Notes on Measurements and SI Units

Basic Idea of Measurement

  • Measurements convey information about matter and energy across macroscopic and microscopic domains.
  • Every measurement provides three kinds of information: the magnitude (number), a standard of comparison (unit), and the uncertainty of the measurement. The number and unit are explicit; the uncertainty is implicit and discussed later.
  • Numbers can be written in decimal form or scientific notation (exponential notation).
    • Example: maximum takeoff weight of a Boeing 777-200ER ≈ 2.98×105 kg2.98 \times 10^5\ \text{kg}
    • Mosquito mass ≈ 2.5×10−6 kg2.5 \times 10^{-6}\ \text{kg}
  • Without units, numbers can be meaningless or dangerous (e.g., drug dosages).

SI Units and Base Units

  • Seven base units (SI):
    • Length: meter=m\text{meter} = \text{m}
    • Mass: kilogram=kg\text{kilogram} = \text{kg}
    • Time: second=s\text{second} = \text{s}
    • Temperature: kelvin=K\text{kelvin} = \text{K}
    • Electric current: ampere=A\text{ampere} = \text{A}
    • Amount of substance: mole=mol\text{mole} = \text{mol}
    • Luminous intensity: candela=cd\text{candela} = \text{cd}
  • SI units are fixed by international agreement (NIST usage since 1964).
  • Other properties use derived units from base units.

SI Prefixes (multipliers of 10)

  • Common prefixes and factors (prefix + symbol + factor):
    • femto: f  10−15f\; 10^{-15}
    • pico: p  10−12p\; 10^{-12}
    • nano: n  10−9n\; 10^{-9}
    • micro: μ  10−6\mu\; 10^{-6}
    • milli: m  10−3m\; 10^{-3}
    • centi: c  10−2c\; 10^{-2}
    • deci: d  10−1d\; 10^{-1}
    • kilo: k  103k\; 10^{3}
    • mega: M  106M\; 10^{6}
  • Everyday SI multiples are powers of 10 (e.g., 1 km = 10^3 m; 1 L = 10^{-3} m^3).
  • Examples:
    • 1 km = 103 m10^3\ \text{m}
    • 1 L = 10−3 m310^{-3}\ \text{m}^3
    • 1 cm^3 = 1 mL = 10−6 m310^{-6}\ \text{m}^3

Length and Distance

  • Meter definition: distance light travels in a fixed fraction of a second (exact).
    • 1 m=the distance light travels in 1299 792 458 s1\ \text{m} = \text{the distance light travels in } \frac{1}{299\,792\,458}\ \text{s}
  • Common length units:
    • 1 m ≈ 39.37 inches; 1 m ≈ 1.094 yards.
    • Shorter: 1 cm = 0.01 m; 1 mm = 0.001 m.
    • Longer: 1 km = 1000 m.

Volume and Derived Units

  • Volume is derived from length: base SI unit for volume is the cubic meter, m3\text{m}^3.
  • Common smaller-derived volume units:
    • 1 L = 1 dm^3; 1 dm = 0.1 m → 1 L = 10−3 m310^{-3}\ \text{m}^3
    • 1 cm^3 = 1 mL; 1 cm = 0.01 m → 1 cm^3 = 10−6 m310^{-6}\ \text{m}^3
  • Density relates mass to volume: ρ=mV\rho = \frac{m}{V}; SI unit: kg/m3\text{kg/m}^3; common practical unit: g/cm3\text{g/cm}^3 or g/L\text{g/L}.

Density and Common Values

  • Density is mass per unit volume; measured for solids, liquids, gases with typical ranges:
    • Solids/liquids: often g/cm3\text{g/cm}^3; gases: g/L\text{g/L}.
  • Example densities (at room conditions):
    • Ice: 0.92 g/cm30.92\ \text{g/cm}^3
    • Water: 1.00 g/cm31.00\ \text{g/cm}^3
    • Copper: ≈ 8.96 g/cm38.96\ \text{g/cm}^3
    • Lead: ≈ 11.3 g/cm311.3\ \text{g/cm}^3
  • Quick check: density of a cube with edge length aa: volume V=a3V = a^3; mass measurements yield density ρ=mV\rho = \frac{m}{V}.
  • Example calculation (lead): mass = 90.7 g90.7\ \text{g}, edge = 2.00 cm2.00\ \text{cm}
    • Volume: V=(2.00 cm)3=8.00 cm3V = (2.00\ \text{cm})^3 = 8.00\ \text{cm}^3
    • Density: ρ=90.7 g8.00 cm3=11.3375 g/cm3≈11.3 g/cm3\rho = \frac{90.7\ \text{g}}{8.00\ \text{cm}^3} = 11.3375\ \text{g/cm}^3 \approx 11.3\ \text{g/cm}^3
  • Water displacement method for density:
    • Iron: mass = 31.48 g; volume change = 4.0 mL; density = ρ=31.48 g4.0 mL=7.90 g/mL\rho = \frac{31.48\ \text{g}}{4.0\ \text{mL}} = 7.90\ \text{g/mL}

Practical Notes on Measurements

  • Always include units with numbers; avoid ambiguity.
  • Temperature scales: Kelvin is the SI unit; Celsius is allowed but offset from Kelvin by 273.15; conversions discussed later.
  • Time: base unit is second; prefixes allow expressing microseconds, megaseconds, etc.
  • When quoting densities, remember common units: g/cm^3 for solids/liquids, g/L for gases.

Quick Reference Summary

  • Base SI units: m,kg,s,K,A,mol,cd\text{m}, \text{kg}, \text{s}, \text{K}, \text{A}, \text{mol}, \text{cd}
  • Volume units: m3,L,cm3,mL\text{m}^3, \text{L}, \text{cm}^3, \text{mL}
  • Density: ρ=mV\rho = \dfrac{m}{V}; common units: kg/m3,g/cm3,g/L\text{kg/m}^3, \text{g/cm}^3, \text{g/L}
  • Key conversions:
    • 1 L=1 dm3=10−3 m31\ \text{L} = 1\ \text{dm}^3 = 10^{-3}\ \text{m}^3
    • 1 cm3=1 mL=10−6 m31\ \text{cm}^3 = 1\ \text{mL} = 10^{-6}\ \text{m}^3
    • 1 m=103 mm, 1 m≈39.37 in1\ \text{m} = 10^{3}\ \text{mm},\ 1\text{ m} \approx 39.37\ \text{in}
  • Important: numbers without units can be dangerous (dosage example demonstrates need for units).