Matter, Measurement, and Problem Solving: Foundations of Quantitative Chemistry

Matter, Measurement, and Problem Solving

Numbers and Chemistry

Chemistry is a quantitative science, meaning that many of its concepts and problems involve numbers and measurements. Understanding how to work with these numbers is essential for success in chemistry.

• Quantitative means involving numerical values.

• Key concepts include units of measurement, quantities that are measured and calculated, uncertainty in measurement, significant figures, and dimensional analysis.

Prefix Multipliers and SI Units

Prefix multipliers are used in the metric system to express very large or very small numbers conveniently. Each prefix represents a specific power of ten.

Prefix	Symbol	Multiplier	Power of Ten
kilo	k	1,000	$10^3$
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	$10^{-12}$

Example: $10^{-3} = 1 \times 10^{-3}$

Unit Conversions

Unit conversions are essential for expressing measurements in different units. This is often done using dimensional analysis.

• To convert between units, multiply by the appropriate conversion factor.

• Always include units in calculations to ensure accuracy.

Example: Convert 23.5 m to km, cm, and nm:

• $23.5\,\text{m} \times \left(\frac{1\,\text{km}}{10^3\,\text{m}}\right) = 0.0235\,\text{km}$

• $23.5\,\text{m} \times \left(\frac{1\,\text{cm}}{10^{-2}\,\text{m}}\right) = 2,350\,\text{cm}$

• $23.5\,\text{m} \times \left(\frac{1\,\text{nm}}{10^{-9}\,\text{m}}\right) = 23,500,000,000\,\text{nm}$

Scientific Notation and Significant Figures

Scientific notation is used to express very large or very small numbers in a compact form. Significant figures reflect the precision of a measurement.

• Scientific Notation: $0.000135 = 1.35 \times 10^{-4}$

• Significant Figures: The number of digits that carry meaning in a measurement.

Examples:

• $0.000135$ has 3 significant figures.

• $0.004753$ has 4 significant figures.

• $785,000$ can have 3, 4, or 6 significant figures depending on how it is written (e.g., $7.85 \times 10^5$, $7.850 \times 10^5$, $7.85000 \times 10^5$).

Derived Units: Volume and Density

Some quantities in chemistry are derived from base units. Two important derived units are volume and density.

• Volume: A measure of space, with units such as $\text{cm}^3$ or liters (L).

• Density: The ratio of a substance’s mass to its volume, with units of mass/volume (e.g., $\text{g/cm}^3$).

Formula for Density:

$\text{Density} = \frac{\text{mass}}{\text{volume}}$

Example: Find the density of a cube with sides of 2.5 cm and a mass of 5.0 g:

• Volume: $V = (2.5\,\text{cm})^3 = 15.625\,\text{cm}^3$

• Density: $D = \frac{5.0\,\text{g}}{15.625\,\text{cm}^3} = 0.32\,\text{g/cm}^3$ (rounded to two significant figures)

Dimensional Analysis

Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move from one unit to another.

• A unit equation is a statement of two equivalent quantities (e.g., $2.54\,\text{cm} = 1\,\text{in}$).

• A conversion factor is a ratio derived from the unit equation (e.g., $\frac{2.54\,\text{cm}}{1\,\text{in}}$ or $\frac{1\,\text{in}}{2.54\,\text{cm}}$).

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

General Form:

Information given $\times$ conversion factor(s) = information sought

Example: Convert 32 cm to inches:

$32\,\text{cm} \times \left(\frac{1\,\text{in}}{2.54\,\text{cm}}\right) = 12.6\,\text{in}$

Common Conversion Factors

• 1 pound = 453.6 g

• 1 kg = 2.2046 lb

• 1 inch = 2.54 cm (exactly)

• 3.8 L = 1 gallon

• 1 mile = 1.609 km

• 1 atmosphere (atm) = 760 mmHg = 760 torr

Units Raised to a Power

When converting units raised to a power (e.g., area or volume), both the number and the unit must be raised to that power.

• For example, to convert $\text{in}^2$ to $\text{cm}^2$:

$2.54\,\text{cm} = 1\,\text{in}$

$(2.54\,\text{cm})^2 = (1\,\text{in})^2$

$6.45\,\text{cm}^2 = 1\,\text{in}^2$

Example: Area of an 8 in by 11 in piece of paper in $\text{m}^2$:

• Convert inches to centimeters, then to meters, and multiply to find area in $\text{m}^2$.

Problems Involving Equations

Some problems require using equations to relate quantities. The approach is similar to dimensional analysis, but the equation provides the relationship.

• Sort: Identify given and find quantities.

• Strategize: Write the equation and plan the solution.

• Solve: Substitute values and solve for the unknown.

• Check: Ensure the answer makes sense and has correct units/significant figures.

Example: Find the density (in $\text{g/cm}^3$) of a metal cylinder with mass $m = 8.3$ g, length $l = 1.93$ cm, and radius $r = 0.55$ cm.

• Volume of cylinder: $V = \pi r^2 l$

• Calculate $V$ and then $D = \frac{m}{V}$

General Problem Solving Strategy

Effective problem solving in chemistry involves a systematic approach:

1. Identify the starting point (given information).

2. Identify the end point (what you must find).

3. Devise a conceptual plan to connect the starting and end points.

4. Sort, strategize, solve, and check your answer for reasonableness.

Additional info:

• Tables and conversion factors are essential references for solving quantitative problems in chemistry.

• Practice with unit conversions and dimensional analysis is crucial for mastering problem solving in chemistry.