Derived Units, Density, Accuracy & Precision, and Significant Figures

Derived Units

Definition and Examples

Derived units are units that are obtained by combining the seven fundamental SI units. These combinations are used to express physical quantities such as area, volume, and density.

• Area: Calculated as length × length, with the SI unit of square meters ($m^2$).

• Volume: Calculated as length × length × length, with the SI unit of cubic meters ($m^3$).

Example: The volume of a cube with sides of 1 cm is $1~cm \times 1~cm \times 1~cm = 1~cm^3$.

Relationship Between $cm^3$ and $mL$

• 1 cubic centimeter ($1~cm^3$) is exactly equal to 1 milliliter ($1~mL$).

• This equivalence is fundamental in chemistry for converting between volume units.

Key Conversion:

• $1~cm^3 = 1~mL$

• $1~L = 1000~mL = 1000~cm^3$

Example: A cube with sides of 10 cm has a volume of $10~cm \times 10~cm \times 10~cm = 1000~cm^3 = 1~L$.

Density

Definition and Formula

Density is a physical property defined as the mass of a substance per unit volume. It is unique for each substance and can be used for identification.

• Formula:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Common units: $g/cm^3$ or $g/mL$ for solids and liquids; $kg/m^3$ for gases.

Example: If a plastic piece has a mass of 2.00 g and a volume of 2.50 mL, its density is $\frac{2.00~g}{2.50~mL} = 0.80~g/mL$.

Density of Common Materials

The density of a substance can help identify it. For example, different plastics have characteristic densities, which can be used in laboratory identification.

Substance	Density (g/cm3)	Substance	Density (g/cm3)
Water (3.98°C)	1.0000	Gold	19.31
Ice (0°C)	0.917	Human fat	0.94
Air (25°C)	0.001185	Human muscle	1.06
Helium (25°C)	0.000164

Additional info: Densities of plastics such as PET, HDPE, PVC, LDPE, PP, and PS are used in recycling and identification.

Temperature Dependence of Density

• Density is temperature dependent; as temperature increases, density generally decreases due to expansion.

• For water, the maximum density (1.000 g/mL) occurs at approximately 4°C.

Example: In laboratory experiments, the temperature of water should be measured to obtain the correct density, rather than assuming 1.00 g/mL.

Worked Example: Calculating Volume from Density and Mass

Given: Chloroform has a density of 1.5 g/mL at 20°C. How many mL are needed for 3.0 g?

• Use the formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Rearrange: $\text{Volume} = \frac{\text{Mass}}{\text{Density}}$

• Calculation: $\text{Volume} = \frac{3.0~g}{1.5~g/mL} = 2.0~mL$

Tip: Always confirm that units cancel appropriately in calculations.

Accuracy and Precision

Definitions

• Accuracy: How close a measured value is to the true or accepted value.

• Precision: How closely repeated measurements agree with each other, regardless of their closeness to the true value.

Example: In dart throwing, if all darts land near the bullseye, the throws are accurate. If all darts land close together but far from the bullseye, the throws are precise but not accurate.

Comparison Table: Accuracy vs. Precision

	High Accuracy	Low Accuracy
High Precision	Measurements are close to the true value and to each other.	Measurements are close to each other but far from the true value.
Low Precision	Measurements are close to the true value but not to each other.	Measurements are neither close to the true value nor to each other.

Significant Figures

Definition and Rules

Significant figures (SFs) are the digits in a measurement that are known with certainty plus one digit that is estimated. They reflect the precision of a measured value.

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Leading zeros are not significant.

• Trailing zeros in a decimal number are significant.

• Exact numbers (such as counted objects) have an infinite number of significant figures.

Examples of Significant Figures

• 0.00661 g: 3 significant figures (6, 6, 1)

• 34200 m: 3, 4, or 5 significant figures depending on notation (e.g., 3.42 × 104 has 3 SFs; 3.4200 × 104 has 5 SFs)

• 34200. m: 5 significant figures (decimal point indicates all digits are significant)

• 50 students: Infinite significant figures (counted, not measured)

Rule of Measurement: Record all certain digits plus one estimated digit (read to 1/10 of the smallest increment on the measuring device).

Why Significant Figures Matter

• They communicate the precision of measurements and calculations.

• They are essential for reporting scientific data accurately.

Additional info: The next lecture will cover how to handle significant figures in calculations, rounding, and unit conversions.