4.1Linear Functions
Section · module m51270Read §4.1 at OpenStax ↗
Study this section
Written from the section, each card quoting the span it came from. The ones you miss come back sooner.
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.
- Applies10.3Phase 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.
- Applies12.4Free 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.
- Applies14.6Buffers
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.
- Needs17.5Collision 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.