Chapter 2: Measurements in Chemistry – Scientific Notation, Significant Figures, and Unit Conversions

Chapter 2: Measurements in Chemistry

Chemistry and Measurement

Chemistry is a quantitative science, meaning that many of its properties and changes can be measured. Accurate measurement is essential for understanding chemical reactions and properties, such as determining how many grams of a reactant are needed for a reaction or calculating the density of a substance.

• Measurement is the process of obtaining the magnitude of a quantity relative to an agreed standard.

• Examples: Measuring mass, volume, temperature, and density.

Scientific Notation

Definition and Purpose

Scientific notation is a method used to express very large or very small numbers in a compact form. It is commonly used in chemistry to handle measurements that span many orders of magnitude.

• Scientific notation expresses numbers as a product of a coefficient and a power of ten.

• The coefficient must be a number between 1 and 9.999... (inclusive).

• Format: Coefficient × 10Exponent

• Example: $1.23 \times 10^8$

Converting to Scientific Notation

• Place the decimal point so that only one nonzero digit remains to its left.

• Count the number of places the decimal point has moved; this number is the exponent.

• If the original number is greater than 1, the exponent is positive; if less than 1, the exponent is negative.

Example:

• 2840000 → $2.84 \times 10^6$

• 0.00004370 → $4.370 \times 10^{-5}$

• 385 → $3.85 \times 10^2$

• 59200 → $5.92 \times 10^4$

Converting to Standard Notation

• Move the decimal point to the right (for positive exponents) or left (for negative exponents) the number of places indicated by the exponent.

• Fill in empty spaces with zeros as needed.

• Remove the decimal point if it is no longer necessary.

Example:

• $3.14 \times 10^{-4}$ → 0.000314

• $9.64 \times 10^5$ → 964000

• $2.01 \times 10^{-2}$ → 0.0201

• $5.75 \times 10^3$ → 5750

Significant Digits (Significant Figures)

Definition and Importance

Significant digits (or significant figures) are the digits in a measurement that are known with certainty plus one digit that is estimated. They reflect the precision of a measurement and the limitations of the measuring instrument.

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Zeros after a decimal point and after a significant digit are significant (they show precision).

• Leading zeros (placeholders) are not significant.

• Counting numbers, conversion factors, and defined quantities have infinite significant figures (no uncertainty).

Example:

• 2.38 cm → 3 significant figures

• 0.000549 g → 3 significant figures

• 12000 J → 2 significant figures (unless otherwise indicated by a decimal point or scientific notation)

• 30004 mg → 5 significant figures

• 1.000 mL → 4 significant figures

Calculations and Significant Figures

• Multiplication/Division: The result should have as many significant figures as the factor with the fewest significant figures.

• Addition/Subtraction: The result should have as many decimal places as the measurement with the fewest decimal places.

• If both operations are present, perform operations in parentheses first, determine significant figures, but do not round until the final answer.

• Only round final answers.

Proper Rounding

• Determine how many significant figures should be in the final answer.

• Look at the digit immediately after the last significant figure:

  • If it is 0-4, leave the last significant figure unchanged.

  • If it is 5-9, round the last significant figure up by one.

Example:

• 4682 cm rounded to 3 sig figs: 4680 cm

• 004791 L rounded to 2 sig figs: 0.0048 L

Basic Measures in Chemistry

Common Quantities

• Length: The distance between two points. SI unit: meter (m).

• Mass: The amount of matter in an object (similar to weight, but independent of gravity). SI unit: kilogram (kg), commonly gram (g) in chemistry.

• Volume: The amount of three-dimensional space an object occupies. SI unit: cubic meter (m3), commonly liter (L) or milliliter (mL) in chemistry.

• Temperature: A measure of how hot or cold something is. SI unit: kelvin (K); Celsius (°C) is also commonly used. Heat is different from temperature; it is a form of energy transfer.

English and Metric Units

• Length: mile, foot, inch (English); meter (metric)

• Volume: gallon, pint, quart (English); liter (metric)

• Mass: pound, ounce (English); gram (metric)

The metric system is a base-10 system, making conversions straightforward by moving the decimal point.

Unit Conversions and Dimensional Analysis

Metric Prefixes

Metric prefixes indicate multiples or fractions of units. The table below summarizes common prefixes:

Prefix	Symbol	Multiplier	Power of Ten
tera-	T	1,000,000,000,000	1012
giga-	G	1,000,000,000	109
mega-	M	1,000,000	106
kilo-	k	1,000	103
hecto-	h	100	102
deca-	da	10	101
deci-	d	0.1	10-1
centi-	c	0.01	10-2
milli-	m	0.001	10-3
micro-	μ	0.000001	10-6
nano-	n	0.000000001	10-9
pico-	p	0.000000000001	10-12
femto-	f	0.000000000000001	10-15

Conversions Among Units

• To convert between metric units, move the decimal point the appropriate number of places based on the prefixes.

• Common equivalencies: 1 mL = 1 cm3 = 1 cc; 1 dm3 = 1 L

• Use conversion factors to relate different units (e.g., 1 in = 2.54 cm).

Example:

• 0.45 g = 450 mg

• 0.00063 L = 0.063 cL

• 3.4 mg = 0.034 cg

• 533 cm = 5.33 m

• 0.631 mg = 0.0631 dg

Dimensional Analysis (Factor-Label Method)

Dimensional analysis is a systematic method for converting between units using conversion factors. Units are treated as algebraic quantities that can be canceled.

• Set up the calculation so that units cancel, leaving the desired unit.

• Multiply by conversion factors as needed.

Example:

• How many inches is 3.0 ft? $3.0~\text{ft} \times \frac{12~\text{in}}{1~\text{ft}} = 36~\text{in}$

• How many cm is 6.0 in? $6.0~\text{in} \times \frac{2.54~\text{cm}}{1~\text{in}} = 15.24~\text{cm}$

Units Raised to a Power

• When converting units raised to a power (e.g., area, volume), raise the conversion factor to that power.

• Example: $1~\text{ft} = 12~\text{in}$, so $1~\text{ft}^2 = (12~\text{in})^2 = 144~\text{in}^2$

Example:

• How many cm2 is 6.00 in2? $6.00~\text{in}^2 \times (2.54~\text{cm}/1~\text{in})^2 = 38.7~\text{cm}^2$

Density

Definition and Formula

Density is a physical property defined as the mass of a substance per unit volume. It is a characteristic property that can be used to identify substances.

• Formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Common units: g/cm3 (solids), g/mL (liquids), g/L (gases)

Example:

• A rock has a mass of 28.4 g and a volume of 8.10 cm3. Density = $\frac{28.4~\text{g}}{8.10~\text{cm}^3} = 3.51~\text{g/cm}^3$

• A sample of Al is measured to be 8.29 cm3 and 22.5 g. Density = $\frac{22.5~\text{g}}{8.29~\text{cm}^3} = 2.71~\text{g/cm}^3$

Density as a Conversion Factor

• Density can be used to convert between mass and volume.

• Set up dimensional analysis using density as a conversion factor.

Example:

• Calculate the mass of a lead sample (D = 11.4 g/cm3) if the volume is 17.5 cm3: $17.5~\text{cm}^3 \times \frac{11.4~\text{g}}{1~\text{cm}^3} = 200~\text{g}$

• Calculate the volume of an iron sample (D = 7.86 g/cm3) if the mass is 185 g: $185~\text{g} \times \frac{1~\text{cm}^3}{7.86~\text{g}} = 23.5~\text{cm}^3$

Floating vs. Sinking

• An object will float if its density is less than the density of the fluid it is placed in.

• An object will sink if its density is greater than the density of the fluid.

Example: Gold has a density of 19.3 g/cm3. If a rock with a volume of 24.5 cm3 has a mass of 219 g, its density is $\frac{219~\text{g}}{24.5~\text{cm}^3} = 8.94~\text{g/cm}^3$, so it is not gold.

Additional info: Some context and examples were inferred and expanded for clarity and completeness.