Notes on Sig Figs, Unit Analysis, and Measurements

Significant Figures, Zeros, and Rounding

Purpose: Determine how many digits in a number are meaningful (sig figs) and how to handle them in calculations.
Key idea: Not all digits are equally important; consider how measurements were made and how precise they are.
Units and conversion factors are central to calculations: convert to the desired unit by multiplying by factors that link units.

Zeros and their significance

Leading zeros: zeros at the start of a number (before the first nonzero digit) are not significant.
- Example: $0.0061$ has two significant figures (the 6 and the 1).
Captive zeros (zeros between nonzero digits): always significant.
Trailing zeros after a decimal point: significant.
Trailing zeros without a decimal point: significance can be ambiguous; decimal presence matters.
If a zero is between a decimal point and a leading nonzero digit (e.g., 0.0… or 00.12): those zeros may be leading or captive depending on position; in general, zeros that are between a nonzero digit and a decimal are captured and often significant if they help define precision.
Practically, compare quantities by the number of sig figs, not just digits.

Exact numbers and definitions

Some conversion factors are exact by definition and do not limit sig figs.
- Examples: $1\text{ in} = 2.54\ \text{cm}$ (exact definition; used as a fixed conversion factor)
- Other exact definitions may appear as tables (e.g., counting constants).
When a factor is exact, you can treat it as having unlimited sig figs for the purpose of the calculation.

Scientific notation and sig figs

Scientific notation helps visualize sig figs: a quantity with $n$ sig figs is written as something like
- $1.00\times 10^{3}$ (3 sig figs)
- $1\times 10^{3}$ (1 sig fig)
When converting between representations, preserve the number of sig figs from the original quantity at each step; do not round at every intermediate step.
Example: converting $1000$ with different precision:
- $1000$ (1 sig fig if no decimal shown) vs $1000.0$ (5 sig figs)

Rounding rules (to a given number of sig figs)

To round to N sig figs, look at the (N+1)th digit:
- If it is 5 or larger, round the Nth digit up.
- If it is less than 5, leave the Nth digit as is.
Example with $1.648\,$
- To 3 sig figs: keep 1.65 (since the next digit 8 >= 5, round up the 4 to 5).
- To 2 sig figs: keep 1.6 (look at the next digit after 1.6; it is 4 < 5, so do not round up).
Important caveat: do not round in every intermediate step of a multi-step calculation; carry extra digits and round only at the end (or as dictated by the least-precise input).

Addition and subtraction with significant figures

For addition/subtraction, the answer should have the same number of decimal places as the quantity with the least decimal places.
Example: $2.8$ (1 decimal place) + $0.28$ (2 decimal places) = $3.08$ ; final should be reported with 1 decimal place ⇒ $3.1$ .
Intuition: more decimal places means more precision; the least precise measurement controls the final decimal accuracy.

Multiplication and division with sig figs

For multiplication/division, the number of sig figs in the result equals the smallest number of sig figs among the inputs.
Example: multiplying three numbers: 3 sig figs, 2 sig figs, and 4 sig figs ⇒ final result has 2 sig figs.
If a calculation involves both multiplication/division and addition/subtraction, keep the track of sig figs for the multiplication/division steps, but apply the decimal-place rule for the addition/subtraction parts as appropriate.
Practice example (from transcript):
- Given numbers with 3, 2, and 4 sig figs, the final should have 2 sig figs.
- A division example: dividing numbers with 4 sig figs by a number with 2 sig figs can yield a result with 2 sig figs.
- If a subsequent subtraction or addition changes decimal places, adjust the final result accordingly (no decimals when the least-precise input dictates none).

Mixed operations (illustrative approach)

For a mixed-operation problem, you’ll typically identify the least number of sig figs across all numbers involved in the final calculation and apply that to the final result.
Example outcome mentioned in the transcript: a mixed-operation result such as $23.85$ (emphasizing keeping track of sig figs across the full calculation).

Temperature scales and conversions

Basic concepts:
- Kinetic energy correlates with temperature: hotter objects tend to have higher kinetic energy.
- Heat transfer typically occurs from hot to cold bodies.
Temperature scales used in laboratory practice:
- Celsius (°C) and Kelvin (K) are common in scientific contexts; Fahrenheit is also used in some contexts.
- Kelvin is the absolute scale; 0 K is the lowest possible temperature.
Key fixed conversions:
- Absolute zero and the relation to Celsius: $0\ \text{K} = -273.15\,^{\circ}\text{C}$
- From Celsius to Kelvin: $K = C + 273.15$
- From Kelvin to Celsius: $C = K - 273.15$
- Freezing and boiling points (at standard pressure):
- Freezing: $0^{\circ}\text{C} = 273.15\ \text{K}$
- Boiling: $100^{\circ}\text{C} = 373.15\ \text{K}$
Fahrenheit conversions (less routine, but provided for completeness):
- From Fahrenheit to Celsius: $C = \tfrac{5}{9}(F-32)$
- From Celsius to Fahrenheit: $F = \tfrac{9}{5}C + 32$

Volume, density, and units

Common volume/mass units:
- Mass: grams (g)
- Volume: milliliters (mL) or cubic centimeters (cm^3)
- Density often uses: $\rho = \frac{m}{V}$ with units such as $\text{g/cm}^3$ or $\text{g/mL}$ (these are equivalent in the sense that 1\,mL = 1\,cm^3).
Density is an intensive property: it does not depend on the amount of substance.
- Example: copper sulfate solution is blue whether you have a small amount or a large amount; color is intensive and remains the same.
Mass is an extensive property: it scales with the amount of substance.
e- If you double the amount of substance, mass doubles.
Practical implication: density remains constant for a given substance under the same conditions, but mass and volume change with amount.

Intensive vs Extensive properties (definitions)

Intensive property: independent of the amount of substance present.
- Example: color, density.
Extensive property: depends on the amount of substance present.
- Example: mass, volume (for homogenous substances, volume correlates with amount).
Note: In upcoming topics (e.g., enthalpy in Chapter 9), enthalpy is extensive.

Unit analysis / Dimensional analysis (conversion-factor approach)

Core idea: use unit factors to convert from a given unit to a desired unit, ensuring units cancel step by step.
How to set up:
- Write the given unit on the bottom (to cancel it) and the desired unit on the top.
- Include conversion factors that link the given unit to the desired unit.
- Cancel units as you go and ensure the final unit is the target unit.
Example workflow:
- To count hours in a year:
- Known: $1\ \text{year} = 365\ \text{days}$ and $1\ \text{day} = 24\ \text{hours}$ .
- Setup: $1\ \text{year} \times \tfrac{365\ \text{days}}{1\ \text{year}} \times \tfrac{24\ \text{hours}}{1\ \text{day}}.$
- Result: the years cancel, leaving hours.
Practical tips:
- Always include units in the setup to verify you’re moving in the right direction.
- If the final unit is not what you expect, re-check the conversion factors and the setup.
- The term “conversion factors” is used interchangeably with “prefix multipliers” or “unit factors.”

Prefix multipliers (memory aid)

You should memorize common prefixes and their multipliers, especially milli- ((10^{-3})) and others as needed.
Example: 1\,mL = 1\,cm^3 (and density units often end up as g/cm^3 or g/mL).

Practice problem: converting grams to ounces

Given: cake recipe calls for 250 g; convert to ounces using 1 oz = 28.25 g.
Setup: $250\ \text{g} \times \tfrac{1\ \text{oz}}{28.25\ \text{g}}$
Units: g cancels, leaving ounces ((\text{oz})).
Calculation: $250 \times \tfrac{1}{28.25} \approx 8.849…$
With the appropriate sig figs (based on the given 250 g and 28.25 g per oz; here 28.25 has four sig figs and 250 g could be two or three depending on context; using two sig figs for 250 g gives 8.8 oz), the result is approximately $8.8\ \text{oz}$ .
Common pitfalls: always report the final unit and ensure the significant figures align with the least precise input.

Quick references from the transcript

1 inch = 2.54 cm (exact)
1 kg = 1000 g
1 mL = 1 cm^3
1 day = 24 hours; 1 year ≈ 365 days
0 °C = 273.15 K; 100 °C = 373.15 K
0 K = -273.15 °C
2.8 + 0.28 = 3.08 → final value should reflect the least decimal places among the operands (1 decimal place in this case → 3.1)
1.648 to 3 sig figs → 1.65; to 2 sig figs → 1.6
For mixed operations, track sig figs across the entire calculation; final answer example given: 23.85 (illustrative)
Example division excerpt: a calculation involving numbers such as 59.35 (4 sig figs) and a divisor with fewer sig figs leads to the final with the least number of sig figs; in the transcript, a final result without decimals (e.g., 24) is obtained after considering decimal places.

Notes on interpretation of the transcript

The speaker emphasizes not rounding at every intermediate step to avoid introducing error.
The emphasis is on matching the final precision to the least precise measurement in the calculation.
The material blends practical lab techniques (unit analysis, conversion factors) with core concepts of significant figures, density, temperature conversions, and the nature of physical properties (intensive vs extensive).
When preparing for exams, be comfortable with:
- Identifying significant figures in numbers, including tricky cases with zeros.
- Applying the rules for addition/subtraction and multiplication/division correctly.
- Performing unit conversions via conversion factors and ensuring proper cancellation.
- Carrying out temperature conversions and understanding the relationships between Celsius, Kelvin, and Fahrenheit.
- Recognizing density as an intensive property and mass as an extensive property.
- Using density as mass per volume: $\rho = \dfrac{m}{V}$ with appropriate units.