Measurement, significant figures and dimensional analysis: use SI units, significant figures and scientific notation, convert units by dimensional analysis, and calculate density and percent error.

What this topic is asking

This part of standard CH.1 is the numeracy of chemistry. Virginia expects you to measure and report quantities in SI units, apply the rules for significant figures, use scientific notation for very large or very small numbers, convert between units by dimensional analysis (the factor-label method), and calculate density and percent error. The on-screen calculator does the arithmetic; the marks are for setting the calculation up correctly and reporting the answer to a sensible precision.

SI units and scientific notation

Chemistry uses the metric (SI) system: mass in grams or kilograms, volume in liters or milliliters, length in meters or centimeters, temperature in kelvin or degrees Celsius. Prefixes scale the base unit: kilo- ($10^3$), centi- ($10^{-2}$), milli- ($10^{-3}$), micro- ($10^{-6}$), nano- ($10^{-9}$).

Significant figures

Significant figures show how precisely a quantity was measured. The rules:

• All nonzero digits are significant ($4.56$ has three).
• Zeros between nonzero digits are significant ($1005$ has four).
• Leading zeros (before the first nonzero digit) are not significant ($0.0042$ has two).
• Trailing zeros after a decimal point are significant ($2.300$ has four).
• Trailing zeros in a whole number with no decimal point are ambiguous; scientific notation removes the ambiguity ($2.0 \times 10^{2}$ has two).

In calculations: when multiplying or dividing, the answer keeps the fewest significant figures of any factor. When adding or subtracting, the answer keeps the fewest decimal places. Round only at the end.

Dimensional analysis

Set the calculation up so the unit you are leaving cancels diagonally. If your units do not cancel to what the question wants, a factor is upside down.

Density and percent error

Two formulas appear constantly:

Density relates a substance's mass to the space it fills (g/mL for liquids and solids, g/L for gases) and is an intensive property that identifies a substance. Percent error reports how far a measured value is from the accepted value, as a percentage; the absolute-value bars keep it positive.

Try this

Q1. Express $0.000\,067$ in scientific notation. [1 point]

• Cue. $6.7 \times 10^{-5}$ (move the decimal five places right, so the exponent is negative five).

Q2. A student measures a boiling point of $98.6\,^{\circ}\text{C}$ for a liquid whose accepted boiling point is $100.0\,^{\circ}\text{C}$. Calculate the percent error. [2 points]

• Cue. $\left|\dfrac{98.6 - 100.0}{100.0}\right| \times 100 = 1.4\%$.