01/14 Foundations of Chemistry and Quantitative Reasoning

Matter, Properties, and Composition

Chemistry begins with Matter, anything that has mass and occupies space. Chemists describe matter using measurable quantities such as mass, volume, temperature, pressure, amount of substance, and concentration.

Matter can be classified by both physical state and composition:

• Physical state: A solid has a defined shape and volume, a liquid has a defined volume but takes the shape of its container, a gas expands to fill its container, and plasma is an ionized state of matter.

• Pure substance: A substance with constant composition. An element contains one type of atom, while a compound contains two or more elements chemically combined in fixed ratios.

• Mixture: Two or more substances physically combined. A homogeneous mixture is uniform throughout; a heterogeneous mixture is not uniform throughout.

A physical change alters a substance's form or state without changing its chemical identity. A chemical change produces one or more new substances. A Physical property can be observed without changing chemical identity, whereas a chemical property describes how a substance can change into other substances, such as through flammability or reaction with an acid.

Chemists also distinguish extensive and intensive quantities. Extensive quantities depend on sample size, while intensive quantities do not. This distinction helps determine whether a measured value can identify a substance independently of the sample amount.

Takeaway: Classify matter by state and composition, and distinguish changes or properties that preserve chemical identity from those that alter it.

Measurement, Accuracy, and Precision

A measurement combines a numerical value with a unit. Writing $12.4\ \mathrm{g}$ communicates more information than writing 12.4 alone because the unit identifies the physical quantity and scale.

Measured values have limited precision. Instruments provide some certain digits and usually one final estimated digit. If a graduated cylinder has markings every $1\ \mathrm{mL}$, a reading may reasonably be recorded to the nearest $0.1\ \mathrm{mL}$, depending on the instrument and observation conditions. Reporting unsupported digits falsely suggests greater precision.

Accuracy and Precision are related but distinct:

• Accuracy describes closeness to an accepted or reference value.

• Precision describes agreement among repeated measurements.

• Random error produces variation or scatter among measurements.

• Systematic error shifts measurements in a consistent direction and can produce precise but inaccurate results.

An uncertainty may be reported with a measurement, for example $25.36\pm0.02\ \mathrm{mL}$. The uncertainty is part of the meaning of the result rather than an optional decoration.

Takeaway: Report measurements with units, appropriate digits, and uncertainty when required; do not confuse consistency among trials with closeness to the accepted value.

SI Units, Prefixes, and Temperature

The International System of Units (SI) provides a consistent framework for scientific measurement. Its seven base quantities include time, length, mass, thermodynamic temperature, electric current, amount of substance, and luminous intensity.

Common chemistry units include:

• Length: meter, $\mathrm{m}$

• Mass: kilogram, $\mathrm{kg}$

• Time: second, $\mathrm{s}$

• Temperature: kelvin, $\mathrm{K}$

• Amount of substance: mole, $\mathrm{mol}$

• Volume: cubic meter, $\mathrm{m^3}$, with liter and milliliter commonly used in chemistry

• Energy: joule, $\mathrm{J}$

• Pressure: pascal, $\mathrm{Pa}$

SI prefixes express powers of ten:

Temperature conversions require both a scale factor and an offset:

Kelvin values are written without a degree symbol, such as $298.15\ \mathrm{K}$. Volume relationships frequently used in chemistry are $1\ \mathrm{L}=10^3\ \mathrm{mL}$ and $1\ \mathrm{mL}=1\ \mathrm{cm^3}$.

Takeaway: Use SI units consistently, apply prefixes as powers of ten, and remember that temperature conversions involving Celsius or Fahrenheit include offsets.

Density and Core Chemical Quantities

Density relates the mass of a sample to the volume it occupies:

The same relationship can be rearranged to solve for either unknown:

For a liquid with mass $39.6\ \mathrm{g}$ and volume $30.0\ \mathrm{mL}$:

The result has three significant figures because both measured quantities have three significant figures.

Other central relationships include:

where $q$ is heat, $m$ is mass, $c$ is specific heat capacity, and $\Delta T$ is temperature change; and

where $n$ is amount of substance, $M$ is molar mass, and $C$ is amount concentration. The mole is the SI unit for amount of substance, and the entities being counted should be identified when ambiguity is possible.

Takeaway: Start with a relationship, rearrange it for the desired quantity, substitute values with units, and verify that the resulting unit matches the target.

Significant Figures and Rounding

Significant figures communicate the precision supported by a measurement. Apply these rules:

1. All nonzero digits are significant. Thus, $347$ has three significant figures.

2. Zeros between nonzero digits are significant. Thus, $1002$ has four significant figures.

3. Leading zeros are not significant. Thus, $0.0045$ has two significant figures.

4. Trailing zeros to the right of a decimal point are significant. Thus, $2.300$ has four significant figures.

5. Trailing zeros in a number without a decimal point may be ambiguous, so scientific notation is clearer. For example, $5.0\times10^2$ has two significant figures, while $5.00\times10^2$ has three.

6. Exact counted quantities and defined conversion factors have unlimited significant figures.

For multiplication and division, retain the same number of significant figures as the factor with the fewest significant figures:

For addition and subtraction, retain the same number of decimal places as the quantity with the fewest decimal places:

Keep extra digits during intermediate calculations and round only at the end. Significant-figure rules communicate measurement precision, but they do not replace a formal uncertainty analysis.

Takeaway: Determine whether the calculation is additive or multiplicative, preserve guard digits during the work, and round the final answer according to the appropriate rule.

Dimensional Analysis and Unit Conversion

Dimensional analysis converts quantities by multiplying by conversion factors equal to one. Each factor is arranged so equivalent units appear in the numerator and denominator, allowing unwanted units to cancel.

For example:

A reliable procedure is:

1. Write the given quantity and unit.

2. Identify the desired unit.

3. Select conversion factors that cancel unwanted units.

4. Calculate the numerical value.

5. Check the units, magnitude, and significant figures.

For a multistep conversion:

Dimensional analysis also tests equations. In $d=vt$, velocity has units $\mathrm{m\,s^{-1}}$ and time has units $\mathrm{s}$, so $vt$ has units of meters, matching distance. An equation with incompatible units cannot be correct as written.

Takeaway: Units are algebraic safeguards. If unwanted units do not cancel or the final unit is not the target unit, revisit the setup before trusting the numerical result.

Quantitative Problem Solving and Laboratory Data

A strong quantitative solution makes its reasoning visible. Use this workflow:

1. Identify the target. State what must be found and include its expected unit.

2. List known values and conditions. Record every value with its unit and significant figures, and distinguish exact quantities from measured quantities.

3. Choose the governing relationship. Examples include $\rho=m/V$, $q=mc\Delta T$, $n=m/M$, and $C=n/V$.

4. Convert units before substituting when useful. For example, use liters with $\mathrm{mol\,L^{-1}}$, kilograms with $\mathrm{J\,kg^{-1}\,K^{-1}}$, or meters with $\mathrm{m\,s^{-1}}$.

5. Substitute with units shown. Treat units algebraically so the source of the final unit is clear.

6. Check the result. Confirm the units, magnitude, sign, distinction between exact and measured values, and final rounding.

For a metal sample with density $7.85\ \mathrm{g\,cm^{-3}}$ and volume $12.0\ \mathrm{cm^3}$:

The cubic-centimeter units cancel, leaving grams, and the result has three significant figures.

When analyzing laboratory data, label graph axes with quantities and units. Place the independent variable on the horizontal axis and the dependent variable on the vertical axis. A straight-line relationship can be represented by:

The slope has units equal to the units of $y$ divided by the units of $x$. For a mass-versus-volume graph, the slope can represent density.

For comparison with an accepted value, Percent error is calculated as:

A low percent error does not necessarily demonstrate precision, and a high percent error does not by itself identify the cause. Interpretation should consider uncertainty, instrument limitations, procedural assumptions, and possible systematic errors.

Takeaway: Organize every problem around a target, a relationship, compatible units, visible unit cancellation, appropriate rounding, and a reasonableness check.

• Intensive quantity