Chapter 1 Vocabulary: Numbers, Sig Figs, and SI Units

Types of Numbers

Measurements can be expressed as two types of numbers:
- Exact numbers: have known certainty. Examples include counting numbers or defined equalities, such as:
- 12\ \text{inches} = 1\ \text{foot}
- 16\ \text{ounces} = 1\ \text{pound}
- 60\ \text{seconds} = 1\ \text{minute}
- Measured numbers: have some degree of uncertainty and come from readings with a device; considered experimental data.
Uncertainty arises from the instrument and the user’s interpretation/readings.
- The quality/precision of the instrument (calibration, smallest division) influences uncertainty (e.g., two rulers with different smallest markings give different precision).
- The user’s interpretation can introduce additional uncertainty, especially in video-based labs where readings may be approximate.
Significant figures (sig figs) codify precision of a measured value. Abbreviations: SF or sig figs.
Exact numbers have unlimited sig figs and do not count toward sig fig precision in calculations.
Scientific notation helps manage sig figs by making the number of significant digits explicit in the coefficient.
- Example: 1.21 \times 10^{4} has 3 sig figs (the digits 1, 2, and 1 in the coefficient).

Uncertainty and Significant Figures (Sig Figs)

Sig figs communicate a measurement to a certain precision: all digits known with certainty plus one uncertain digit (the last one).
How many sig figs a measurement has depends on the measuring device and context.
Examples of sig figs rules in practice:
- Nonzero integers are always significant.
- Example: 123\ \text{g} has 3 sig figs.
- Zeros can be tricky:
- Leading zeros are not significant (e.g., 0.0123\ m has 3 sig figs: 1, 2, 3).
- Captive zeros (zeros between nonzero digits) are significant (e.g., 101\ L\ has 3 sig figs).
- Trailing zeros:
  - If there is no decimal point, trailing zeros are not necessarily significant (e.g., 100\ g has 1 sig fig).
  - If a decimal point is present, trailing zeros count (e.g., 100.0\ g has 4 sig figs, though the exact count depends on notation; convention often treats 100.0 as 4 sig figs only if written with a decimal clarifying precision).
Exact numbers have unlimited sig figs; do not count toward sig figs of measurements.
In scientific notation, all digits in the coefficient are significant (e.g., 1.21\times 10^{4} has 3 sig figs).

Rounding Rules (sig figs context)

When rounding, if the digit to be dropped is less than 5, keep the last kept digit unchanged.
If the digit to be dropped is greater than 5, round the last kept digit up by one.
If the digit to be dropped is exactly 5, the transcript notes don’t specify a rule; standard practice is to round up (or apply a chosen tie-breaking rule). In practice, many courses round 5 up.
In series calculations, carrying one extra sig fig during intermediate steps helps minimize rounding errors.

Sig Figs in Calculations

Multiplication and division:
- The final result should have the same number of sig figs as the factor with the smallest number of sig figs.
- Example: 38.65\times 105.93\approx 4.094195\times 10^{3}
- Sig figs: 38.65 has 4; 105.93 has 5; smallest = 4; Final rounded to 4 sig figs: 4.094\times 10^{3} (i.e., 4094).
Division example:
- Example: 125 \div 9 = 13.888\ldots
- Three sig figs in the result, so report as 13.9 (three sig figs).
Addition and subtraction:
- The result is limited by the term with the fewest decimal places, not the number of sig figs.
- Example: with numbers that have 3, 1, and 2 decimal places respectively, round your final answer to 1 decimal place (the smallest).
- Worked example (conceptual):
- If you have numbers like 56.789\ +\ 102.2\ +\ 300.99, the smallest number of decimal places is 1 (102.2 has one). The sum is 459.979, which rounds to 460.0 at 1 decimal place.
Practical tip: carry one extra sig fig during intermediate steps; round only at the end to the required precision.

Worked Example Problems (conceptual)

Multiplication example:
- Given: two numbers with sig figs, multiply them, then round to the smallest number of sig figs among the factors.
Division example:
- Given: two numbers; divide, then round to the smallest number of sig figs among the factors.
Addition/subtraction example:
- Given: several measured numbers with decimals; align decimal places, sum, then round to the smallest number of decimal places among the addends.

Units and SI Base Units

Base units (SI) are defined units used in science; there are seven base units. In chemistry, the commonly used five base units are:
- Mass: kilogram (kg)
- Length: meter (m)
- Time: second (s)
- Temperature: kelvin (K) (note: Celsius is often used in labs, but Kelvin is the SI base for temperature; conversion between C and K is K = C + 273.15)
- Amount of substance: mole (mol)
Temperature conversion:
- K = C + 273.15
Prefixes modify base units to express larger or smaller quantities. Common prefixes to know (red-box essentials):
- kilo (k): 10^{3}
- centi (c): 10^{-2}
- milli (m): 10^{-3}
- micro (μ): 10^{-6}
- nano (n): 10^{-9}
Notation example: 1 kilogram = 1000 grams; 1 meter = 100 centimeters; 1 liter = 1000 milliliters.
Scientific notation representations for these prefixes are provided to show the magnitude relationship (e.g., 1 cm = 0.01 m).

Derived SI Units

Derived units come from combinations of base units (multiplied or divided):
- Volume: V = length × width × height (for regular solids). For liquids, use liters and milliliters; 1 L = 1000 mL; 1 cm^3 = 1 mL.
- Energy (joule): J = kg·m^2/s^2.
- Density: ρ = m / V; common density units include g/mL, g/cm^3, kg/m^3, etc.
- Note: 1 cm^3 = 1 mL; 1 L = 1000 mL; density with water: pure water has density about 1.00 g/mL at room temperature.
Density and buoyancy:
- Substances with density less than water (1 g/mL) tend to float; greater density tend to sink.
Practical example units:
- Mass: grams (g) for lab-scale, but kilograms (kg) are used in larger scales; mass can be converted between g and kg.
- Volume for liquids: liters (L) and milliliters (mL).
- Distance/length: centimeters (cm) or meters (m).
Important relationship: 1 mL is equivalent to 1 cm^3.

Volume and Density: Methods and Calculations

Volume of regular-shaped solids: V = l × w × h (for simple rectangular blocks, etc.).
Volume of irregular solids: volume by water displacement (Archimedes principle).
- Steps:
  1) Fill a graduated cylinder with a known volume of water (Vinital).
  2) Submerge the object and record the final volume (Vfinal).
  3) Volume of object: Vobject = Vfinal − Vinital.
Density calculation:
- \rho = \frac{m}{V}
- Mass in grams, volume in mL or cm^3 yields density in g/mL or g/cm^3.
Practice problem (density by displacement):
- Example: A 48 g metal piece causes the water level to rise from 25 mL to 33 mL.
- Volume displaced: V = 33 - 25 = 8\ \text{mL}
- Density: \rho = \frac{48\ \text{g}}{8\ \text{mL}} = 6\ \text{g/mL}
Specific gravity (SG):
- SG = \frac{\rho{substance}}{\rho{water}}
- Since ρwater ≈ 1 g/mL, SG is unitless.
- Applications: fermentation monitoring in beer/wine production (e.g., to track sugar conversion to alcohol).

Equivalence, Equality, and Conversion Factors

Equivalencies (equalities) are exact relationships used to convert units.
- Examples: 60\ \text{seconds} = 1\ \text{minute},\quad 12\ \text{inches} = 1\ \text{foot}, 1\ \text{L} = 1000\ \text{mL}, 1\ \text{m} = 100\ \text{cm}.
- The word "per" indicates a division relationship (an equality): e.g., 60 s per minute.
Converting using conversion factors:
- A conversion factor is an equality expressed as a fraction used to multiply a quantity to convert units.
- Set up so that old units cancel and new units remain.
- Each conversion factor has two equalities (one in each direction).
- Example setup:
- To convert from cm to m: 100\ \text{cm} = 1\ \text{m}; set up as a fraction to cancel cm and leave m.
Density as an equality:
- \rho = 3.8\ \text{g/mL} = \frac{3.8\ \text{g}}{1\ \text{mL}}
- This can be used as a conversion factor depending on the problem.
Practice and resources:
- A dedicated practice lab on dimensional analysis and conversion factors is available (Brightspace) for extra help with equalities and conversion factors.
- A separate handout on dimensional analysis is also provided.

Process Example Problem (Dimensional Analysis)

Problem: A 195 lb patient with blood volume 7.5 quarts. If density of blood is 1.06 g/mL, what is the mass of the patient’s blood?
- Step 1: Identify the starting point (measured data): 7.5 quarts.
- Step 2: Use equalities to convert to mL (non-useful data should be ignored if not relevant):
- 1\ \text{qt} = 946\ \text{mL}
- So, 7.5\ \text{qt} = 7.5 \times 946 = 7095\ \text{mL}
- Step 3: Convert mL to grams using density: density = 1.06\ \text{g/mL}
- Mass: m = \rho \times V = 1.06\ \frac{\text{g}}{\text{mL}} \times 7095\ \text{mL} = 7520.7\ \text{g}
- Step 4: Express the result with the appropriate sig figs based on the starting measurement (7.5 qt has 2 sig figs): final mass ≈ 7.5 \times 10^{3}\ \text{g} (two sig figs).
Step 5: If needed, convert to more convenient units (e.g., kilograms): 7.5\times 10^{3}\ \text{g} = 7.5\ \text{kg} (depending on desired unit).
Practical notes:
- It’s important to identify the starting data and discard extraneous information (e.g., patient weight in pounds may be irrelevant to the density/volume calculation).
- Keep track of sig figs to determine the final report precision.

Additional Example Problems (Practical Conversions)

Prefix and unit conversion:
- Student height: 175 cm → meters:
- 1\ \,\text{m} = 100\ \text{cm}
- 175 cm → 1.75 m (3 sig figs)
Using conversion-factor-formatted steps (recommended): write every step as a fraction with explicit units to show cancellation and the direction of conversion.

Homework and Practice Problems (Chapter 1)

Homework Question 1: Four measurements — identify the correct number of significant figures for each measurement. No need to provide reasoning, but you may show it if you want.
Homework Question 2: Four calculations — determine the final answer with the correct number of significant figures. Do these on separate paper and upload to the assignment.
Homework Question 3: A Synthroid tablet has 75 micrograms. How many tablets are required to reach a total dosage of 0.150 mg (i.e., 0.15 mg)?
- Key facts:
- 1 mg = 1000 μg, so 0.150 mg = 150 μg.
- Each tablet provides 75 μg, so number of tablets = 150 μg / 75 μg = 2 tablets (2 sig figs).
Note: The instructor also mentions extra practice labs and handouts on dimensional analysis and conversion factors available on Brightspace for additional help.

Summary of Key Concepts to Remember

Exact numbers vs measured numbers; only measured numbers have significant figures and uncertainty.
The last digit in a measurement is the uncertain digit, unless specified otherwise.
Sig figs depend on the measuring device precision; exact numbers have unlimited sig figs.
Rounding rules depend on the operation: multiplication/division use sig figs; addition/subtraction use decimal places.
The base SI units and common prefixes translate to practical lab use (g, kg, m, L, mL, s, K, mol).
Volume, density, and specific gravity connect mass, volume, and material properties; displacement method for irregular shapes.
Conversion factors are exact equalities used to convert units; structure problems to ensure units cancel properly.
Real-world applications: density determines sinking/floating and specific gravity is used in processes like fermentation.

Quick Reference Formulas (in LaTeX)

Exact equalities (examples):
- 12\ \text{inches} = 1\ \text{foot}
- 60\ \text{seconds} = 1\ \text{minute}
- 1\ \text{L} = 1000\ \text{mL}
Temperature conversion: K = C + 273.15
Volume of regular solids: V = l \times w \times h
Density: \rho = \frac{m}{V}
Specific gravity: SG = \frac{\rho}{\rho_{water}}
Energy (joule): J = \text{kg}\cdot\text{m}^2/\text{s}^2
Conversion factor setup (example): 1\ \text{qt} = 946\ \text{mL} and use as a fraction to cancel units
Volume relationship: 1\ \text{cm}^3 = 1\ \text{mL}
Prefix magnitudes (typical): \text{kilo} = 10^3,\; \text{centi} = 10^{-2},\; \text{milli} = 10^{-3},\; \text{micro} = 10^{-6},\; \text{nano} = 10^{-9}