Measurement and Problem Solving in Introductory Chemistry

Measurement and Problem Solving

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It consists of two parts: a decimal part (between 1 and 10) and an exponential part (10 raised to an integer exponent).

• Decimal part: A number between 1 and 10.

• Exponential part: 10 raised to an exponent, n, which indicates how many places the decimal point has been moved.

• Positive exponent: Indicates multiplication by 10 n times (large numbers).

• Negative exponent: Indicates division by 10 n times (small numbers).

• Conversion steps: Move the decimal to get a number between 1 and 10, then multiply by 10 to the appropriate power.

• Example: $5983 = 5.983 \times 10^3$; $0.00034 = 3.4 \times 10^{-4}$

Uncertainty in Measurement

All measurements have some degree of uncertainty, which is reflected in the way numbers are reported. The last digit in a measurement is always estimated, indicating the uncertainty.

• Precision: More digits indicate greater precision.

• Uncertainty: The last reported digit is uncertain.

• Example: Reporting a temperature increase as 0.6°C means the actual value could be between 0.5°C and 0.7°C.

Significant Figures

Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated. The rules for determining significant figures are essential for reporting scientific data accurately.

• All nonzero digits are significant.

• Interior zeros (between nonzero digits) are significant.

• Trailing zeros after a decimal point are significant.

• Trailing zeros before a decimal point are significant.

• Leading zeros (before the first nonzero digit) are NOT significant.

• Trailing zeros at the end of a number without a decimal point are ambiguous.

• Exact numbers (from counting or definitions) have unlimited significant figures.

Significant Figures in Calculations

When performing calculations, the number of significant figures in the result depends on the operation:

• Multiplication/Division: The result has the same number of significant figures as the factor with the fewest significant figures.

• Addition/Subtraction: The result has the same number of decimal places as the quantity with the fewest decimal places.

• Rounding: Round only the final answer, not intermediate steps. If the digit dropped is 5 or more, round up; otherwise, round down.

SI Units and Prefix Multipliers

The International System of Units (SI) is the standard for scientific measurements. It includes base units for length (meter), mass (kilogram), time (second), and temperature (kelvin). Prefix multipliers are used to express multiples or fractions of these units.

• Base units: meter (m), kilogram (kg), second (s), kelvin (K)

• Prefix multipliers: kilo- (103), centi- (10-2), milli- (10-3), micro- (10-6), etc.

• Choosing prefixes: Select the prefix that makes the number easy to read and write.

Volume as a Derived Unit

Volume is a derived unit, calculated by raising a unit of length to the third power (e.g., cubic meters, cubic centimeters). Common units include liters (L) and milliliters (mL).

Problem-Solving and Unit Conversions

Many chemistry problems involve converting between units. Dimensional analysis is a systematic approach that uses conversion factors to ensure units are properly converted and canceled.

• Conversion factor: A ratio of equivalent quantities (e.g., $1 \text{ in.} = 2.54 \text{ cm}$).

• Solution map: A visual outline of the steps needed to solve a problem, focusing on units.

• General strategy: Sort information, strategize with a solution map, solve, and check the answer for sense and correct units.

Unit Conversion in Numerator and Denominator

Some problems require converting both the numerator and denominator units (e.g., converting mi/gal to km/L). Each part must be converted separately using appropriate conversion factors.

Converting Units Raised to a Power

When converting units raised to a power (e.g., cm3 to in3), the conversion factor must also be raised to that power.

Density

Density is a physical property defined as the mass of a substance divided by its volume. It is commonly used to identify substances and as a conversion factor between mass and volume.

• Formula: $d = \frac{m}{V}$

• Units: grams per cubic centimeter (g/cm3) or grams per milliliter (g/mL)

• Example: A liquid with mass 27.2 g and volume 22.5 mL has density $d = \frac{27.2\,g}{22.5\,mL} = 1.21\,g/mL$

Common Densities

Different substances have characteristic densities. For example, water has a density of 1.0 g/cm3, while gold has a density of 19.3 g/cm3.

Substance	Density (g/cm3)
Charcoal, oak	0.57
Ethanol	0.789
Ice	0.92
Water	1.0
Glass	2.6
Aluminum	2.7
Titanium	4.50
Iron	7.86
Copper	8.96
Lead	11.4
Gold	19.3
Platinum	21.4

Review and Learning Outcomes

• Express numbers in scientific notation.

• Report measured quantities with the correct number of digits and significant figures.

• Apply rules for significant figures in calculations.

• Convert between units, including those in numerators and denominators, and those raised to a power.

• Calculate and use density as a conversion factor.

Importance of Units in Science

Using correct units is critical in science and engineering. A famous example is the loss of NASA's Mars Climate Orbiter in 1999 due to a unit conversion error, highlighting the importance of clear communication and correct unit usage in scientific work.