Significant Figures, Unit Conversions, and Scientific Calculations in General Chemistry 2.0

Significant Figures and Scientific Calculations

Understanding Significant Figures

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated. They are crucial in scientific calculations to reflect the precision of measured values.

• Definition: Significant figures indicate the precision of a measured or calculated quantity.

• Rules for Counting Significant Figures:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros are significant only if there is a decimal point.

• Example: 75.052 g has 5 significant figures; 1.852 g has 4 significant figures.

Significant Figures in Calculations

When performing calculations, the number of significant figures in the result depends on the operation:

• Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

• Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: Subtracting 73.18 g from 75.052 g (both to the hundredths place) gives a result with 2 significant figures: $75.052 - 73.18 = 1.9$ g.

Unit Conversions

Converting Between Units

Unit conversions are essential in chemistry to express measurements in different units. This often involves multiplying by conversion factors that relate two units.

• Conversion Factor: A ratio that expresses how many of one unit are equal to another unit.

• Example: To convert 0.0921 nm to pm:

  • 1 nm = 1000 pm

  • $0.0921 \text{ nm} \times \frac{1000 \text{ pm}}{1 \text{ nm}} = 92.1 \text{ pm}$

Density and Unit Conversion

Density is a physical property defined as mass per unit volume. It can be converted between units using appropriate conversion factors.

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

• Example: To convert the density of gold from $18.9 \text{ g/cm}^3$ to $\text{lb/ft}^3$:

  • Use the following conversion factors:

    • 1 lb = 453.592 g

    • 1 ft = 30.48 cm, so $1 \text{ ft}^3 = (30.48 \text{ cm})^3 = 28,316.8 \text{ cm}^3$

  • $18.9 \text{ g/cm}^3 \times \frac{1 \text{ lb}}{453.592 \text{ g}} \times 28,316.8 \text{ cm}^3/\text{ft}^3 = 1,180 \text{ lb/ft}^3$ (rounded to 3 significant figures)

Scientific Notation and Calculations

Using Scientific Notation

Scientific notation is used to express very large or very small numbers in a compact form. It is especially useful in chemistry for dealing with measurements at the atomic or molecular scale.

• Format: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

• Example: $2.3357 \times 10^7$

Calculations with Scientific Notation

• Multiplication: Multiply the coefficients and add the exponents.

  • $(2.3357 \times 10^7) \times (6.92 \times 10^6) = (2.3357 \times 6.92) \times 10^{7+6}$

• Division: Divide the coefficients and subtract the exponents.

  • $(3.0223 \times 10^4) / (1.01 \times 10^2) = (3.0223 / 1.01) \times 10^{4-2}$

Practice Problems and Applications

Sample Problems

1. Significant Figures in Subtraction: $75.052 \text{ g} - 73.18 \text{ g} = 1.9 \text{ g}$ (2 significant figures)

2. Significant Figures in Division: $0.11 / 3.97 = 0.025$ (rounded to 2 significant figures)

3. Significant Figures in Addition: $98.00725084 + 2.8052381 = 100.81$ (rounded to 2 decimal places)

4. Significant Figures in Multiplication/Division: $27.92 / 14.3 = 1.952$ (rounded to 3 significant figures)

5. Scientific Notation Multiplication: $(2.3357 \times 10^7) \times (6.92 \times 10^6) = 1.616 \times 10^{14}$ (rounded to 4 significant figures)

Unit Conversion Examples

• Blood Glucose Concentration: Convert 122 mg/dL to g/L:

  • $122 \text{ mg/dL} \times \frac{1 \text{ g}}{1000 \text{ mg}} \times \frac{10 \text{ dL}}{1 \text{ L}} = 1.22 \text{ g/L}$

• Area Conversion: Convert $27.2 \text{ cm}^2$ to $\text{in}^2$:

  • 1 in = 2.54 cm, so $1 \text{ in}^2 = (2.54 \text{ cm})^2 = 6.4516 \text{ cm}^2$

  • $27.2 \text{ cm}^2 \times \frac{1 \text{ in}^2}{6.4516 \text{ cm}^2} = 4.22 \text{ in}^2$

Summary Table: Significant Figures Rules

Operation	Rule for Significant Figures	Example
Addition/Subtraction	Result has same number of decimal places as the least precise measurement	12.32 + 1.1 = 13.4
Multiplication/Division	Result has same number of significant figures as the least precise measurement	2.5 × 3.42 = 8.6