Numbers, Units & Measurement

Chemistry is a quantitative science, and every measurement has three parts: a number, a unit, and an implied uncertainty. Without units, a number is meaningless — and potentially dangerous (imagine a drug dose of “100” without specifying milligrams vs. grams).

The International System of Units (SI) defines seven base units from which all other units are derived:

• Length: meter (m) — about 39.37 inches, or 3 inches longer than a yard
• Mass: kilogram (kg) — about 2.2 pounds; the gram (g) = 10^-3 kg
• Time: second (s)
• Temperature: kelvin (K) — an absolute scale where 0 K is the lowest possible temperature
• Amount of substance: mole (mol)
• Electric current: ampere (A)
• Luminous intensity: candela (cd)

In chemistry, the four most commonly used base units are the meter, kilogram, second, and mole. Other units (like volume and density) are derived from these.

SI units scale by powers of 10 using prefixes attached to the base unit name. This makes it easy to express very large or very small quantities without writing long strings of zeros.

The highlighted prefixes — kilo-, centi-, milli-, micro-, nano-, and pico- — appear most frequently in general chemistry.

Examples: 1 kilometer = 1000 m = 10³ m. 1 milliliter = 0.001 L = 10^-3 L. 4 nanograms = 4 × 10^-9 g.

Key volume relationships to memorize: 1 L = 1 dm³ = 1000 mL, and 1 mL = 1 cm³. These equivalences appear constantly in chemistry calculations.

Scientific notation expresses a number as a coefficient between 1 and 10 multiplied by a power of 10. It’s essential in chemistry because measurements span enormous ranges — from atomic masses (~10^-23 g) to Avogadro’s number (~10²³).

Format: C × 10ⁿ, where 1 ≤ C < 10.

• Large numbers: move the decimal left, exponent is positive. 298,000 = 2.98 × 10⁵
• Small numbers: move the decimal right, exponent is negative. 0.0000025 = 2.5 × 10^-6

Scientific notation also removes ambiguity about significant figures. Is 1,300 two sig figs or four? Writing 1.3 × 10³ makes it clearly two; 1.300 × 10³ makes it clearly four.

When multiplying in scientific notation, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents. Always adjust the result so the coefficient stays between 1 and 10.

Understanding the distinction between exact and inexact numbers is crucial for applying significant figure rules correctly.

Exact numbers have no uncertainty — they are known with infinite precision:

• Counted quantities: 12 eggs in a dozen, 24 students in a class
• Defined relationships: 1 ft = exactly 12 in., 1 in. = exactly 2.54 cm, 1 kg = exactly 1000 g

Inexact numbers come from measurements and always carry some uncertainty:

• A balance reading of 6.72 g (uncertain in the last digit)
• A thermometer showing 22.3 °C (estimated tenths place)

The key practical rule: exact numbers never limit significant figures in a calculation. If you divide 45.6 g by exactly 3 samples, your answer has three sig figs (from 45.6), not one (from 3).

The significant figures (sig figs) in a measurement include all certain digits plus the first uncertain (estimated) digit. To count them, focus on zeros — nonzero digits are always significant.

Rules for zeros:

1. Leading zeros (before the first nonzero digit) are never significant. They only locate the decimal point. Example: 0.00832 has 3 sig figs.
2. Captive zeros (between nonzero digits) are always significant. Example: 1.008 has 4 sig figs.
3. Trailing zeros after a decimal point are always significant. Example: 2.200 has 4 sig figs.
4. Trailing zeros with no decimal point are ambiguous. Example: 1300 could be 2, 3, or 4 sig figs — use scientific notation to clarify (1.30 × 10³ = 3 sig figs).

A quick approach: start counting from the first nonzero digit on the left and count all digits to the right, including zeros. The only exception is ambiguous trailing zeros without a decimal point.

Calculated results can only be as certain as the least certain measurement that went into them. Two rules govern how to round:

Addition and subtraction: round the result to the same number of decimal places as the measurement with the fewest decimal places.

• 1.0023 + 4.383 = 5.3853 → round to 5.385 (3 decimal places, matching 4.383)
• 486 − 421.23 = 64.77 → round to 65 (0 decimal places, matching 486)

Multiplication and division: round the result to the same number of significant figures as the measurement with the fewest sig figs.

• 0.6238 × 6.6 = 4.11708 → round to 4.1 (2 sig figs, matching 6.6)

Rounding rule: if the dropped digit is < 5, round down; if > 5, round up; if exactly 5 (with nothing or only zeros after it), round to the nearest even digit. This “round half to even” rule eliminates bias over many calculations.

Dimensional analysis (the factor-label method) is the most reliable way to convert between units. The core principle: units undergo the same math as numbers, and identical units in the numerator and denominator cancel.

The method:

1. Start with the given quantity and its unit
2. Multiply by one or more conversion factors — fractions equal to 1 (e.g., 2.54 cm / 1 in.)
3. Orient each factor so unwanted units cancel and desired units remain
4. Multiply and divide the numbers; verify the final unit is correct

Example: convert 34 in. to cm.

34 in. × (2.54 cm / 1 in.) = 86 cm

Multi-step conversions simply chain more factors. To convert quarts to milliliters: qt → L → mL, using 1 L = 1.0567 qt and 1 L = 1000 mL. Dimensional analysis works for any unit conversion — it’s not limited to length or volume.

Density is the ratio of mass to volume (d = m/V) and has units like g/cm³ or g/mL. Because it relates mass and volume, density acts as a built-in conversion factor between the two.

For example, the density of gold is 19.3 g/cm³. This means:

• Given volume → find mass: multiply by density. A 2.00 cm³ gold sample has mass = 2.00 × 19.3 = 38.6 g
• Given mass → find volume: divide by density (or multiply by the inverse). A 50.0 g gold sample has volume = 50.0 / 19.3 = 2.59 cm³

Density varies with temperature (substances expand when heated), so it is always reported at a specified temperature. Common reference densities: water = 1.00 g/mL at 25 °C, air ≈ 1.2 g/L at 25 °C and 1 atm.

In lab, density is typically determined by measuring mass (with a balance) and volume (by displacement of water for irregular solids, or directly for liquids).

Chemistry uses three temperature scales. Unlike most unit conversions, temperature conversions involve both multiplication and addition because the scales have different zero points.

Kelvin ↔ Celsius (same degree size, shifted zero):

• T(K) = T(°C) + 273.15
• T(°C) = T(K) − 273.15

Fahrenheit ↔ Celsius (different degree size and different zero):

• T(°F) = (9/5) × T(°C) + 32
• T(°C) = (5/9) × [T(°F) − 32]

Key reference points: water freezes at 0 °C = 32 °F = 273.15 K; water boils at 100 °C = 212 °F = 373.15 K; normal body temperature ≈ 37 °C = 98.6 °F = 310 K.

The kelvin scale is absolute — 0 K is the lowest theoretically achievable temperature. Note: the unit is “kelvin” (lowercase) with symbol K (no degree sign).

Use this sequence to avoid the most frequent errors in measurement and conversion problems:

1. Identify whether each number is exact or measured. Exact numbers (definitions, counts) have unlimited significant figures and never limit your answer.
2. Count significant figures correctly. Leading zeros are never significant. Captive zeros (between nonzero digits) are always significant. Trailing zeros are significant only if a decimal point is present.
3. Apply the right sig-fig rule for the operation. Addition/subtraction: round to the fewest decimal places. Multiplication/division: round to the fewest significant figures.
4. Set up dimensional analysis with units visible at every step. Cancel units explicitly; if units don’t cancel to give the target unit, the setup is wrong.
5. Check reasonableness. A person’s mass should be tens of kilograms, not thousands; a room temperature should be near 20–25 °C.

Common mistakes: treating leading zeros as significant, applying the multiplication rule to addition problems, forgetting to convert °C to K before using gas-law or thermodynamic equations, and dropping units partway through a multi-step conversion so that an error in setup goes undetected.