Understanding Significant Figures

Significant figures, often abbreviated as "sig figs," are critical in the field of chemistry for reporting measurements of physical quantities like mass and volume.
The precision of a measurement is influenced by the instruments or glassware used, and this precision must be communicated in reported values and calculations.
Proper communication of significant figures allows other scientists to understand how reliable the data is, reflected in their calculations or findings (e.g., mass of aluminum cube).

Any non-zero digit in a reported measurement is always significant.
- Example: In the measurement "45", both the 4 and 5 are significant.

Zeros that appear between two non-zero digits are considered significant.
- Example: In "405", the zero is significant because it is sandwiched between 4 and 5.

Zeros to the left of the first non-zero digit are not significant.
- Example: In "0.045", the leading zero before the 4 is not significant; it only has two significant figures, which are 4 and 5.

Zeros to the right of non-zero digits are significant only if there is a decimal point present.
- Example: In "0.405", the last zero is significant because there is a decimal point; thus, it has four sig figs.

Various measurements should be evaluated for sig figs:
1. Measurement "4050" has four sig figs since there's a trailing zero with a decimal point (reported as "4050.").
2. Measurement "200" has one sig fig if no decimal is present; however, "200.0" would have four sig figs.
3. Measurement "4200" without a decimal point has two significant figures; in "4200.", however, all four digits are significant.

In scientific notation, only the number before the power of 10 is significant.
- Example: For the measurement $(4.50 imes 10^3)$ , it has three significant figures.
The exponent and the base "10" are exact numbers and do not affect the sig fig count.

The precision of a calculated value is limited by the least precise value involved in the calculation.

For multiplication and division, the final result should contain the same number of significant figures as the measurement with the fewest sig figs.
- Example: Given values $(4.6 ext{ mL})$ (3 sig figs) and $(0.78 ext{ g/mL})$ (2 sig figs), compute:
- Calculation yields $(3.588)$ , but report as $(3.6 ext{ g})$ (2 sig figs, due to 0.78 g/mL).

Note: Conversion factors (e.g., $(1,000 ext{ mL} = 1 ext{ L})$ ) are exact values and do not affect the sig fig count of the measurement used in the calculation.
- Example: Converting $(25.04 ext{ mL})$ to liters remains significant with four sig figs: $(0.02504 ext{ L})$ .

For addition and subtraction, the result must be reported with the same number of decimal places as the value with the least decimal places.
- Example: $(3.7 ext{ mL})$ has one digit after the decimal, limiting the answer $(4.0 ext{ mL})$ to one digit after the decimal as well.

When performing calculations:
1. The answer for the operation of subtraction ( $4.5 ext{ ATM}$ ) must fit the sig fig rule of one decimal point dictated by a number like $0.3 ext{ ATM}$ .
2. In division, as seen in $8.62 ext{ mol}$ , the answer must keep the sig fig rule determined by the least precise measurement involved (three sig figs from $504 ext{ g}$ ).

Proficiency in recognizing and applying the rules of significant figures should become second nature for students and professionals in science.
Significant figures serve as a communication tool for the precision of measurements throughout all calculations in scientific contexts and methodologies.