PT

Chapter 2 Measurements and Calculations — Vocabulary Flashcards

Scientific Notation

  • Technique to express very large or very small numbers as a product of a number between 1 and 10 and a power of 10.
  • Expresses a number as $a \times 10^{n}$ where $1 \le a < 10$.
  • Examples: 345 = 3.45 \times 10^{2}, 0.0671 = 6.71 \times 10^{-2}.
  • How to read/write:
    • Moving the decimal point left → positive power of 10.
    • Moving the decimal point right → negative power of 10.

Units

  • Unit = the scale/standard used to represent a measurement.
  • Common systems: English, Metric; SI is preferred for science.
  • SI Base Units:
    • Mass: \text{kg} (kilogram)
    • Length: \text{m} (meter)
    • Time: \text{s} (second)
    • Temperature: \text{K} (kelvin)
    • Electric current: \text{A} (ampere)
    • Amount of substance: \text{mol} (mole)
  • Prefixes change the size of a unit (e.g., kilo, milli, centi, micro).
  • Common conversions:
    • 1 mL = 1 cm$^{3}$
    • 1 L = 1 dm$^{3}$

Measurements of Length, Volume, and Mass

  • Length: fundamental SI unit is the meter (\text{m}).
  • Volume: 1\ \text{m}^{3}; commonly in cm$^{3}$; 1\ \text{mL} = 1\ \text{cm}^{3}; 1\ \text{L} = 1\ \text{dm}^{3}.
  • Mass: fundamental SI unit is the kilogram (\text{kg});
    • 1\text{ kg} = 2.2046\text{ lbs}; 1\text{ lb} = 453.59\text{ g}.

Uncertainty in Measurement

  • A digit that must be estimated is uncertain.
  • A measurement always has some degree of uncertainty.
  • Certain digits are recorded first, followed by uncertain/estimated digits.
  • Example: length to the nearest hundredth of a centimeter may yield 2.85 cm with the last digit (5) uncertain.

Significant Figures

  • Rules for counting significant figures:
    • Nonzero integers are always significant (e.g., 3456 has 4 sig figs).
    • Leading zeros are not significant (e.g., 0.048 has 2 sig figs).
    • Captive zeros between nonzero digits are significant (e.g., 16.07 has 4 sig figs).
    • Trailing zeros are significant only if a decimal point is present (e.g., 9.300 has 4 sig figs; 150 has 2 sig figs).
    • Exact numbers have unlimited sig figs (e.g., 1 in = 2.54 cm; 9 pencils).
  • Exponential notation/sig figs: e.g., 300 = 3.00 \times 10^{2} has 3 sig figs.
  • Rounding rules (brief):
    • Rule 1: If the digit to be removed is <5, keep the preceding digit.
    • If the digit to be removed is >=5, increase the preceding digit by 1.
    • Rule 2: Carry extra digits through calculations, then round at the end.
  • Sig figs in operations:
    • Multiplication/Division: result has sig figs equal to the smallest number of sig figs in any operand (e.g., 1.342 \times 5.5 = 7.381 \Rightarrow 7.4).
    • Addition/Subtraction: result limited by the smallest number of decimal places.

Dimensional Analysis (Problem Solving)

  • Use conversion factors to switch between units.
  • Choose factors by direction of change; cancel unwanted units.
  • Multiply the quantity by the conversion factor until the desired units remain.
  • Check for correct significant figures and reasonableness.

Temperature Conversions

  • Three scales: Fahrenheit (F), Celsius (C), Kelvin (K).
  • Key conversion formulas:
    • Fahrenheit to Celsius: C = \tfrac{5}{9}(F-32)
    • Celsius to Fahrenheit: F = \tfrac{9}{5}C + 32
    • Celsius to Kelvin: K = C + 273.15
    • Kelvin to Celsius: C = K - 273.15
  • Example: normal dog body temperature ≈ 102°F corresponds to about 312\ \text{K}.

Density

  • Density: mass per unit volume; common units \text{g/cm}^{3} or \text{g/mL}.
  • Volume by water displacement (for irregular solids).
  • Density formula: \rho = \dfrac{m}{V}.
  • Example: mineral mass 17.8 g, volume 2.35 cm$^{3}$ ⇒ \rho = \dfrac{17.8}{2.35} \approx 7.57\ \text{g/cm}^{3}.
  • Example: mass of 49.6 mL liquid with density 0.85 g/mL ⇒ m = \rho V = 0.85 \times 49.6 \approx 42.16\ g (2 sig figs: ~42 g).
  • Example: copper density 8.96 g/cm$^{3}$, mass 75.0 g in 50.0 mL water raises volume to ~58.4 mL (displaced volume ~8.4 mL).