Chemistry: Sig Figs & Scientific Notation

Understanding Significant Figures

• Significant Figures (Sig Figs): Digits in a number that contribute to its accuracy.
  • Zeros at the end of a number without a decimal point are not significant.
  • Example: 9,000 has one significant figure unless specified with a decimal, e.g., 9,000. has four significant figures.
  • In numbers like 52,000.390, there would be five significant figures due to the decimal point.

Rules for Counting Significant Figures

1. Leading Zeros: Zeros before the first non-zero digit are not significant.
   • Example: In 0.02901, the leading zeros do not count; hence, it has five significant figures.
2. Captive Zeros: Zeros between non-zero digits are always significant.
3. Trailing Zeros with Decimal Point: Zeros at the end of a number with a decimal point are significant.
   • Example: 120.010 has six significant figures.

Exact Numbers and Their Significance

• Exact Numbers: Numbers that arise from counting or defined relationships (e.g., conversion factors) are considered to have an infinite number of significant figures.
  • Example: Counting 2 apples gives an exact figure, thus infinite sig figs.

Scientific Notation

• Purpose: A method to express very large or very small numbers conveniently.
• Format: $a imes 10^b$, where:
  • $a$ is a number between 1 and 10.
  • $b$ indicates the number of decimal places to move.
• The part of the number before the multiplication symbol may have significant figures, but the exponent itself is considered exact.

Writing Numbers in Scientific Notation

• To convert a number into scientific notation:
  1. Move the decimal point so that one non-zero digit is left of the decimal.
  2. Count how many places the decimal was moved and adjust the exponent accordingly (positive for large numbers, negative for small).
  • Example: $0.00006789 = 6.789 imes 10^{-5}$.

Rounding Significant Figures

• Rounding rules depend on the digit immediately following the last significant figure to keep:
  • If that digit is less than 5, round down.
  • If that digit is 5 or greater, round up.
• Example: For 1.00629 rounding to four sig figs, we drop the last two digits to get 1.006.
• If the number were 1.00669, we would round it to 1.007.

Using Calculators for Scientific Notation

• Understand how to enter values in scientific notation on your specific calculator.
• Most scientific calculators use the format: number e exponent (e.g., 4.0 e 6 for $4.0 imes 10^6$).

Operations with Significant Figures

• When adding or subtracting, the result should be recorded with the same number of decimal places as the measurement with the least decimal places.
  • Example: $67.98 - 1.5 = 66.48$ is reported as 66.5 because 1.5 has one decimal place.
• When multiplying or dividing, the result should have the same number of significant figures as the measurement with the least significant figures.

Estimating and Reporting Measurements

• Measurement includes a degree of uncertainty, and the last digit in a measurement reflects this estimation.
  • Example: Reporting temperature as 26.2 means certainty in the first two digits, but uncertainty in the last (the estimated digit).