I. Introduction

Dimensional analysis is the habit of treating units as algebra. You track units/dimensions (mass, length, time, charge, temperature, amount of substance) through calculations to convert quantities, check whether equations can be correct, and often reconstruct relationships between variables when you don’t remember a formula.

II. Dimensions vs Units

A physical quantity has a number and a unit (such as, 3.50 m/s). The unit is a label you can rewrite in terms of base dimensions.
Common base dimensions:

  • M = mass
  • L = length
  • T = time
  • I = electric current
  • Θ = temperature
  • N = amount of substance (moles)
    Examples of Derived Dimensions:
  • velocity: LT−1L T^{-1}
  • acceleration: LT−2L T^{-2}
  • force: MLT−2M L T^{-2}
  • energy: ML2T−2M L^{2} T^{-2}
  • pressure: ML−1T−2M L^{-1} T^{-2}
  • momentum: MLT−1M L T^{-1}
  • power: ML2T−3M L^{2} T^{-3}
  • charge: ITI T
  • voltage: ML2T−3I−1M L^{2} T^{-3} I^{-1}

    Rules of doing consistent math with units/dimensions:
    1. you can only add or subtract quantities with the same dimensions
    2. both sides of an equation must have the same dimensions
    3. arguments of exp, ln, sin, cos, etc. must be dimensionless

      Dimensional Analysis Applications
  • Convert units cleanly (factor-label method)
  • Catch wrong equations early because if a proposed equation has mismatched units, it can’t be right. Also finding derived units from universal constants.

III. Factor-Label Method

This method just means doing unit conversions with factors that equal 1. An example is multiplying by (100cm/1m) to convert a quantity from turning meters to centimeters.
Step-by-step

  1. Write the given quantity with its unit.
  2. Multiply by a conversion fraction equal to 1.
  3. Place units so they cancel top-to-bottom.
  4. Repeat until only the target unit remains.
  5. Do arithmetic last.

IV. Seen in Chemistry

A. Stoichiometry

  • Most stoichiometry problems follow a unit path like:
  • given units → moles → mole ratio → desired units Example skeleton:
  • convert grams to moles using molar mass (g/mol)
  • use the balanced equation to convert moles of A to moles of B
  • convert moles of B to grams, liters, molecules, etc.
    B. Molarity and dilution
  • Molarity: M = mol/L So: mol = M·L L = mol/M
  • Dilution: M1 V1 = M2 V2
    Unit check: (M)(V) = (mol/L)(L) = mol Both sides reduce to moles, so the equation is dimensionally consistent.

V. Significant Figures and Unit Conversions

Dimensional analysis keeps the units correct. Significant figures keep the precision honest by reflecting the limitations of measuring tools.
Good habits:

  • keep extra digits in intermediate steps and round at the end
  • defined conversion factors (like 1 in = 2.54 cm) are exact and do not limit significant figures