Numbers in Chemistry: Scientific Notation, Significant Figures, and Unit Conversions

Scientific Notation

Definition and Structure

Scientific notation is a method used in chemistry to express very large or very small numbers in a compact form. It consists of two main parts:

• Decimal part: A number between 1 and 10.

• Exponential part: 10 raised to an exponent, n.

For example, $1.2 \times 10^{-10}$, where 1.2 is the decimal part and $10^{-10}$ is the exponential part.

Interpreting Exponents

• Positive exponent (n): Indicates multiplication by 10, n times. $1 \times 10^1 = 10$ $1 \times 10^2 = 100$ $1 \times 10^3 = 1000$

• Negative exponent (–n): Indicates division by 10, n times. $10^{-1} = \frac{1}{10} = 0.1$ $10^{-2} = \frac{1}{10 \times 10} = 0.01$ $10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001$

Converting Numbers to Scientific Notation

To convert a number to scientific notation:

1. Move the decimal point to obtain a number between 1 and 10 (the decimal part).

2. Multiply that number by 10 raised to the power that reflects the movement of the decimal point (the exponent).

• If the decimal point is moved to the left, the exponent is positive.

• If the decimal point is moved to the right, the exponent is negative.

Examples:

• $5083 = 5.083 \times 10^3$

• $0.00034 = 3.4 \times 10^{-4}$

• $6,022,000,000,000,000,000,000 = 6.022 \times 10^{21}$

Significant Figures

Definition and Importance

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated. The number of significant figures reflects the precision of a measurement.

• Certain digits: Digits that are known exactly.

• Estimated digit: The last digit, which is uncertain.

Example: In 45.872, the first four digits (45.87) are certain, and the last digit (2) is estimated.

Rules for Identifying Significant Figures

• All nonzero digits are significant.

• Interior zeros (zeros between nonzero digits) are significant.

• Trailing zeros (zeros to the right of a nonzero digit) after a decimal point are significant.

• Trailing zeros before a decimal point are significant.

• Leading zeros (zeros to the left of the first nonzero digit) are not significant; they only indicate the position of the decimal point.

• Trailing zeros at the end of a number, but before an implied decimal point, are ambiguous and should be avoided.

Exact Numbers

Exact numbers have an unlimited number of significant figures. These include:

• Counting discrete objects (e.g., 1 dozen = 12)

• Integral numbers in equations

• Defined quantities (e.g., 1 in = 2.54 cm exactly)

Examples of Significant Figures

Number	Significant Figures	Explanation
0.0035	2	Leading zeros are not significant
1.080	4	Trailing zero after decimal is significant
2371	4	All nonzero digits are significant
2.97 × 103	3	All digits in decimal part are significant
100.00	5	Trailing zeros after decimal are significant
100,000	Ambiguous	Trailing zeros before implied decimal are ambiguous
1 dozen = 12	Unlimited	Exact number

Significant Figures in Calculations

Rounding Rules

• Round down if the last digit dropped is 4 or less.

• Round up if the last digit dropped is 5 or more.

• For multi-step calculations, round only the final answer.

Multiplication and Division Rule

• The result should have the same number of significant figures as the input factor with the fewest significant figures.

Example:

• $4.02 \times 89.665 \times 0.10 = 36.1$ (rounded to 2 significant figures)

Addition and Subtraction Rule

• The result should have the same number of decimal places as the quantity with the fewest decimal places.

Example:

• $5.74 + 9.21 = 14.95$ (rounded to 2 decimal places)

• $4.8 - 3.965 = 0.835$ (rounded to 1 decimal place: 0.8)

Unit Conversions

Dimensional Analysis

Unit conversion is a fundamental skill in chemistry, allowing quantities to be expressed in different units. The process uses conversion factors and dimensional analysis.

• Conversion factor: A ratio constructed from two quantities known to be equivalent.

• Dimensional analysis: The method of using units as a guide to solving problems.

Steps for Unit Conversion

1. Write every number with its associated unit.

2. Multiply or divide by conversion factors to cancel units and obtain the desired unit.

3. Check that the final units are correct and the magnitude makes sense.

Constructing Conversion Factors

Conversion factors are constructed from equalities:

• For example, 1 in = 2.54 cm (exactly)

• Conversion factor: $\frac{2.54\,\text{cm}}{1\,\text{in}}$ or $\frac{1\,\text{in}}{2.54\,\text{cm}}$

Example of Unit Conversion

• Convert 44.7 cm to inches: $44.7\,\text{cm} \times \frac{1\,\text{in}}{2.54\,\text{cm}} = 17.6\,\text{in}$

Conversion factors can be inverted because they are equal to 1 and the inverse of 1 is 1.

Common Conversion Factors Table

Equality	Conversion Factor	Exact?
1 in = 2.54 cm	$\frac{1\,\text{in}}{2.54\,\text{cm}}$ or $\frac{2.54\,\text{cm}}{1\,\text{in}}$	Yes
1 lb = 453.6 g	$\frac{1\,\text{lb}}{453.6\,\text{g}}$ or $\frac{453.6\,\text{g}}{1\,\text{lb}}$	No (4 sig. figs)

Additional info: Dimensional analysis is a powerful tool for checking the correctness of calculations and ensuring that the final answer has the appropriate units.