Chapter 1 Notes: Particles of Matter and Measurement Tools

A Particulate View – Zooming In

• The smallest particle of an element that retains the chemical characteristics of that element is an atom.
• A molecule is an assembly of two or more atoms that are held together in a characteristic pattern by chemical bonds.

Ions

• Some compounds consist of positively or negatively charged particles called ions.
• Cations — positively charged ions: ext{Ca}^{2+}, ext{Na}^{+}
• Anions — negatively charged ions: ext{Cl}^{-}, ext{OH}^{-}

Rounding Procedure

• Step 1: Look at the leftmost digit to be dropped.
• Step 2: If this digit is 5 or greater: add 1 to the last digit to be retained; drop all digits farther to the right.
• Step 3: If this digit is less than 5: drop all digits farther to the right.
• Examples:
  • $1.2151$ rounded to three significant figures is 1.22
  • $1.2143$ rounded to three significant figures is 1.21

Exponential Notation (Scientific Notation)

• Also called exponential notation; used for very large or very small numbers.
• Example: Avogadro’s Number: 6.02\times 10^{23}
• Example: Radius of a bromine atom: 1.14\times 10^{-10}\text{ mm}
• Standard Form / Scientific Notation: $N\times 10^n$ where N is between 1 and 9 and n is an integer (positive or negative).
• 2300 = 2.3\times 10^{3}
• The exponent $n$ indicates how many times the coefficient is multiplied by ten.

Standard Notation vs Scientific Notation

• Standard form: 24600 = 2.46\times 10^{4}
• Example: 12{,}000{,}000 = 1.2\times 10^{7}
• 0.0342 = 3.42\times 10^{-2}
• Move the decimal to place after the first nonzero digit:
  • 24653.24 = 2.465324\times 10^{4}
  • 0.000246 = 2.46\times 10^{-4}

How to write in Scientific Notation (Practice)

• Write the following numbers in scientific notation:
  • 0.000653
  • 350{,}000
  • 0.02700

Prefixes for SI Units

• Prefixes correspond to powers of 10 and can substitute for those powers of 10.
• Examples:
  • 7.9\times 10^{-6}\,s = 7.9\,\mu s
• Sizes: Red blood cell ≈ 10\,\mu m; A strand of DNA ≈ 2\,\text{nm} wide.

Scientific Notation and Metric Prefixes

• Because each prefix has a power of 10, you can substitute the prefix for the power of 10.
• Example: 7.9\times 10^{-6}\,s = 7.9\,\mu s

Write measurements without scientific notation using the SI prefixes

• Convert the following:
  • 4.851\times 10^{-9}\text{ g} = 4.851\text{ ng}
  • 3.16\times 10^{-2}\text{ m} = 3.16\text{ cm}
  • 8.93\times 10^{-12}\text{ s} = 8.93\text{ ps}

Practice 16

• Write the following measurements without scientific notation using the appropriate SI prefix:
  • 4.851\text{ ng}
  • 3.16\text{ cm}
  • 8.93\text{ ps}

Practice 17 (Scientific Notation Conversions)

• Convert using scientific notation:
  • 6.20\text{ km} \rightarrow \text{m}:\quad 6.20\times 10^{3}\text{ m}
  • 2.54\text{ cm} \rightarrow \text{m}:\quad 2.54\times 10^{-2}\text{ m}
  • 1.98\text{ ns} \rightarrow \text{s}:\quad 1.98\times 10^{-9}\text{ s}
  • 5.23\mu\text{g} \rightarrow \text{g}:\quad 5.23\times 10^{-6}\text{ g}

Counting Significant Figures (Basics)

• Nonzero integers are always significant:
  • 3456 → 4 sig figs
  • 7.35 → 3 sig figs
• Zeros:
  • Leading zeros are not significant: 0.0392 → 3 sig figs
  • Leading zeros in decimals do not count; trailing zeros before a decimal point may not be significant unless decimal point is present.
  • Trailing zeros are not significant unless they come after a decimal point:
  • 3700 → 2 sig figs
  • 140.00 → 5 sig figs
• Captive zeros are always significant:
  • 16.07 → 4 sig figs
• 20.007 → 5 sig figs

Practice: Significant Figures in Measurements

• Examples to determine sig figs:
  • 0.04550 g → 4 sig figs
  • 100 lb → 2 or 3? (depends on context; often 2 sig figs if written as 100 with no decimal point)
  • 101.05 mL → 5 sig figs
  • 350.0 g → 4 sig figs

Significance in Mathematical Operations (cont. 1)

• Multiplication/Division:
  • The number of sig figs in the result = the number in the least precise measurement used in the calculation.
  • Example: 6.38 \times 2.0 = 12.76 \rightarrow 13 (2 sig figs)
  • Example: 16.84 \div 2.54 = 6.6299 \rightarrow 6.63 (3 sig figs)

Significance in Mathematical Operations (cont. 2)

• Addition/Subtraction:
  • The number of significant figures in the result depends on the number of decimal places in the least accurate measurement.
  • Examples:
  • 6.8 + 11.934 = 18.734 \rightarrow 18.7 (3 sig figs, based on decimal places)
  • 37.657 - 2.1 = 35.557 \rightarrow 35.6

Temperature Scales

• Fahrenheit (°F), Celsius (°C), Kelvin (K)
• Conversions:
  • K = °C + 273.15
  • °C = \tfrac{5}{9}(°F - 32)
  • °F = \tfrac{9}{5}°C + 32

Unit Conversion (Unit Conversion 3)

• Unit conversion steps: 1) Determine your starting point, 2) Determine the target units, 3) Choose a conversion factor, 4) Multiply/divide to cancel units.
• Example questions:
  • 3.6 feet to inches: 1\text{ ft} = 12\text{ in} -> 3.6\text{ ft} = 3.6 \times 12 = 43.2\text{ in}
  • 0.547 lb to grams: 1\text{ lb} = 453.6\text{ g} -> 0.547\text{ lb} = 0.547 \times 453.6\text{ g}

Equality: Conversion Factor

• Equality: 1\text{ km} = 0.6214\text{ mi}
• Unit conversion factor: type of equation: Initial units × (Desired units / Initial units) = Desired units
• Converting a value from one unit to another relies on equality factors.

Sample Exercise: Unit Conversions

• The masses of diamonds are usually expressed in carats (1 g = 5 carats exactly).
• The Star of Africa diamond mass: 530.2 carats.
• What is its mass in grams? 530.2\text{ carats} \times \left(\frac{1\text{ g}}{5\text{ carats}}\right) = 106.04\text{ g}

Practical Unit Conversions in Everyday Context

• A bottle of vitamin C: one 200.0 mg tablet per day. How many grams per week?
  • 200.0\text{ mg/day} × 7 days = 1400\text{ mg} = 1.4\text{ g}

Speed, Distance, and Time Conversions

• The speed of sound in air is about 343 m/s. What is this in miles per hour?
  • 1\text{ mi} = 1609\text{ m}, 1\text{ min} = 60\text{ s}, 1\text{ h} = 60\text{ min}
  • 343\frac{\text{m}}{\text{s}} \times \frac{3600\text{ s}}{1\text{ h}} \times \frac{1\text{ mi}}{1609\text{ m}} = \approx 767\frac{\text{mi}}{\text{h}}

Density (d = m/V)

• Definition: Density is the ratio of mass to volume: d = \frac{m}{V}
• Common units:
  • Solids: \text{g/cm}^3
  • Liquids: \text{g/mL}
  • Gases: \text{g/L}

Worked Problems: Using Significant Figures in Calculations

• Example 1: Gold nugget by water displacement

  • Mass = 30.01 g; Volume displaced = final volume − initial volume = 62.6 mL − 56.3 mL = 6.3 mL
  • Density = \rho = \frac{m}{V} = \frac{30.01\text{ g}}{6.3\text{ mL}} ≈ 4.7635\text{ g/mL}
  • The volume is known to two significant figures; report density to two sig figs: \rho \approx 4.8\text{ g/mL}
  • Compare to density of gold: d_{\text{Au}} = 19.3\frac{\text{g}}{\text{mL}}; nugget is not pure gold.

• Example 2: Oil of wintergreen density

  • Mass = 28.1 g; Volume = 23.7 mL
  • Density: d = \frac{28.1\text{ g}}{23.7\text{ mL}} ≈ 1.1857\text{ g/mL}
  • Report to three significant figures: d ≈ 1.19\text{ g/mL}

• Example 3: Gasoline density

  • Density = 0.718 g/mL; Mass = 454 g
  • Volume: V = \frac{m}{d} = \frac{454\text{ g}}{0.718\text{ g/mL}} ≈ 632.312\text{ mL}
  • Report to the nearest milliliter: V ≈ 632\text{ mL}

Derived Units

• Derived units are combinations of fundamental units.
• Examples:
  • Speed: v = \frac{\text{distance}}{\text{time}}
  • Volume: V = \text{length} \times \text{width} \times \text{height} with units of \text{m}^3

Quantity and SI Units

• Quantity: a property that can be measured (e.g., length, mass, time).
• SI Units (examples):
  • Length/Distance: m
  • Time: s
  • Mass: kg
• Derived quantities and their SI units:
  • Area: m^2
  • Volume: m^3
  • Density: \frac{kg}{m^3}
  • Speed: \frac{m}{s}
  • Acceleration: \frac{m}{s^2}
  • Force: kg \cdot m / s^2 = \text{N} (newton)
  • Pressure: kg/(m\cdot s^2) = \text{Pa} (pascal)
  • Energy: kg\cdot m^2/s^2 = \text{J} (joule)

Density (Summary)

• Density definition: d = \frac{m}{V}
• Common density units:
  • Solids: \text{g/cm}^3
  • Liquids: \text{g/mL}
  • Gases: \text{g/L}

Additional Notes and Real-World Relevance

• Understanding units and conversions is essential for consistency across measurements, experiments, and reporting.
• Significant figures reflect measurement precision and impact reported results and uncertainties.
• Derived units help connect basic measurements to practical quantities like speed, density, and energy.