Honors Chemistry Unit 2 Review: Scientific Measures & Calculations

The scientific method is a cycle for testing ideas with evidence. You start with an observation, ask a question, do background research, and form a hypothesis. A hypothesis is a testable proposed explanation that leads to predictions. For example, you might hypothesize that higher temperature increases sucrose solubility, then predict that hotter water will dissolve more sucrose under the same conditions. Next you run an experiment, analyze the data, and decide whether the results support the hypothesis. If they don't, you revise the hypothesis and test again.

In a simple controlled experiment, you change one independent variable while keeping other relevant conditions constant. The independent variable is what you change on purpose (water temperature). The dependent variable is what you measure in response (grams of sugar dissolved). Controlled variables stay the same in every trial, so changes in them cannot explain changes in the dependent variable. Examples include the amount of water, stirring time, and type of sugar.

Observations come in two kinds. Qualitative observations describe properties like color or smell. Quantitative observations are numbers with units, like 25.0 mL.

Watch the vocabulary here. A hypothesis is a proposed explanation still being tested. A theory is a well-tested explanation backed by many experiments, like atomic theory. A scientific law describes what happens, often as an equation, without explaining why, like the law of conservation of mass. A theory never "graduates" into a law, because the two do different jobs.

Scientific notation writes a nonzero number as a coefficient that is at least 1 but less than 10 (ignoring any negative sign), multiplied by a power of 10. The Avogadro constant is exactly $6.02214076 \times 10^{23}\ \text{mol}^{-1}$, often rounded to $6.02 \times 10^{23}\ \text{mol}^{-1}$. That rounded value is 602,000,000,000,000,000,000,000 per mole in ordinary notation. Small numbers work the same way, so 0.00052 m becomes $5.2 \times 10^{-4}$ m. Numbers with a size of 10 or more get positive exponents, and numbers between 0 and 1 get negative ones, whatever their sign. So $-0.0035$ is $-3.5 \times 10^{-3}$.

To multiply, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents. To add or subtract, first rewrite the numbers so their exponents match.

The metric system scales units by powers of 10. SI base units include the meter, the kilogram, and the second. Volume is derived from length, but chemistry usually uses liters and milliliters. Common prefixes are kilo- ($10^3$), centi- ($10^{-2}$), milli- ($10^{-3}$), micro- ($10^{-6}$), and nano- ($10^{-9}$). For example, 250 mL equals 0.250 L.

Worked example. Multiply $(4.0 \times 10^{5})(3.0 \times 10^{-2})$. The coefficients give $4.0 \times 3.0 = 12$, and the exponents give $5 + (-2) = 3$, so you get $12 \times 10^{3}$. A coefficient of 12 is too big, so move the decimal one place left and raise the exponent by one. The answer is $1.2 \times 10^{4}$.

Significant figures are the digits in a measurement that you actually know, plus one estimated digit. Say a ruler has millimeter marks and you read 4.37 cm. The 4 and the 3 are certain, and the 7 is your best estimate between marks. Writing 4.3700 cm would claim precision the ruler can't give.

To count sig figs, start with nonzero digits, which always count. Zeros between nonzero digits count too, so 509 has 3. Leading zeros never count, so 0.0025 has 2. Trailing zeros to the right of a decimal point count, so 50.00 has 4 significant figures. Trailing zeros in a whole number without a decimal point can be ambiguous, so 500 could represent 1, 2, or 3 significant figures. Scientific notation removes the ambiguity, since $5 \times 10^2$, $5.0 \times 10^2$, and $5.00 \times 10^2$ have 1, 2, and 3 significant figures, respectively.

Calculated answers follow two different rules. For addition and subtraction, round to the fewest decimal places. For multiplication and division, round to the fewest significant figures. Exact numbers, like counted objects or defined values such as 1 m = 100 cm, have unlimited sig figs and never limit your answer.

Worked example. A sample has a mass of 12.7 g and a volume of 4.3 mL. The calculator gives $12.7 \div 4.3 = 2.9534...$ g/mL. Division follows the sig fig rule. Since 4.3 has only 2 sig figs, the density is 3.0 g/mL. Keep the trailing zero, because it shows the second sig fig.

Accuracy is how close a measurement is to the true or accepted value. Precision is how close repeated measurements are to each other. The two are independent. Suppose the accepted density of aluminum is 2.70 g/cm³ and your three trials read 2.95, 2.96, and 2.95 g/cm³. Your data is precise but not accurate.

The type of error explains the difference. Systematic error shifts every reading in the same direction, like a balance that wasn't zeroed and reads 0.15 g high every time. It hurts accuracy and may not appear as scatter. Repeating the same uncorrected procedure does not reduce it, but checking the instrument against a known standard can reveal it. Random error comes from small unpredictable variations, like reading a meniscus slightly differently each time. It hurts precision, and averaging more trials reduces its effect.

Percent error measures accuracy:

$\text{percent error} = \frac{|\text{experimental} - \text{accepted}|}{\text{accepted}} \times 100\%$

Worked example. You measure the density of copper as 8.65 g/cm³, and the accepted value is 8.96 g/cm³. The difference is $|8.65 - 8.96| = 0.31$ g/cm³. Divide by the accepted value to get $0.31 \div 8.96 = 0.0346$. Multiply by 100%, and the percent error is 3.5%. Some teachers drop the absolute value bars so the sign shows whether you were high or low, so check your class's convention.

Dimensional analysis, also called the factor-label method, converts units by multiplying by conversion factors. A conversion factor is a fraction made from two equal amounts, like $\frac{1000 \text{ m}}{1 \text{ km}}$. The top and bottom are equal, so the fraction equals 1. Multiplying by it changes the units without changing the amount.

Choose the version of each factor that puts the unit you want to cancel on the opposite side of the fraction. If you start with km, km goes in the denominator of the factor. You can chain as many factors as you need. Then check that every unit cancels except the one you're solving for. If the leftover units are wrong, the setup is wrong, even if the arithmetic is right.

Worked example. Convert 5.00 inches to millimeters.

$5.00 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{10 \text{ mm}}{1 \text{ cm}} = 127 \text{ mm}$

Inches cancel first, then centimeters, which leaves mm. Both 2.54 and 10 are exact definitions, so the measured 5.00 sets the answer at three sig figs.

You'll use this same setup for the rest of the course. Molar mass will become a conversion factor between grams and moles, and a stoichiometry problem is just a longer chain of factors.