Units and Measurements Chem chap #1

1.1.1 Units and Measurements

• Measurements have two essential components:
  • A magnitude (a number) indicating how much, how large, or how long, etc.
  • A unit specifying what is being measured (e.g., g, L, s).
  • Both components are necessary; without a unit, a measurement is ambiguous (e.g., the mass is 25 vs 25 g).
• Common SI base units used in chemistry:
  • Mass: $\text{mass} \rightarrow \text{Unit}: \; \text{kg} \;\text{Symbol}: \; \text{kg}$
  • Length: $\text{length} \rightarrow \text{Unit}: \; \text{m} \;\text{Symbol}: \; \text{m}$
  • Temperature: $\text{temperature} \rightarrow \text{Unit}: \; \text{K} \;\text{Symbol}: \; \text{K}$
  • Amount of substance: $\text{mole} \rightarrow \text{Unit}: \; \text{mol} \;\text{Symbol}: \; \text{mol}$
  • Time: $\text{time} \rightarrow \text{Unit}: \; \text{s} \;\;\text{Symbol}: \; \text{s}$
  • Energy: $\text{energy} \rightarrow \text{Unit}: \; \text{J} \;\;\text{Symbol}: \; \text{J}$
• Temperature in chemistry is usually expressed as °C or K. Relationship between them:
  • $T(\mathrm{K}) = T(^{\circ}\mathrm{C}) + 273.15$
• SI unit system is based on standardized definitions to ensure consistency across measurements and calculations.
• Note: In chemistry, temperature is commonly reported in °C or K, not °F.

1.1.2 SI Prefixes

• SI prefixes represent powers of ten to handle very large or very small numbers. Some commonly used prefixes:
  • kilo (k): $10^3$
  • centi (c): $10^{-2}$
  • milli (m): $10^{-3}$
  • micro (\mu): $10^{-6}$
  • nano (n): $10^{-9}$
• Examples:
  • $1\ \text{km} = 1000\ \text{m}$
  • $1\ \text{cm} = 0.01\ \text{m}$
  • $1\ \text{mL} = 1\ \text{cm}^3$
• Practical note: prefixes are used to express very large or very small quantities conveniently and to keep numerical values within a readable range.

1.1.3 Scientific Notation

• Scientific notation expresses numbers in the form $a \times 10^{n}$ where 1 \leq |a| < 10.
• Examples:
  • $3.2 \times 10^4 = 32000$
  • $4.5 \times 10^{-6} = 0.0000045$
• Purpose: simplifies calculations and clarifies precision, especially for very large or very small values.

1.1.4 Volume and Derived SI Units

• Volume is the amount of space occupied by a substance or object.
• Volume can be calculated from dimensions: $V = \text{length} \times \text{width} \times \text{height}$
• Derived SI unit for volume: base unit is $\text{m}^3$, but lab use often employs more convenient units:
  • $1\ \text{L} = 1\ \text{dm}^3$ (a litre is a cubic decimeter)
  • $1\ \text{mL} = 1\ \text{cm}^3$
• Density is a derived quantity obtained from mass and volume:
  • $\rho = \frac{m}{V}$
  • SI unit: $\mathrm{kg\, m^{-3}}$; common units for solids/liquids: $\mathrm{g\, cm^{-3}}$ or $\mathrm{g\, L^{-1}}$ for gases

1.1.5 Density and Example Calculations

• Density definition: $\rho = \dfrac{m}{V}$
• Example values: pure substances exhibit characteristic densities; solids/liquids commonly range from about $0.7\ \mathrm{g\, cm^{-3}}$ (gasoline) to about $19\ \mathrm{g\, cm^{-3}}$ (gold). Air has density about $1.2\ \mathrm{g\, L^{-1}}$.
• Example calculation: for a cube with edge length $a$ and mass $m$,
  • Volume: $V = a^3$
  • Density: $\rho = \dfrac{m}{V}$
  • If a cube has edge length $2.00\ \text{cm}$ and mass $90.7\ \text{g}$, then
  • $V = (2.00\ \text{cm})^3 = 8.00\ \text{cm}^3$
  • $\rho = \dfrac{90.7\ \text{g}}{8.00\ \text{cm}^3} \approx 11.3\ \text{g cm}^{-3}$

1.1.6 Unit Conversions and Dimensional Analysis

• Conversions use conversion factors, which are ratios that relate one unit to another and equal 1 in value, e.g.,
  • $1\ \text{L} = 1000\ \text{mL}$, so a conversion factor can be written as $\dfrac{1000\ \text{mL}}{1\ \text{L}} = 1$.
• Dimensional analysis (factor-label method): multiply starting value by appropriate conversion factors to cancel unwanted units and leave the desired unit.
• Examples:
  • Convert 2.5 h to seconds:
  • $2.5\ \text{h} \times \dfrac{60\ \text{min}}{1\ \text{h}} \times \dfrac{60\ \text{s}}{1\ \text{min}} = 9000\ \text{s}$
  • Convert velocity: hummingbird 14.7 m/s to km/h:
  • $14.7\ \mathrm{m\, s^{-1}} \times \dfrac{3.6\ \mathrm{km\, h^{-1}}}{1\ \mathrm{m\, s^{-1}}} = 52.92\ \mathrm{km\, h^{-1}} \approx 52.9\ \mathrm{km\, h^{-1}}$
• Additional notes:
  • 1 L = 1000 mL; 1 cm^3 = 1 mL; 1 m^3 = 1000 L; etc.

1.1.7 Significant Figures: Basics and Rules

• Significant figures (sig figs) convey precision of a measurement.
• Rules for significance:
  • Nonzero digits are always significant.
  • Leading zeros (before the first nonzero digit) are not significant.
  • Captive zeros (between nonzero digits) are always significant.
  • trailing zeros (after the last nonzero digit) are significant only if a decimal point is present or explicitly indicated; otherwise trailing zeros can be ambiguous.
• Scientific notation makes sig figs explicit: all digits shown in the coefficient are significant.
• Examples:
  • 0.00203 g → 3 sig figs (the leading zeros are not significant)
  • 3.20 cm → 3 sig figs (the trailing zero is significant because it comes after the decimal point)
  • 1.02 V → 3 sig figs (the zero is a captive zero and significant)
  • 1.000 g → 4 sig figs (trailing zeros after the decimal point are significant)
  • 0.00832407 = 8.32407 × 10^(-3) → 6 sig figs (the digits in 8.32407 are all significant)
• Some numbers can be written as 5.070 × 10^n, 400 km, or 5.0000 g; ambiguity about sig figs can be removed by using scientific notation.

1.1.7 (continued): Significant Figures in Calculations (A practical guide)

• When reporting final results from calculations, use the least precise value to determine the precision of the result:
  • Addition/subtraction: round to the fewest decimal places among terms.
  • Multiplication/division: round to the fewest significant figures among factors.
  • Logarithms (log or ln): the number of decimal places in the result equals the number of significant figures in the original number.
  • Antilogarithms (10^x): the number of significant figures in the result equals the number of decimal places in the exponent.
  • Rounding rule: if the dropped digit is < 5, round down; if ≥ 5, round up. Do not round until the final step to avoid cumulative rounding error.
• Example rules (conceptual):
  • For addition/subtraction: e.g., 12.11 + 0.3 → 12.4 (1 decimal place since 0.3 has 1 decimal place)
  • For multiplication/division: e.g., 3.24 × 7.1 → 23 (2 sig figs; the limiting value is 2 sig figs from 7.1)
  • For logs: e.g., log(31) ≈ 1.491 → 1.49 (2 sig figs → 2 decimal places) when the original has 2 sig figs
  • For antilogs: e.g., 10^2.0 has 2 decimal places in exponent, giving 2 sig figs in the result
  • Always round only in the final result of a calculation to avoid accumulation of rounding errors

1.1.8 More Examples: Addition/Subtraction and Multiplication/Division with Sig Figs

• Addition/subtraction example:
  • a) $2.334\ \text{mL} + 0.31\ \text{mL}$
  • Rule: fewest decimal places among terms is 2 (since 0.31 has 2 decimals). Result rounded to 2 decimals: $2.64\ \text{mL}$
• Subtraction example:
  • b) $56.533\ \text{m} - 55.8752\ \text{m}$
  • Rule: fewest decimal places is 3 decimals in 56.533; result rounded to 3 decimals: $0.658\ \text{m}$
• Multiplication example:
  • a) $0.6238\ \text{cm} \times 6.6\ \text{cm}$
  • Result: $0.6238 \times 6.6 = 4.11708\ \text{cm}^2$ with two significant figures in the end because 6.6 has 2 sig figs, giving $4.1\ \text{cm}^2$
• Final note: when combining different rules, apply each operation’s rule at the appropriate step and round only in the final result.

1.1.9 Additional Practice: Sig Figs in Multiplication

• Example: $2.334\ \text{cm} \times 0.320\ \text{cm} = 0.74688\ \text{cm}^2$
• Sig figs: factors have 4 and 3 sig figs respectively; the product should have 3 sig figs (the fewest among factors):
  • $0.747\ \text{cm}^2$
• This section also covers how to determine density from displacement measurements in an experiment, applying the same sig fig rules to both volume and mass.

1.1.9 (Experimental Determination of Density): Water Displacement Method

• Principle: the volume of an irregular object is equal to the volume of water displaced when the object is submerged.
• Steps:
  • Record initial water volume, approximate to the nearest 0.1 mL (or the instrument’s precision).
  • Submerge the object and record the final volume.
  • Displaced volume: $\Delta V = V<em>{\text{final}} - V</em>{\text{initial}}$ (rounded to the instrument’s precision).
  • Mass of the object: measure with a balance; mass value carries significant figures as per the balance precision.
  • Density: $\rho = \dfrac{m}{\Delta V}$ (round according to sig figs for multiplication/division).
• Practical note: digital instruments contribute a last-digit estimate; report all significant digits up to the instrument’s precision.

1.1.10 More Density by Water Displacement (Irregular Material)

• Example procedure: weigh an irregular, shiny material; measure its volume by water displacement as described above.
• Volume displaced: difference between the final and initial water volumes, rounded to the nearest 0.1 mL per the measurement precision.
• Density calculation: use the ratio $\rho = \dfrac{m}{V}$, and round to the appropriate number of significant figures based on the input measurements (typically two to four sig figs depending on mass and volume precision).
• Example outcome (typical): if mass ≈ 51.842 g and displaced volume ≈ 2.7 cm^3, then
  • $\rho ≈ \dfrac{51.842\ \text{g}}{2.7\ \text{cm}^3} \approx 19.2\ \text{g cm}^{-3}$
  • Report with appropriate sig figs (often 2–3 sig figs depending on data precision).

1.1.11 Summary and Practical Implications

• The SI system provides a standardized language for reporting measurements across chemistry and related fields.
• Common base units: meter (m), kilogram (kg), second (s), kelvin (K), mole (mol), joule (J).
• Derived units like volume (m^3 or L) and density (g/cm^3 or g/L) are built from base units.
• Dimensional analysis (factor-label method) ensures unit consistency in calculations; conversion factors express equivalences like $1\ \text{L} = 1000\ \text{mL}$ and $1\ \text{cm}^3 = 1\ \text{mL}$.
• Significant figures convey measurement precision and must be preserved in calculations to avoid overstating certainty.
  • Multiplication/division: final answer has as many sig figs as the least precise factor.
  • Addition/subtraction: final answer is rounded to the least precise decimal place among the terms.
  • Logs/antilogs and rounding rules govern how to handle digits after the decimal point and during the calculation process.
• Real-world relevance: precise reporting of measurements affects data interpretation, experiments, quality control, and decision-making in chemistry and related disciplines.
• Ethical and practical considerations: avoid over-claiming precision; report measurements honestly with their uncertainties; document methodology to ensure reproducibility of results.