1: Units and dimensions

Section 1: Foundations: 1: Units and Dimensions

Accurate measurement and clear communication of physical quantities are fundamental to chemistry. This relies on a consistent system of units and the rigorous application of dimensional analysis. The Système International d'Unités (SI), the modern metric system, provides this global standard.

SI Base Units: These seven fundamental units define all others:

• Length: meter (m)
• Mass: kilogram (kg)
• Time: second (s)
• Electric current: ampere (A)
• Temperature: kelvin (K) - Crucial note: absolute temperature differences are in K ($\Delta T = 1\ \text{K}$), while the Celsius scale (°C) is defined as $T(^\circ\text{C}) = T(\text{K}) - 273.15$.
• Amount of substance: mole (mol) - Defined as exactly $6.02214076 \times 10^{23}$ elementary entities (atoms, molecules, ions, etc.).
• Luminous intensity: candela (cd) - Less frequently used in core chemistry.

Derived Units: These are formed by combining base units according to physical laws. Essential examples in chemistry include:

• Volume: cubic meter ($\text{m}^3$). Commonly used subunits: liter ($\text{L}$) $= 0.001\ \text{m}^3$, milliliter ($\text{mL}$) $= 0.001\ \text{L} = 10^{-6}\ \text{m}^3$.
• Density: kilogram per cubic meter ($\text{kg/m}^3$) or grams per milliliter ($\text{g/mL}$).
• Pressure: pascal ($\text{Pa}$) $= \text{N/m}^2 = \text{kg/(m}\cdot\text{s}^2)$. Frequently used: bar ($10^5\ \text{Pa}$) and atmosphere ($\text{atm} \approx 1.013\ \text{bar}$).
• Energy: joule ($\text{J}$) $= \text{N}\cdot\text{m} = \text{kg}\cdot\text{m}^2\text{/s}^2$. Also common: electronvolt ($\text{eV}$) and calorie ($\text{cal}$).
• Force: newton ($\text{N}$) $= \text{kg}\cdot\text{m/s}^2$.
• Concentration: moles per cubic meter ($\text{mol/m}^3$). Most common: moles per liter ($\text{mol/L}$ or $\text{M}$).

Dimensional Analysis: This powerful technique involves treating units as algebraic quantities. Its core purposes are:

1. Unit Conversions: Systematically converting a quantity from one unit to another using conversion factors (ratios equal to 1, like $1\ \text{L} / 1000\ \text{mL}$). Multiply the original value by appropriate conversion factors, canceling units step-by-step until the desired unit remains.
2. Checking Equations: Verifying the dimensional homogeneity of an equation. All terms in a physically meaningful equation must have the same overall dimensions (combination of base units like mass, length, time). If the dimensions don't match, the equation is incorrect. For example, checking if an expression for energy (dimensions: $M L^2 T^{-2}$) matches both sides of an equation.
3. Deriving Units: Determining the units of a derived quantity from the equation defining it (e.g., pressure $=$ force/area implies units $\text{N/m}^2$).

Significant Figures (Digits): These indicate the precision of a measured or calculated value. Key rules:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant (e.g., 101 has 3 sig figs).
• Leading zeros (before the first non-zero digit) are never significant (e.g., 0.0025 has 2 sig figs).
• Trailing zeros after a decimal point are significant (e.g., 2.50 has 3 sig figs).
• Trailing zeros without a decimal point are ambiguous (e.g., 2500 may have 2, 3, or 4 sig figs; use scientific notation like $2.50 \times 10^3$ for clarity).
• Calculations:
  • Multiplication/Division: The result has the same number of significant figures as the measurement with the fewest significant figures.
  • Addition/Subtraction: The result has the same number of decimal places as the measurement with the fewest decimal places.
• Exact Numbers: Counted numbers or defined conversion factors (e.g., 12 eggs, $100\ \text{cm/m}$) have infinite significant figures and do not limit the result's precision.