Units of Measurement and Significant Figures

Every numerical chemistry answer depends on units and correct significant figures. This topic is directly examined (1–2 marks) and quietly decides accuracy in every calculation across the syllabus.

Core ideas

SI base units: mass — kilogram (kg), length — metre (m), time — second (s), temperature — kelvin (K), amount — mole (mol), current — ampere (A), luminous intensity — candela (cd).

Temperature conversions:
K = °C + 273.15 · °F = (9/5)°C + 32

Density = mass/volume; SI unit kg m⁻³ (commonly g cm⁻³ or g mL⁻¹).

Significant figures — all certain digits plus one uncertain digit. Rules: (1) non-zero digits are significant; (2) zeros between non-zeros are significant; (3) leading zeros are NOT significant (0.005 → 1 s.f.); (4) trailing zeros after a decimal ARE significant (2.500 → 4 s.f.); (5) in exponential form N × 10ⁿ, only N counts.

Calculation rules: in multiplication/division, the result keeps the fewest significant figures; in addition/subtraction, the result keeps the fewest decimal places.

Precision vs accuracy: precision = closeness of repeated measurements to each other; accuracy = closeness to the true value.

Dimensional analysis (factor-label method): multiply by unit conversion factors so unwanted units cancel.

Exam Tricks & Tips

• 🎯 Leading zeros never count; trailing zeros count only if a decimal point is present (100 = 1 s.f., 100. = 3 s.f., 1.00×10² = 3 s.f.).
• 🎯 Multiplication/division → fewest significant figures; addition/subtraction → fewest decimal places. Don't mix the two rules.
• 🎯 Exact numbers (counted objects, defined constants) have infinite significant figures and never limit the answer.
• 🎯 Use scientific notation to make s.f. unambiguous.
• 🎯 Round only the final answer, not intermediate steps.
• ❌ Common mistake: writing 0.00456 as having 5 significant figures — the three leading zeros are not significant, so it has only 3.

Expected exam pattern

Expect a "how many significant figures" MCQ, a unit-conversion or dimensional-analysis numerical, and possibly a precision-vs-accuracy statement to identify.

Quick recap

Learn the seven SI base units, K = °C + 273.15, the significant-figure rules, and the two calculation rules (× ÷ → fewest s.f.; + − → fewest decimals). Precision ≠ accuracy.

Cover the answer, recall, then check. 12 cards on measurement and significant figures.

Q1. State the SI base unit of amount of substance.
A1. The mole (mol).

Q2. Convert 25 °C to kelvin.
A2. K = 25 + 273.15 = 298.15 K.

Q3. What is the SI unit of density?
A3. kg m⁻³ (often expressed as g cm⁻³).

Q4. Are leading zeros significant? Example.
A4. No. 0.0025 has only 2 significant figures.

Q5. How many significant figures in 2.500?
A5. Four — trailing zeros after a decimal point are significant.

Q6. How many significant figures in 100 (no decimal)?
A6. One (ambiguous); write 1.00×10² for three.

Q7. Rule for significant figures in multiplication/division?
A7. The answer keeps the least number of significant figures of any factor.

Q8. Rule for addition/subtraction?
A8. The answer keeps the least number of decimal places.

Q9. How many significant figures do exact/counted numbers have?
A9. Infinite — they never limit the precision of the result.

Q10. Difference between precision and accuracy?
A10. Precision = agreement among repeated measurements; accuracy = closeness to the true value.

Q11. What is the factor-label (dimensional analysis) method?
A11. Multiplying by unit conversion factors so unwanted units cancel and only the desired unit remains.

Q12. In 6.02 × 10²³, which part determines significant figures?
A12. Only the coefficient 6.02 (three significant figures); the power of ten does not count.

A measurement without a unit is meaningless, and a measurement reported with the wrong number of digits is dishonest. This topic gives you the grammar of quantitative chemistry — SI units, scientific notation, significant figures, and the factor-label method that silently earns marks in every numerical.

Definition: A physical quantity is expressed as (numerical value) × (unit). The SI system fixes seven base units so every measurement is reproducible worldwide.

Beginner — the seven SI base units

Quantity	Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount	mole	mol
Current	ampere	A
Luminous intensity	candela	cd

All other units are derived (e.g. volume m³, density kg/m³, force newton = kg·m·s⁻²). Prefixes scale them: kilo (10³), deci (10⁻¹), centi (10⁻²), milli (10⁻³), micro (10⁻⁶), nano (10⁻⁹), pico (10⁻¹²).

Intermediate — scientific notation and significant figures

Scientific notation writes any number as N × 10ⁿ where 1 ≤ N < 10 (e.g. 0.00023 = 2.3 × 10⁻⁴).

Significant figures (sig figs) = all certain digits plus one uncertain last digit. Rules:

• All non-zero digits are significant (245 → 3).
• Zeros between non-zeros count (2.03 → 3).
• Leading zeros do not count (0.0025 → 2).
• Trailing zeros count only if a decimal point is present (2.50 → 3; 250 → ambiguous, 2 or 3).
• Exact/counted numbers (e.g. 12 eggs, Avogadro's constant defined) have infinite sig figs.

Calculation rules:

• Addition/subtraction — answer keeps the fewest decimal places.
• Multiplication/division — answer keeps the fewest significant figures.

Advanced — precision, accuracy, dimensional analysis

Accuracy = closeness to the true value; precision = closeness of repeated readings to one another. High precision ≠ high accuracy (a mis-calibrated balance is precise but inaccurate).

Dimensional analysis (factor-label method): multiply by unit ratios equal to 1 so unwanted units cancel. This is the safest way to convert.

Worked example

Convert 5 feet to metres (1 inch = 2.54 cm), and report sig figs for 2.5 × 1.25.

5 ft × (12 in / 1 ft) × (2.54 cm / 1 in) × (1 m / 100 cm) = 1.524 m → the inch factor is exact, 5 ft has 1 sig fig, so report 2 m (or 1.5 m if 5 is exact context).

2.5 × 1.25 = 3.125 → fewest sig figs is 2 → 3.1.

Real-world / exam application

Every JEE/NEET numerical expects the final answer rounded to correct sig figs; examiners deduct for over-reporting digits. Density, molarity, and gas-law problems all begin with unit conversion — the factor-label method prevents the classic "forgot to convert mL to L" error.

Exam tricks & mnemonic

Mnemonic for prefixes (large→small): "King Henry Died By Drinking Chocolate Milk" (Kilo, Hecto, Deca, Base, Deci, Centi, Milli). For sig figs, remember: leading zeros are just placeholders — they never count.

Key formulas / rules

• SI base units: m, kg, s, A, K, mol, cd.
• Temperature conversions: K = °C + 273.15; °F = (9/5)°C + 32.
• Density: d = mass/volume (g/cm³ or kg/m³).
• Significant figures: ×/÷ keep the least number of sig figs; +/− keep the least number of decimal places.
• Scientific notation: N × 10ⁿ with 1 ≤ N < 10.
• Precision vs accuracy: precision = reproducibility; accuracy = closeness to true value.

Dimensional (factor-label) analysis converts units by multiplying with unit ratios equal to 1.

Worked Example

Problem: A metal sample has a mass of 3.28 g and occupies a volume of 25.0 mL. Calculate its density and report the answer to the correct number of significant figures.

Solution:
Step 1 — Write the formula for density:
density = mass / volume.

Step 2 — Substitute the values:
density = 3.28 g / 25.0 mL = 0.1312 g/mL.

Step 3 — Apply the significant-figure rule for division: the answer keeps as many significant figures as the measurement with the fewest. Here 3.28 has 3 s.f. and 25.0 has 3 s.f., so the answer must have 3 s.f.

Step 4 — Round 0.1312 to 3 significant figures. The fourth figure is 2 (< 5), so drop it without changing the third:
0.1312 → 0.131 g/mL.

Answer: The density is 0.131 g/mL (3 significant figures).