Measurement & Units

The SI system is built on seven base units from which all other units are derived. In chemistry, the most commonly used are: the kilogram (kg) for mass, the metre (m) for length, the second (s) for time, the kelvin (K) for thermodynamic temperature, the mole (mol) for amount of substance, and the ampere (A) for electric current. The candela (cd) for luminous intensity is rarely needed in general chemistry. Derived units such as the pascal (Pa = kg·m⁻¹·s⁻²) for pressure, the joule (J = kg·m²·s⁻²) for energy, and the litre (L = 10⁻³ m³) for volume are combinations of these base units.

SI prefixes scale units by powers of ten, making it convenient to express very large or very small quantities. Common prefixes in chemistry include: nano- (n, 10⁻⁹), micro- (μ, 10⁻⁶), milli- (m, 10⁻³), centi- (c, 10⁻²), deci- (d, 10⁻¹), kilo- (k, 10³), mega- (M, 10⁶), and giga- (G, 10⁹). For example, a bond length might be expressed as 154 pm (picometres, 10⁻¹² m), while a large industrial reaction might involve megajoules (MJ) of energy. Mastering prefix conversions is essential for avoiding errors of many orders of magnitude.

Dimensional analysis converts a measurement from one unit to another by multiplying by conversion factors — fractions equal to 1 in which the numerator and denominator represent the same quantity in different units. Units cancel algebraically, just like variables in algebra. For example, to convert 5.00 km to metres: 5.00 km × (1000 m / 1 km) = 5000 m. The km units cancel, leaving metres. This method can be chained across multiple conversions and is a powerful self-checking mechanism: if the final units are wrong, an error in the setup is immediately apparent. It also applies to derived units — converting mL/min to L/s, or g/cm³ to kg/m³.

Significant figures (sig figs) reflect the precision of a measured value. The rules for counting sig figs are: (1) All nonzero digits are significant. (2) Zeros between nonzero digits are significant (e.g., 1007 has 4 sig figs). (3) Leading zeros are never significant (0.0045 has 2 sig figs). (4) Trailing zeros after a decimal point are significant (3.500 has 4 sig figs). (5) Trailing zeros in a whole number without a decimal point are ambiguous — scientific notation resolves this (3.50 × 10³ clearly has 3 sig figs). For multiplication and division, the answer has as many sig figs as the measurement with the fewest. For addition and subtraction, the answer is rounded to the least number of decimal places among the measurements.

Accuracy describes how close a measurement is to the true or accepted value, while precision describes how reproducible or consistent repeated measurements are. A set of measurements can be precise (clustered together) but inaccurate (far from the true value), as might happen with a miscalibrated balance. Conversely, measurements can be accurate on average but imprecise if there is large random error. In experimental chemistry, both accuracy and precision matter: systematic errors reduce accuracy, while random errors reduce precision. Significant figures communicate precision, not accuracy — a reported value of 12.34 g implies the measurement was made to the nearest 0.01 g.

• \rho — Density (kg\,m^{-3} \text{ (SI)}; \text{ often } g\,cm^{-3} \text{ or } g\,mL^{-1} \text{ in chemistry}): Mass per unit volume; a derived SI quantity combining the base units of kilogram and metre.
• m — Mass (kg): The SI base unit of mass is the kilogram. In lab settings, grams (g) are common: 1 kg = 1000 g.
• V — Volume (m^3 \text{ (SI)}; \text{ often } cm^3 \text{ or } mL \text{ in chemistry (1 mL = 1 cm}^3\text{)}): The three-dimensional space occupied by a substance. Volume is a derived unit: 1 m³ = 10⁶ cm³ = 10³ L.

Problem. A student measures the mass of an aluminum cylinder as 54.32 g and its volume by water displacement as 20.1 mL. (a) Calculate the density of the cylinder in g/mL to the correct number of significant figures. (b) Convert this density to kg/m³ using dimensional analysis. The accepted density of aluminum is 2.70 g/cm³. Comment on the accuracy of the result.

Answer. ρ = 2.70 g/mL = 2700 kg/m³ (3 significant figures); result matches the accepted value for aluminum, indicating accurate measurement.

• The Mole and Molar Mass — The mole is one of the seven SI base units and is central to all stoichiometric calculations. Understanding SI units and dimensional analysis is prerequisite to converting between grams, moles, and number of particles.
• Density and Its Calculations — Density (g/mL or kg/m³) is a derived SI unit requiring both mass and volume measurements. Applying sig fig rules and dimensional analysis directly in density problems reinforces measurement concepts.
• Experimental Error and Percent Error — Accuracy and precision are discussed alongside significant figures. Percent error quantifies the accuracy of a measurement relative to an accepted value, extending measurement concepts into data analysis.
• The Ideal Gas Law — PV = nRT requires careful unit consistency (Pa vs atm, m³ vs L, K vs °C). Dimensional analysis and SI unit conversions are essential for correct application of this fundamental gas law.