SI Units, Metric Conversions, Derived Units, and Temperature Calculations

Measurement Error and Constant Values

• Measurement quantities carry inherent uncertainty and error (e.g., measuring out $3.28\,\text{mL}$ of hydrogen peroxide requires tracking signal errors).

• Physical constants do not contribute to propagation of error in calculations.

• Planck's constant is a defined constant; multiplying measurements by Planck's constant does not require accounting for additional uncertainty from the constant itself.

The International System of Units (SI)

• A unit is defined as a fixed standard of measurement.

• The International System of Units (known as SI) is a specific standard metric system from which all other units are derived.

• Quantities and their corresponding SI base units:

  • Mass: Gram family, base unit gram ($\text{g}$).

  • Length: Meter ($\text{m}$).

  • Temperature: Represented by concept symbol $T$. Standard SI base unit is Kelvin ($\text{K}$), though Celsius ($^\circ\text{C}$) is also used.

  • Amount of Substance: Mole ($\text{mol}$).

  • Time: Second ($\text{s}$).

  • Volume: Liter ($\text{L}$).

Metric System Prefixes and Scaling

• Prefixes are attached to SI base units to indicate orders of magnitude:

  • Mega ($\text{M}$): $10^5$

  • Kilo ($\text{k}$): $10^3$ ($1000$)

  • Centi ($\text{c}$): $10^{-2}$ ($1/100$)

  • Milli ($\text{m}$): $10^{-3}$ ($1/1000$)

  • Micro ($\mu$): $10^{-6}$

  • Nano ($\text{n}$): $10^{-9}$

• Metric Scale Structure:

  • Most metric step conversions differ by a factor of $1000$ ($10^3$).

  • Conversion example: $1\,\text{kg} = 1000\,\text{g}$.

  • Conversion example: $1\,\text{g} = 1000\,\text{mg}$.

  • Conversion example: $1\,\text{m} = 1000\,\text{mm}$.

  • Centi represents a factor of $100$: $1\,\text{m} = 100\,\text{cm}$.

  • Millimeter to centimeter relation: $10\,\text{mm} = 1\,\text{cm}$.

• Conversion Guidelines:

  • Move systematically along magnitude steps. Do not jump straight across multiple orders of magnitude (e.g., convert $\text{mm} \rightarrow \text{m} \rightarrow \text{km}$ rather than $\text{mm} \rightarrow \text{km}$ directly).

  • Directional rule on the metric line: Moving to the right corresponds to multiplication; moving to the left corresponds to division.

  • Base/central units on the metric line include gram ($\text{g}$), liter ($\text{L}$), and meter ($\text{m}$).

  • Condensed metric conversion reference handouts are provided for quizzes, but must be fully memorized for examinations.

Derived Units and Spatial Volume

• Volume can be defined in two ways:

  • Liquid volume: Represented directly by liters ($\text{L}$).

  • Spatial volume: Calculated geometrically using the equation: $\text{Volume} = \text{Area} \times \text{Height}$

• Derived units are formed by mathematically combining standard base units.

• Primary Derived Units:

  • Area: Calculated as $\text{Length} \times \text{Length}$. Expressed as $\text{m}^2$, $\text{cm}^2$, or $\text{nm}^2$ $\text{Area} = \text{m} \times \text{m} = \text{m}^2$

  • Spatial Volume: Calculated as $\text{Area} \times \text{Length}$. Expressed as $\text{m}^3$ or $\text{cm}^3$ $\text{Volume} = \text{m}^2 \times \text{m} = \text{m}^3$

  • Density: Calculated as $\frac{\text{Mass}}{\text{Volume}}$. Standard SI representation is $\frac{\text{kg}}{\text{m}^3}$, but practical laboratory units are $\frac{\text{g}}{\text{mL}}$ or $\frac{\text{g}}{\text{cm}^3}$ $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

  • Speed: Calculated as $\frac{\text{Distance}}{\text{Time}}$. Expressed in SI units as $\frac{\text{m}}{\text{s}}$ $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Unit Conversion and Dimensional Analysis

• A conversion factor is an expression of relationship between two equal values with different units.

• Conversion factors always equal $1$ because the numerator and denominator represent identical values: $12\,\text{inches} = 1\,\text{foot}$ $1000\,\text{mg} = 1\,\text{g}$ $1000\,\text{g} = 1\,\text{kg}$

• Significant figures in conversions:

  • Conversion factors are exact numbers and carry zero uncertainty.

  • Exact conversion factors do not limit the number of significant figures in a calculation.

Conversion Practice Problems

• Problem 1: Order the following measurements from smallest to largest: $2.3 \times 10^2\,\text{mg}$ $2.3 \times 10^{-13}\,\text{g}$ $2.3 \times 10^{-3}\,\text{kg}$

  • Step 1: Convert $2.3 \times 10^2\,\text{mg}$ to base unit grams ($\text{g}$): $2.3 \times 10^2\,\text{mg} \times \frac{1\,\text{g}}{1000\,\text{mg}} = 0.23\,\text{g}$

  • Step 2: Check significant figures. The starting term $2.3 \times 10^2$ has $2$ significant figures. The answer $0.23\,\text{g}$ maintains $2$ significant figures.

  • Step 3: Convert $2.3 \times 10^{-3}\,\text{kg}$ to base unit grams ($\text{g}$): $2.3 \times 10^{-3}\,\text{kg} \times \frac{1000\,\text{g}}{1\,\text{kg}} = 2.3\,\text{g}$

  • Step 4: Compare values in grams: $2.3 \times 10^{-13}\,\text{g} < 0.23\,\text{g}\ (2.3 \times 10^2\,\text{mg}) < 2.3\,\text{g}\ (2.3 \times 10^{-3}\,\text{kg})$

• Problem 2: Calculate how many cubic meters ($\text{m}^3$) are in $1\,\text{L}$.

  • Essential bridge conversion factor between volume and length units: $1\,\text{mL} = 1\,\text{cm}^3$

  • Conversion setup: $1\,\text{L} \times \frac{1000\,\text{mL}}{1\,\text{L}} \times \frac{1\,\text{cm}^3}{1\,\text{mL}} \times \left(\frac{1\,\text{m}}{100\,\text{cm}}\right)^3$

  • Expanding the cubed conversion factor: $\left(\frac{1\,\text{m}}{100\,\text{cm}}\right)^3 = \frac{1\,\text{m}^3}{1000000\,\text{cm}^3}$

  • Calculation: $1 \times 1000 \times 1 \times \frac{1}{1000000} = 1 \times 10^{-3}\,\text{m}^3$

Temperature Conversions

• Temperature measures how hot or cold a substance is.

• Three standard temperature units: Fahrenheit ($^\circ\text{F}$), Celsius ($^\circ\text{C}$), and Kelvin ($\text{K}$).

• Conversion equations: $\text{Kelvin} = ^\circ\text{C} + 273.15$ $^\circ\text{F} = \frac{9}{5}(^\circ\text{C}) + 32$

• Problem 3: Convert normal body temperature $98.6\,^\circ\text{F}$ into Celsius and Kelvin.

  • Step 1: Solve for Celsius: $98.6\,^\circ\text{F} = \frac{9}{5}(^\circ\text{C}) + 32$ $98.6 - 32 = 66.6$ $66.6 = \frac{9}{5}(^\circ\text{C})$ $^\circ\text{C} = 66.6 \times \frac{5}{9} = 37.0\,^\circ\text{C}$

  • Significant figures check: $98.6$ contains $3$ significant figures, so Celsius is recorded as $37.0\,^\circ\text{C}$ ($3$ significant figures).

  • Step 2: Solve for Kelvin: $\text{Kelvin} = 37.0 + 273.15 = 310.15\,\text{K}$

  • Significant figures check: Rounding to match precision yields $310\,\text{K}$ ($3$ significant figures).

Questions & Discussion

• Question: What calculator input method should be used for scientific notation?

  • Answer: Enter terms using the dedicated scientific notation exponent key (EE or EXP for $\times 10^x$) rather than the constant $e$. Enclose exponential expressions in parentheses when performing division or exponentiation to maintain correct order of operations.

• Question: How do you determine whether to multiply or divide when moving across the metric conversion line?

  • Answer: When moving to the right along the scale (towards smaller unit sizes), multiply. When moving to the left along the scale (towards larger unit sizes), divide.

• Question: Can final temperature values be expressed as whole numbers?

  • Answer: Yes, provided the final value correctly reflects the required number of significant figures dictated by the initial measurement precision.

• Question: What is the policy regarding Problem of the Day (POD) and homework notebooks?

  • Answer: Perform error propagation calculations by retaining all unrounded intermediate digits throughout the calculation steps, rounding only at the final step. Homework notebooks containing all completed practice problems are checked in person on the scheduled examination day.