Significant Figures and Measurement

Significant Figures in Measurements

Definition of Significant Figures: Significant figures (or significant digits) represent the precision of a measured quantity. They are important in scientific communication and ensure clarity concerning the certainty of measurements.

Nonzero Digits: All nonzero digits are significant.
- Example: In 123.45, all digits count, resulting in 5 significant figures.
Zeros Between Nonzero Digits: Any zeros that are placed between nonzero digits are significant.
- Example: In 106.7, there are 4 significant figures.
Zeros at the End of a Decimal Number: Zeros that occur at the end of a decimal number are significant.
- Example: The number 30.0 has 3 significant figures, due to the trailing zero.
Zeros in Scientific Notation: In scientific notation, zeros in the coefficient are considered significant, regardless of the presence of a decimal point.
- Example: In the number 1.80 × 10^4, there are 3 significant figures due to the 1, 8, and 0.
Leading Zeros: Zeros at the beginning of a decimal number are not considered significant; they merely act as placeholders.
- Example: In 0.00539, the leading zeros do not count, resulting in 3 significant figures.

Measured Quantities: Students are tasked to categorize the following quantities based on their significant figures:
- 0.00 59 m → 3 significant figures
- 1.80 × 10^4 kg → 3 significant figures
- 0.8250 L → 4 significant figures
- 110.09 g → 5 significant figures
- 36,000,000 km → 2 significant figures (presuming no decimal)
- 30.0 °C → 3 significant figures

Students check for pairs of numbers with the same number of significant figures:
- 410000 s and 4.10 × 10^5 s → 5 significant figures
- 281 K and 281.0 K → 4 significant figures (the decimal in 281.0 makes the last zero significant)
- 0.00539 g and 5.39 × 10^-3 g → 3 significant figures
- 4.07 × 10^-2 L and 4.70 × 10^5 L → 3 and 3 significant figures, respectively

Determination of Zeros: Rules for whether zeros are significant:
- All Zeros Significant:
  - Example: 2.01 km (3 significant figures)
- No Zeros Significant:
  - Example: 27000 g (2 significant figures, no decimal)
  - Example: 0.0015 cm (2 significant figures, leading zeros are not counted)

Problem: A nurse practitioner prepares an injection of promethazine, labeled as 20. mg/mL, and is supposed to administer a dose of 15.0 mg.
- Calculation: Determine the volume in milliliters that corresponds to this dose:
- Use the conversion factor:
  ext{Volume (mL)} = rac{15.0 ext{ mg}}{20.0 ext{ mg/mL}} = 0.75 ext{ mL}
Significant Figures: The final answer should be expressed to 2 significant figures. Thus, the volume drawn is 0.75 mL.

Problem: For ampicillin, prescribed at a dose of 35 mg/kg for a 66 lb child. Each capsule contains 250 mg.
- Step 1: Convert lbs to kg:
  ext{mass (kg)} = 66 ext{ lb} imes rac{1 ext{ kg}}{2.20 ext{ lb}} = 30.0 ext{ kg}
- Step 2: Calculate necessary mg dosage:
  ext{Required dose (mg)} = 35 ext{ mg/kg} imes 30.0 ext{ kg} = 1050.0 ext{ mg}
- Step 3: Find number of capsules:
  ext{Number of capsules} = rac{1050 ext{ mg}}{250 ext{ mg/capsule}} = 4.2 ext{ capsules}
- Final answer should be rounded to the nearest whole number: 4 capsules.

Problem: Dosage of asthma medication is 3.0 mg/kg. What is the required dose for a child weighing 38 lbs?
- Step 1: Convert lbs to kg:
  ext{mass (kg)} = 38 ext{ lb} imes rac{1 ext{ kg}}{2.20 ext{ lb}} = 17.27 ext{ kg}
- Step 2: Calculate necessary mg dosage:
  ext{Required dose (mg)} = 3.0 ext{ mg/kg} imes 17.27 ext{ kg} = 51.81 ext{ mg}
- Rounded to 2 significant figures: 52 mg

In practical implementations, especially in fields requiring precision (like medicine), maintaining the correct number of significant figures avoids dosages that could lead to under or overdosing patients.
Always round cautiously and track units to ensure that all calculations uphold the rules of significant figures throughout.