Class Notes on Chemical Calculations and Measurement Precision

Major Chemical Conversion Factors

In this chemistry course, three major conversion factors are used constantly: percentage, density, and molar mass.
Molar Mass: This is considered the most important of the three for chemical calculations.
Density: Defined as the ratio of mass to volume ( $\text{Density} = \frac{\text{mass}}{\text{volume}}$ ).
Percentage: This is measured by taking the quantity of a specific component and dividing it by the quantity of the whole substance.

Scientific Measurement and Instrument Precision

The primary goal in lab settings is to report measurements to the proper degree of precision. Accuracy is important, but learning to read instruments correctly is vital for medical and scientific careers.
The Fundamental Rule of Measurement: Always record a measurement to one-tenth smaller than the degree of precision (the smallest marking) shown on the instrument.
The Meniscus: When liquid is poured into a glass cylinder, water molecules stick to the glass more than to each other, creating a curved surface. Always read the measurement from the bottom of the meniscus.
Graduated Cylinders Examples:
- 10 mL Cylinder: Typically marked in tenths ( $0.1\,mL$ ). Therefore, measurements must be recorded to the hundredths place (e.g., $2.93\,mL$ or $3.00\,mL$ ).
- 50 mL or 100 mL Cylinder: Typically marked in ones ( $1\,mL$ ). Measurements must be recorded to the tenths place (e.g., $15.5\,mL$ ).
- 500 mL or 1000 mL (1 L) Cylinder: Typically marked in tens ( $10\,mL$ ). Measurements must be recorded to the ones place (e.g., $151\,mL$ ).
Significance of Trailing Zeros: Trailing zeros behind a decimal point are significant because they indicate the precision of the measurement tool. In a calculation, entering $3$ vs. $3.00$ in a calculator yields the same numerical result, but writing $3.00$ on a report signifies a specific measurement methodology.

Rules for Significant Figures in Calculations

Addition and Subtraction Rule: The answer must be reported to the smallest degree of precision found in the values used, which is determined by the "least precise" column.
- Example: Adding $150$ (assuming the zero is non-significant/placeholder) and $42$ results in $190$ , because the hundreds place or tens place was the limit of precision.
- Example: If a digital scale reads to the thousandths place ( $0.001\,g$ ), even if a subtraction results in a simple value like $0.2$ , it must be recorded as $0.200\,g$ to maintain the recorded precision of the scale.
Multiplication and Division Rule: The final answer is limited by the value with the fewest number of significant figures (sig figs).
- Example: $\frac{2.976\,g}{3.00\,mL}$ involves a four-sig-fig mass and a three-sig-fig volume. The answer must be three sig figs: $0.992\,g/mL$ .
- Example: $14.73\,mL \times 0.992\,g/mL = 14.61216\,g$ . Since the density has three sig figs, the answer is rounded to $14.6\,g$ .
Scientific Notation: Used to eliminate ambiguity in significant figures, especially with trailing zeros. In scientific notation, only one non-zero digit is placed before the decimal (e.g., $4.45 \times 10^5$ ).
Mixed Operations: When a problem involves both subtraction/addition and multiplication/division, perform the operations inside parentheses first according to their specific rules, then proceed to the next step using the new rules (e.g., subtract first to find the degree of precision, then use that sig fig count for the final multiplication).

Exact Numbers and Definitions

Exact Numbers: These do not affect the number of significant figures in a calculation. They have an infinite number of sig figs.
Examples of Exact Numbers:
- Definitions: Conversions like $1\,g = 1000\,mg$ or $1\,dozen = 12$ items.
- Integer Counts: The number of radii in a diameter is exactly $2$ .
- Mathematical Constants: $\pi$ (Pi) is an irrational number that is treated as exact in context, though its representation ( $3.14159...$ ) can be expanded as needed.
- Molar Mass (Class Standard): For this specific course, molar mass values taken to the nearest tenth are often treated as exact to simplify sig fig tracking for students.

Periodic Table and Atomic Structure

The Mole: A unit of number representing $6.02 \times 10^{23}$ items (Avogadro's Number). This is used for atoms and molecules because they are so small.
Periodic Table Layout (Recommendation: ptable.com):
- Metals: Located on the left and center. They tend to lose electrons to form positive ions.
- Non-metals: Located on the right (green sections).
- Metalloids: Elements with properties of both metals and non-metals.
- Noble Gases: Located on the far right; they are generally unreactive/inert.
Periodic Table Notations:
- Atomic Number: The integer that counts the number of protons in an element (e.g., Iron is $26$ ). This defines the element.
- Atomic Symbol: One or two letters. The first letter is always capitalized, and the second is always lowercase (e.g., $Fe$ for Iron).
- Mass Number: This represents the sum of protons and neutrons. It is not listed directly on the periodic table because it varies between isotopes.
Isotopes: atoms of the same element (same proton count) with different numbers of neutrons (different mass numbers).
Molar Mass Calculation: Derived from a weighted average of all stable isotopes of an element based on their natural percentage of abundance.

Practical Lab Protocols and Calculator Use

Protocol Adherence: Following scientific rules is compared to medical procedures, such as injecting an IV into a vein rather than an artery. Following the correct rule/procedure is a non-negotiable skill.
Calculator Entry for Scientific Notation: Use the $E$ or $EE$ button (representing "times 10 to the power of"). Do not use the multiplication operator and the number $10$ . Ensure proper use of the change-sign ( $+/\text{-}$ ) key as opposed to the subtraction operator when entering negative exponents.
Lab Reports: The "Math and the Calculator" exercise covers these precision rules. While not submitted as a physical report, the content forms the basis for an upcoming lab report quiz due on Friday.

Questions & Discussion

Question: How do you handle a conversion where the units are different (e.g., milligrams to milliliters using density in grams/milliliter)?
Answer: This becomes a two-step problem. First, convert the given units to match the conversion factor (e.g., milligrams to grams by dividing by $1000$ ), then apply the density factor. Mathematically, it is expressed as: $\text{milligrams} \times \frac{1\,g}{1000\,mg} \times \frac{1\,mL}{\text{density in grams}}$ .
Question: How do you round in a scientific notation problem if the next digit is higher than five?
Answer: If rounding to a specific number of sig figs, look at the first digit to the right of your cutoff. If it is five or greater, round up the last significant digit. Always include placeholder zeros if you are not using scientific notation to ensure the magnitude of the number is preserved.