Measurements

Chapter Outline

After completing this chapter, the student should be able to:

Define the basic terms and concepts related to common laboratory measurements, including volume, mass, and weight.
Calculate using scientific notation when needed.
Properly round any calculations involving measurements by using the rules for significant figures.
Use the fundamental units of the Système International (SI) in making measurements and in doing calculations.
Use any common temperature scale and convert between scales as needed.

Measurements are an integral part of everyday life and key to scientific, technical, and business-related jobs.
Measurements can be grouped into categories, such as:
- Length: Describes size or distance (e.g., "I am 5 feet 10 inches tall").
- Weight/Mass:
  - Weight is the force of gravitational attraction on an object (e.g., "I weigh 155 pounds").
  - Mass is the quantity of matter in an object (e.g., "My car has a mass of 1800 kilograms"). Significant distinction:
  - Mass (constant globally) vs. Weight (varies with gravity).
- Volume: Measured capacity (e.g., "This bottle holds 2 liters of soda").
- Time: Common units include second, minute, hour, etc.
- Measurement Units Varieties: Various units exist under each measurement category, which may be familiar or unfamiliar.

Large and small numbers in science can be cumbersome; scientific notation simplifies this.
Express a number as a imes 10^n, where:
- a is between 1 and 10.
- n is an integer.
Converting Decimal Numbers to Scientific Notation:
- Example Step: Move decimal right of the first non-zero digit; count moves for exponent.
- Example: 240,000 = 2.4 imes 10^5.

Write in scientific notation:
- 1. 25,300,000 = 2.53 imes 10^7
- 2. 0.000206 = 2.06 imes 10^{-4}
- 3. 1,225 = 1.225 imes 10^3
- 4. 2.45 = 2.45 imes 10^0

To change from scientific notation to decimal:
- x imes 10^n; determine decimal movement whether left (negative) or right (positive).
- Example: 7.23 imes 10^3 converts to 7230.

Multiplication: Multiply the values and add exponents:
- Example: (6.2 imes 10^2)(2.1 imes 10^3) = (6.2 imes 2.1) imes 10^{2+3} = 13.02 imes 10^5 (not in correct form, convert: 1.302 imes 10^6).
Division: Divide the values and subtract exponents:
- Use similar steps as multiplication but apply the rule for exponents accordingly.

Special functions exist for handling scientific notation on calculators (e.g., EE button).
Enter values following the calculator-specific mode to ensure proper result displays.

Measured numbers carry uncertainty and precision, determined by measuring devices and skills.
Significant Figures: Total digits known with certainty plus one estimated digit (e.g., 2.54 mL may represent a range between 2.535 and 2.544 mL, showing uncertainty).
Rules for Significant Figures:
1. Non-zero digits are significant.
2. Zeroes between non-zero digits are significant. (Example: 7,005 has 4 sig. figs)
3. Zeros after a decimal point and significant digits are significant (e.g., 41.0 has 3 sig. figs).
4. Ambiguous zeroes without a decimal point are not significant (e.g., 5000 could be 1 sig. fig).
5. Leading zeroes aren’t significant (e.g., 0.0083 has 2 sig. figs).

Rule 1: If the first digit to be dropped is 4 or less, drop it (e.g., 74.593 rounds to 74.59).
Rule 2: If the first digit is 5 or greater, drop it and increase the previous digit (e.g., 1.0268 rounds to 1.027).
Significant Figures in Multiplication/Division: The answer reflects the least precise measurement's sig. figs.
Significant Figures in Addition/Subtraction: The final precision matches the least precise measurement's decimal places.

(SI)

The SI unit system includes various units for mass, length, time, etc. and introduces prefixes to express multiples/divisors of base units.
Common prefixes in the SI include:
- Giga (10^9), Mega (10^6), Kilo (10^3), Hecto (10^2), Deka (10^1), Deci (10^{-1}), Centi (10^{-2}), Milli (10^{-3}), Micro (10^{-6}), Nano (10^{-9})
Conversion Factors: Used for changing unit measures (e.g., 1L = 1000 mL).
Decimal Bumping Method: Changing measurement by moving decimal placement corresponding to the power of 10 in SI prefixes.

Three common scales: Fahrenheit (°F), Celsius (°C), and Kelvin (K).
Fahrenheit Scale:
- Water freezes at 32°F, boils at 212°F.
- Divided into 180 degrees between freezing/boiling points.
Celsius Scale:
- Water freezes at 0°C, boils at 100°C; divided into 100 degrees.
- Implication: 1°C is 1.8°F.
Kelvin Scale: Absolute zero (0 K, equivalent to -273°C), no negative temperatures.
Conversion Formulas:
1. TC = rac{TF - 32}{1.8}
2. TF = 1.8 TC + 32
3. TK = TC + 273

Scientific notation conversions, decimal notation conversions, significant figures, unit conversions.
Example Practice: Convert between temperature scales and calculate average weights, temperature responses to various measurements to understand the implications of measurement precision.