k against 1/T (not against T) is what makes the graph straight, or that the slope equals −Eₐ/R rather than Eₐ itself.