Scientific Notation, Metrics, and Significant Figures

Fundamental SI Units

The most common fundamental units include:

• Mass: kilogram (kg)
• Length: meter (m)
• Temperature: kelvin (K)
• Amount of substance: mole (mol)

Metric Prefixes

To scale these fundamental units up or down, we use metric prefixes. A common mnemonic to remember the order is "King Henry Doesn't Usually Drink Chocolate Milk":

Significant Figures: Communicating Precision

When you record a measurement, you are communicating how precise your tool was. Every measurement contains a certain number of digits that are known with certainty, plus one final digit that is uncertain. All of these digits combined are called significant figures (sig figs).

The Atlantic/Pacific Rule for Zeros

Zeros can be tricky. To determine if a zero is significant, use the Atlantic/Pacific rule:

1. Pacific Side (Decimal is Present): If a decimal point is present, start from the left (the Pacific side) and move toward the right. Start counting at the first non-zero digit you encounter. All zeros following that digit are significant.
2. Atlantic Side (Decimal is Absent): If there is no decimal point, start from the right (the Atlantic side) and move toward the left. Start counting at the first non-zero digit you encounter. All zeros preceding that digit are significant.

Examples:

• $2.700$ (Decimal present $\rightarrow$ Pacific): Start at 2, count the zeros. 4 sig figs.
• $960$ (Decimal absent $\rightarrow$ Atlantic): Start at 6, move left. The zero is not significant. 2 sig figs.
• $0.0096700$ (Decimal present $\rightarrow$ Pacific): Start at 9, count the following zeros. 5 sig figs.

Rules for Calculations and Rounding

When performing math, you must round your final answer to reflect the precision of your original measurements.

• Addition and Subtraction: Round the result to the same number of decimal places as the value with the fewest decimal places.
  • Example: $1.445 + 7.6 = 9.0$ (rounded to one decimal place).
• Multiplication and Division: Round the result to the same number of significant figures as the value with the fewest significant figures.
  • Example: $3.21 \times 1.4489 = 4.65$ (rounded to three sig figs).

Rounding Rule: If the digit to be dropped is less than 5, leave the number alone. If it is 5 or greater, round up.

Dimensional Analysis and Conversions

Dimensional Analysis is a method used to convert from one unit to another by using conversion factors. This is often done through a series of fractions where the units you want to remove are canceled out, leaving only the desired unit.

Temperature Conversions

In science, we often need to convert between Celsius ($^\circ\text{C}$) and Kelvin (K). The Kelvin scale is unique because it is an absolute scale where $0\text{ K}$ represents "absolute zero"—the point where all molecular movement stops.

The relationship is defined as: $0\text{ K} = -273.15 ^\circ\text{C}$

To convert from Celsius to Kelvin: $T_{\text{(K)}} = T_{\text{(}^\circ\text{C)}} + 273.15$

To convert from Kelvin to Celsius: $T_{\text{(}^\circ\text{C)}} = T_{\text{(K)}} - 273.15$

Examples:

• $36.3 ^\circ\text{C} \rightarrow 309.45\text{ K}$
• $72.0\text{ K} \rightarrow -201.15 ^\circ\text{C}$

Density Calculations

Density is a physical property that describes the mass of a substance per unit of its volume. It is a crucial concept for identifying substances and understanding their physical characteristics.

The formula for density ($D$) is: $D = \frac{M}{V}$

Where:

• $M$ = Mass
• $V$ = Volume

Common units for density include $\text{g/cm}^3$ or $\text{g/mL}$.

Worked Example: If a mineral has a mass of $17.8\text{ g}$ and a volume of $2.35\text{ cm}^3$, its density is: $\begin{aligned} D &= \frac{17.8\text{ g}}{2.35\text{ cm}^3} \\ D &= 7.57\text{ g/cm}^3 \end{aligned}$

If you need to find the mass ($M$) and you already know the density and volume, you can rearrange the formula: $M = D \times V$

For example, a $49.6\text{ mL}$ sample of a liquid with a density of $0.85\text{ g/mL}$ has a mass of: $\begin{aligned} M &= (0.85\text{ g/mL}) \times (49.6\text{ mL}) \\ M &= 42\text{ g} \end{aligned}$