ch 2 Exact vs. Measured Values, Significant Figures & Unit Conversions
Exact vs. Measured Quantities
Exact (definition / counting)
Arise from pure counting (e.g., number of discrete items like children, eggs) or from precisely defined relationships; no measuring device is involved in their determination. Such definitions are agreed upon by convention or derived from fundamental constants.
Examples: 6 bananas (counted items), 12 books (counted items), 1000\,\text{m}=1\,\text{km} (a definition in the metric system), 1\,\text{ft}=12\,\text{in} (an exact definition), 1\,\text{in}=2.54\,\text{cm} (this is an internationally defined equivalence, making it exact). Also, integers in formulas (e.g., in A = \pi r^2, the '2' is exact).
Carry infinite significant figures because there is no uncertainty associated with them. They therefore never limit the precision of a calculation based on significant figure rules; their precision is considered perfect.
Measured
Obtained directly or indirectly with an instrument; always subject to experimental uncertainty and limitations of the measuring device. The last digit in a measured value is always an estimate.
Examples: 2.4\,\text{gal} dispensed by a pump (limited by pump accuracy), the length of a fingernail from a ruler (limited by ruler markings), a blood–mineral concentration (limited by analytical instrument precision), personal body mass (limited by scale precision), a speed of 0.01\,\text{m·s}^{-1} (limited by timing/distance measurement uncertainty).
Significant-figure rules apply only to these values to correctly reflect their inherent uncertainty.
Significant Figures (Sig Figs)
Purpose \r
All non-zero digits are significant. They directly represent a measurement.
11.275 - Zeros between non-zero digits (trapped zeros) are significant. These zeros are part of the measurement and provide information about its precision.
10275 - Leading zeros (to the left of the first non-zero digit) are never significant. They are purely placeholders to indicate the magnitude of the number, especially when using decimals, and do not convey information about the precision of the measurement.
0.00275 ***
Trunca te only at the end. Final answer is limited to the smallest number of decimal places in the inputted measurements.
Example: Suppose you have a cylindrical object with a measured radius and height. You want to calculate its density.
Calculate the volume: Use the formula for the volume of a cylinder, V = \pi r^2 h. If r=2.0\,\text{cm} (2 sig figs) and h=5.00\,\text{cm} (3 sig figs), then V = \pi (2.0\,\text{cm})^2 (5.00\,\text{cm}) = 62.831…\,\text{cm}^3. The 2.0\,\text{cm} measurement has 2 significant figures, which is the least number of sig figs among the measured values, so the volume should be rounded to 2 sig figs: 63\,\text{cm}^3.
Calculate the density: If the mass of the cylinder is 125.0\,\text{g} (4 sig figs), then
ho = \dfrac{m}{V} = \dfrac{125.0\,\text{g}}{63\,\text{cm}^3} = 1.9841…\,\text{g/cm}^3. The volume (63 cm³) has 2 significant figures (the least), so the density should be rounded to 2 sig figs: 2.0\,\text{g/cm}^3. Notice that trailing zeros became significant when the value was written in scientific notation to represent its proper precision.
Rounding Rules (to a Desired Sig-Fig Count)
Locate the significant digit that will be the final digit you are allowed to keep based on the required significant figures count.
Look at the digit one place immediately to the right of that final retained digit:
If this "next digit" is 5 or greater (>= 5) -> round up that final retained digit.
If this "next digit" is 4 or less (<= 4) -> leave the final retained digit unchanged.
After rounding, if you are rounding to the left of the decimal, replace any non-significant digits between the rounded digit and the decimal point with zeros to maintain magnitude.
Examples to 3 sig figs:
26.272\rightarrow26.3 (because the "next digit" = 7, which is >= 5).
26.252\rightarrow26.3 ("next digit" = 5, so round up).
26.221\rightarrow26.2 ("next digit" = 2, which is <= 4, so leave unchanged).
125,780 \rightarrow 126,000 (to 3 sig figs: 1, 2, 6, with zeros as placeholders).
Sig Figs in Calculations
Accuracy and precision of a calculated result are limited by the least accurate and least precise measurement used in the calculation.
Multiplication & Division
The final answer must contain the least number of significant figures among all the measured values used in the calculation. This rule reflects that the result cannot be more precise than the least precise input measurement (propagates relative error).
a\,(2\,\text{sf}) \times b\,(3\,\text{sf}) \times c\,(4\,\text{sf})\;\longrightarrow\; \text{answer = 2 sf}.
Example: 3.45\,\text{cm} \times 1.2\,\text{cm} = 4.14\,\text{cm}^2. Since 1.2\,\text{cm} has 2 sig figs (the least), the answer is rounded to 4.1\,\text{cm}^2.
Exact conversion factors or counted numbers never affect the significant-figure limit. They are considered to have infinite significant figures.
Addition & Subtraction
The final answer is limited by the least number of decimal places among the measured terms. This rule reflects that the uncertainty in the sum or difference is determined by the term with the largest absolute uncertainty (propagates absolute error).
60.2\text{ (1 dp)} - 52.3812\text{ (4 dp)} = 7.8188, which rounds to 7.8\text{ (1 dp)}. The answer cannot have more decimal places than the measurement with the fewest decimal places.
Example: 12.1\,\text{g} + 1.234\,\text{g} + 8.12\,\text{g} = 21.454\,\text{g}.
12.1\,\text{g} (1 decimal place)
1.234\,\text{g} (3 decimal places)
8.12\,\text{g} (2 decimal places)
Least decimal places is 1 (from 12.1\,\text{g}), so the answer is rounded to 21.5\,\text{g}.
Unit Conversion & Dimensional Analysis
Dimensional analysis (or the factor-label method) is a systematic approach to solving problems that involve unit conversions. It ensures that units are correctly converted by multiplying by carefully chosen conversion factors.
A conversion factor is an exact equivalence written as a fraction where the numerator and denominator represent the same quantity but in different units.
Example: \dfrac{1000\,\text{mL}}{1\,\text{L}} or \dfrac{1\,\text{L}}{1000\,\text{mL}}. Both are valid and equal to 1.
Choose the orientation (fraction form) that cancels the given unit (placing it in the denominator) and introduces the desired unit (placing it in the numerator); chain as many conversion factors as needed in sequence. This process ensures that the units algebraically cancel out until only the desired unit remains.
Multi-Step Example (Convert 254\,\text{m} \u2192 ft)
Start with the given value and units: 254\,\cancel{\text{m}} (This is a measured value, 3 sig figs).
Multiply by the required conversion factors, ensuring units cancel:
\times\dfrac{100\,\cancel{\text{cm}}}{1\,\cancel{\text{m}}} (metric conversion, exact)
\times\dfrac{1\,\cancel{\text{in}}}{2.54\,\cancel{\text{cm}}} (defined conversion, exact)
\times\dfrac{1\,\text{ft}}{12\,\cancel{\text{in}}} (defined conversion, exact)Perform the calculation: 254 \times 100 \div 2.54 \div 12 = 833.333… \,\text{ft}.
Apply significant figure rules. Since all conversion factors are exact, the number of significant figures in the answer is determined solely by the initial measured value, 254\,\text{m} (3 sig figs).
Calculator gives 833.333… \,\text{ft}, then rounded to 3 sig figs: 8.33\times10^{2}\,\text{ft}.
Metric-Prefix Reminder (exact)
All relationships between base units and their prefixed versions in the metric system are exact definitions.
1\,\text{kg}=1000\,\text{g} (kilo means 10^3)
1\,\text{g}=1000\,\text{mg} (milli means 10^{-3})
1\,\text{m}=100\,\text{cm} (centi means 10^{-2})
1\,\text{L}=10^6\,\mu\text{L} (micro means 10^{-6})
These exact relationships have infinite significant figures and do not limit the precision of calculations.
Sample Conversion ( 0.0003617\,\text{kg} \u2192 mg )
0.0003617\,\cancel{\text{kg}} \times \dfrac{1000\,\cancel{\text{g}}}{1\,\cancel{\text{kg}}} \times \dfrac{1000\,\text{mg}}{1\,\cancel{\text{g}}} = 361.7\,\text{mg}
The input value 0.0003617\,\text{kg} has 4 significant figures (leading zeros are not significant).
All conversion factors are exact and thus have infinite significant figures.
Therefore, the answer retains 4 significant figures (from the measured input).
Temperature Conversions (equations supplied on exam handout)
Temperature scales (Celsius, Fahrenheit, Kelvin) measure the degree of hotness or coldness. Their conversions are precise relationships, but the measured temperature values themselves are subject to sig-fig rules.
Formulas:
K = {^\circ C} + 273.15 (Kelvin is an absolute temperature scale, where 0\,\text{K} is absolute zero, the theoretical lowest possible temperature. The value 273.15 is a defined constant for converting Celsius to Kelvin.)
{^\circ F} = ({^\circ C}\times1.8) + 32 (Fahrenheit and Celsius scales have different zero points and different size degrees. The 1.8 comes from the ratio of degrees between freezing and boiling points: 180^\circ \text{F} vs 100^\circ \text{C}, so 180/100 = 1.8. The 32 accounts for the different freezing points: 0^\circ \text{C} = 32^\circ \text{F}).
Rearrange algebraically as needed to solve for the unknown temperature:
^\circ C = \dfrac{^\circ F - 32}{1.8}.
Plug in the known value, and solve for the unknown. When rounding, follow the rules for addition/subtraction (for K to C conversion) or multiplication/division and then addition/subtraction (for F to C/C to F conversion) based on the measured temperature's significant figures or decimal places. Remember that 1.8 and 32 are generally treated as exact in these conversions for introductory chemistry, especially if they are part of a supplied formula. However, the measured temperature will limit the final answer's precision.
Density
Density (\rho) is a fundamental intensive physical property; meaning it depends only on the substance's identity and state, not on the amount of substance present. It relates a substance's mass to the volume it occupies.
Definition: \rho = \dfrac{\text{mass}}{\text{volume}} = \dfrac{m}{V}.
For solids and liquids: \rho is typically reported in units of \dfrac{\text{g}}{\text{mL}} or \dfrac{\text{g}}{\text{cm}^{3}} (since 1\,\text{mL} = 1\,\text{cm}^{3} exactly). These values usually remain relatively constant with typical changes in temperature and pressure.
For gases: \rho is often reported in \dfrac{\text{g}}{\text{L}}. A 1 Liter volume is commonly chosen because gas densities are much lower than solids or liquids, so using grams per milliliter would result in very small, inconvenient numbers. Gas densities are highly sensitive to temperature and pressure changes.
Because \rho is a constant for a given substance under specified conditions, it can serve as a conversion factor between mass and volume. For example, if you know the density of a liquid is 1.20\,\text{g/mL}, you can use \dfrac{1.20\,\text{g}}{\text{1\,mL}} or \dfrac{\text{1\,mL}}{1.20\,\text{g}} to convert between its mass and volume. The significant figures of the density value will affect the calculation.
Summary of Calculation Ethics & Good Practice
Always distinguish between definition/counted quantities (exact) and measurements (uncertain) before assigning significant figures. Incorrectly identifying exact numbers as measured can lead to undue rounding errors or false precision.
Preserve measurement integrity: Never report more precision than justified by the least precise measurement. At the same time, never truncate too early during intermediate steps; carry at least one or two extra "guard digits" beyond the required significant figure count through intermediate calculations to prevent cumulative rounding errors. Only round to the correct number of significant figures at the very final step of a multi-step calculation.
Clearly show unit cancellations in dimensional analysis to provide a logical pathway for the solution and to avoid dimensional errors (e.g., ending up with \text{m}^2 instead of \text{m}).
When using scientific notation, the coefficient part must accurately mirror the original precision in terms of the number of significant figures.
When publishing or sharing experimental data, preferentially use SI units (International System of Units, e.g., meters, kilograms, seconds, Celsius/Kelvin) unless the specific context or discipline dictates otherwise. This promotes international consistency and ease of comparison.
Connections & Context
Builds on Ch. 1 metric prefixes and SI base units: This topic reinforces the fundamental understanding of units and their