Chapter 1 · The Chemical World
1.4Analyzing and Interpreting Data
12 min · two checks
Predict
If a graph of mass versus volume is a straight line through the origin, what kind of relationship is that?
The idea
- Read a proportional relationship off a graph.
- Tell interpolation from extrapolation and say which is riskier.
Data become an argument only after you look for a pattern. Suppose you measure the mass of five samples of the same liquid and plot mass on the vertical axis against volume on the horizontal axis. If the points fall on a straight line that passes through (0, 0), mass and volume are directly proportional: doubling the volume doubles the mass. The slope, rise over run, has units of mass over volume. That slope is the density.
An inverse relationship bends the other way: as one quantity grows, the other shrinks so that their product stays roughly constant. When you estimate a value between measured points you are interpolating, which is usually safe if the trend is smooth. Extending the line past the last point is extrapolation. It is a hypothesis about unmeasured territory, not a measurement. Also check the axis scale before calling a line “steep.” A zoomed axis can dramatize a tiny change.
Keep these
- A straight line through the origin means y = kx. The slope is k, with units.
- Interpolation estimates inside the data; extrapolation guesses outside it.
- Read units, scale, and scatter before you trust the story a graph seems to tell.
Worked path
A graph of mass (g) versus volume (mL) for ethanol is linear through the origin. One point is 15.8 g at 20.0 mL. Estimate the density and the mass of 55.0 mL.
Observe
The line through the origin means mass and volume are proportional. Density is the slope.