Notes on Chapter 2: Chemistry and Measurement

Overview

• Chemistry is the study of matter, its properties, and the changes matter undergoes.
• Chemical principles operate in everyday life (food preparation, environment) and in complex processes.
• Substances’ properties can be tailored by controlling composition and structure.

Units of Measurement and SI System

• The metric system is widely used; SI (Système International d’Unités) has seven base units from which others are derived.
• A different base unit is used for each quantity.
• Key idea: every measurement consists of a number and a unit; the number is meaningless without the unit.
• Example: Proper aspirin dosage is 325 (milligrams or kilograms or pounds); a 100 m dash time could be 10.0 (seconds or minutes).

SI Base Units and Common Quantities

• Base units in the metric system include:
  • Mass: gram (g)
  • Length: meter (m)
  • Time: second (s)
  • Temperature: degrees Celsius (°C) or Kelvin (K)
  • Amount of a substance: mole (mol)
  • Volume: liter (L) or cubic centimeter (cm^3) (note: volume is derived in SI as m^3; see later)
• Volume is commonly expressed as cm^3 or L:
  • 1 L = 1000 mL
  • 1 mL = 1 cm^3 = 1 cc
• Important nuance: 1 L is a cube 1 decimeter on each side; 1 mL is a cube 1 centimeter on each side.
• Volume is technically a derived unit in SI: m^3 = (m)(m)(m); common practical units are L and mL.

Prefixes and Conversions (Metric System)

• Prefixes convert base units to larger or smaller scales. Examples:
  • 1 G = 1 × 10^9 (Giga, e.g., 1 Gm = 10^9 m)
  • 1 m = 1 × 10^−3 (milli, e.g., 1 mm = 1×10^−3 m)
  • 1 k = 1 × 10^3 (kilo, e.g., 1 km = 10^3 m)
  • 1 n = 1 × 10^−9 (nano, e.g., 1 nm = 1×10^−9 m)
• You must know the meaning of each prefix and the corresponding equivalence statement.
• Fill-in-the-blank exercise illustrates these equivalences.

Temperature: Celsius and Kelvin

• The Celsius scale is based on water properties: 0 °C is the freezing point of water; 100 °C is the boiling point.
• The Kelvin scale is the SI unit of temperature; based on gas properties; there are no negative Kelvin temperatures; absolute zero is 0 K.
• Conversion: $K = °C + 273.15$
• Converting between scales:
  • °C to K: $K = °C + 273.15$
  • K to °C: $°C = K - 273.15$
• Absolute temperatures are compared directly in K; temperature differences (ΔT) are the same in °C and K (ΔT in °C equals ΔT in K).

Temperature and Measurements: Comparing Scales

• Temperature is the measure of the average kinetic energy of particles in a sample.
• Commonly used: Celsius and Kelvin.
• For conversion tasks, use the relations above and compare both absolute temperatures and temperature changes.

Volume and Derived Units

• Volume is not a base SI unit; it is derived from length: $V = L^3$ (for a cube) or, more generally, volume is mass per unit volume related in density calculations.
• Common metric units for volume: liter (L) and milliliter (mL).
• 1 L = 1000 mL; 1 mL = 1 cm^3; thus 1 L = 1000 cm^3.

Density and Other Density-Related Concepts

• Density is a physical property defined as mass per unit volume:
  d = rac{m}{V}
• Density can be expressed in any mass/volume units; common units are g/mL or g/cm^3 (they are numerically equivalent).
• Note the common mistake: equal masses do not imply equal volumes (e.g., 1 kg of air vs 1 kg of iron have different volumes).
• In density problems, density is the conversion factor between mass and volume.

Problem-Solving with Density

• Example problem: Calculate the volume of 65.0 g of methanol if density is 0.791 g/mL.
  • Use the conversion: $1\,mL = 0.791\,g$ or equivalently $V = \frac{m}{d} = \frac{65.0\,g}{0.791\,g/mL} \approx 82.2\,mL$
  • We can express density two ways: $1\,mL = 0.791\,g\quad \text{or} \quad 1\,g = \frac{1}{0.791}\,mL$

Significant Figures (SF) and How to Report Numbers

• Definitions:
  • Exact numbers have no uncertainty (e.g., 12 eggs in a dozen, 1000 g in 1 kg).
  • Inexact numbers come from measurements and have uncertainty.
• Rules for significant figures:
  1) All nonzero digits are significant. Example: $3.25\text{ s.f.}$, $1896\text{ s.f.}$, $35\text{ s.f.}$
  2) Zeros between significant figures are significant. Examples: $3.005\text{ s.f.}$, $106\text{ s.f.}$, $20,002\text{ s.f.}$
  3) Leading zeros are never significant. Examples: $0.0057\text{ s.f.}$, $0.292\text{ s.f.}$, $0.00008\text{ s.f.}$
  4) Trailing zeros are significant if a decimal point is present: $8000.\text{ (4 s.f.)}, \ 2.3900\text{ (5 s.f.)}$
• Rules for addition/subtraction vs multiplication/division:
  • Addition/Subtraction: the result should have the same number of decimal places as the measurement with the fewest decimal places. Example: $106.7\ g + 0.25\ g + 0.195\ g = 107.1\ g$ (1 decimal place in the result)
  • Multiplication/Division: the result should have the same number of significant figures as the measurement with the fewest significant figures. Example: $(7.0\,\text{cm})(0.002\,\text{cm}) = 0.014\,\text{cm}^2$ with appropriate SF rounding
• Practical tips:
  • Do not round during intermediate steps; apply SF rules only after the final calculation.
  • Use the PEMDAS rule when handling multi-step calculations.
  • Exact numbers do not affect SF in calculations.

Scientific Notation (SciNot) and Calculator Use

• Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10:
  • Large numbers: exponent positive
  • Small numbers: exponent negative
• Notation form: N = a \times 10^{n}, \quad 1 \le a < 10
• Examples: 3820 → $3.820 \times 10^{3}$ (to 4 s.f.); 0.0478 → $4.78 \times 10^{-2}$
• Calculator usage:
  • Many calculators have an exponent key (EXP or EE). Example: 7.45E-17 corresponds to $7.45 \times 10^{-17}$
  • To enter: enter 7.45, press EXP/EE, enter -17.

Dimensional Analysis (Unit Analysis)

• A method to convert quantities by multiplying by conversion factors based on equivalence statements.
• Core idea: cancel units to obtain the desired unit.
• Example equivalence: 1 in = 2.54 cm. Two possible conversion factors: $\frac{1\,\text{in}}{2.54\,\text{cm}} \quad \text{or} \quad \frac{2.54\,\text{cm}}{1\,\text{in}}$
• Process:
  • Identify the original and desired units.
  • Write the necessary conversion factors.
  • Arrange the factors so that unwanted units cancel.
• Example: Convert $2.50\ \text{kg}$ to g.
  • Step: use $1\,\text{kg} = 1000\,\text{g}$
  • Calculation: $2.50\ \text{kg} \times \frac{1000\ \text{g}}{1\ \text{kg}} = 2500\ \text{g}$
  • Note on significant figures: 2.50 kg has 3 significant figures; express result accordingly (e.g., $2.50\times 10^{3}\ \text{g}$) if precision must be explicit.

Single vs Multi-Factor Dimensional Analysis Problems

• Single-factor example: Convert 2.50 kg to g; setup shows that the kg cancels leaving g.
• Multi-factor example: Convert 300.0 mL to gallons using two factors:
  • Step 1: $300.0\ \text{mL} \times \frac{1\ \text{L}}{1000\ \text{mL}} = 0.300\ \text{L}$
  • Step 2: $0.300\ \text{L} \times \frac{1\ \text{gal}}{3.785\ \text{L}} = 7.93\times 10^{-1}\ \text{gal}$
  • Final: approximately $0.0793\ \text{gal}$ with proper significant figures.

Practical Applications: Health, Medicine, and Toxicology

• Dimensional analysis is frequently used in health-related calculations (drug dosages, concentrations).
• Important concept: dose is often expressed as mg per kg of body mass; LD50 (lethal dose for 50% of test animals) is given in mg/kg and used to compare toxicity:
  • Example: caffeine LD50 ≈ $192\ \text{mg/kg}$.
• Toxicology: LD50 values for common substances provide a toxicity scale; lower LD50 means higher toxicity (e.g., Parathion ≈ $3\ \text{mg/kg}$).
• A typical toxicity table lists substances and their LD50 in mg/kg (in rats) to compare relative danger.

Health and Medicine: Targeted Calculations and Real-World Examples

• Understanding what you will be given and what you must determine:
  • Mass and conversion between mass units (lb, g, kg)
  • Distance and conversion between units (inch, cm, m, km)
  • Volume conversions (L, mL, cm^3, gal, etc.)
  • Temperature conversions (°C, K)
  • Compound vs element, homogeneous vs heterogeneous, accuracy vs precision
• Examples and common problem patterns include:
  • Converting dose amounts (mg to mL using concentration)
  • Calculating number of tablets from total dose
  • Converting between mass, volume, and concentration for solutions
• Dose-related conversion factors often come from problem statements (e.g., 250 mg tablet or 80 mg in 2.5 mL syrup) and serve as built-in conversion factors.

Worked Drug-Dose Problems (Representative Scenarios)

• Isotretinoin dosing: 0.5 mg per kg body weight; tablets come in 20 mg tablets; convert body weight to kg, then compute required mg, then number of tablets.
  • Step 1: Convert weight from lb to kg: $1\ \text{lb} = 0.454\,\text{kg}$
  • Step 2: Required mg: $\text{mg needed} = 0.5\ \frac{\text{mg}}{\text{kg}} \times m_{kg}$
  • Step 3: Number of tablets: $\text{tablets} = \frac{\text{mg needed}}{20\ \text{mg/tablet}}$
• Example with a 133 lb patient:
  • Convert weight: $133\ \text{lb} \times 0.454\frac{\text{kg}}{\text{lb}} = 60.382\ \text{kg}$
  • Mg needed: $60.382\ \text{kg} \times 0.5\frac{\text{mg}}{\text{kg}} = 30.191\ \text{mg}$
  • Tablets needed: $\frac{30.191\ \text{mg}}{20\ \text{mg/tablet}} = 1.509\text{ tablets} \approx 1.5\text{ tablets}$
• Another isotretinoin problem variation: compute mg first, then convert to number of tablets (same method).

Specific Measurement Conversions (Representative Problems)

• abc Problem: A hospital has 125 deciliters of blood plasma. Convert to milliliters:
  • 1 dL = 100 mL; Result: $125\ \text{dL} \times \frac{100\ \text{mL}}{1\ \text{dL}} = 12500\ \text{mL}$
• Self-check patterns (for practice):
  • a) 300 g to mg → $300\,\text{g} \times \frac{10^3\,\text{mg}}{1\,\text{g}} = 3.00\times 10^5\,\text{mg}$
  • b) 2 mL to µL → $2\,\text{mL} \times \frac{10^3\,\text{µL}}{1\,\text{mL}} = 2.00\times 10^3\,\text{µL}$
  • c) 5.0 cm to m → $5.0\,\text{cm} \times \frac{1\,\text{m}}{10^2\,\text{cm}} = 0.050\,\text{m}$
  • d) 2 ft to m (1 ft = 0.3048 m) → $2\,\text{ft} \times 0.3048\,\frac{\text{m}}{\text{ft}} = 0.6096\,\text{m}$
  • e) 35 kg to g → $35\,\text{kg} \times \frac{10^3\,\text{g}}{1\,\text{kg}} = 3.50\times 10^4\,\text{g}$
  • f) 0.35 cup to mL (1 cup = 236.6 mL) → $0.35\text{ cup} \times \frac{236.6\,\text{mL}}{1\text{ cup}} = 82.81\,\text{mL}$
• Two-factor conversion example: Gallons in 300.0 mL
  • Setup: $300.0\ \text{mL} \times \frac{1\ \text{L}}{1000\ \text{mL}} \times \frac{1\ \text{gal}}{3.785\ \text{L}} = 0.0793\ \text{gal}$
  • Emphasize arranging factors so that units cancel properly.

Special Topics and Miscellany

• Time, decimal places, and rounding: keep track of the fewest decimal places in addition/subtraction; preserve SF in final results.
• Exact numbers and unit handling: treat units with the same care as algebraic symbols (e.g., 1x + 1x = 2x).
• Dimensional analysis notes: when performing multiple steps, press Enter after each division on a calculator as needed; track unit cancellations explicitly.
• Practical health example: drug concentration conversions leverage mg to mL or mg to tablet counts based on reported concentrations or tablet contents.

Quick Reference: Common Formulas and Rules (Summary)

• Temperature conversion: $K = °C + 273.15$
• Density: $d = \dfrac{m}{V}$
• Volume relationships: $1\ \text{L} = 1000\ \text{mL}, \quad 1\ \text{mL} = 1\ \text{cm}^3$
• Mass conversions: $1\ \text{kg} = 1000\ \text{g}$
• Scientific notation: N = a \times 10^{n}, \quad 1 \le a < 10, with appropriate exponents for large or small numbers
• Dimensional analysis core idea: equivalent statements drive conversion factors; cancel units to obtain the desired unit
• SF rules (short):
  • Nonzero digits are significant; zeros between significant digits are significant; leading zeros are not; trailing zeros are significant if a decimal point is present
  • Addition/subtraction: align decimal places; least precise decimal place controls
  • Multiplication/division: limit to the fewest significant figures among inputs
• Exact numbers do not limit SF in calculations unless explicitly defined by a measurement (e.g., counting numbers, defined constants)

Remembered Takeaways

• Understand the distinction between base vs derived units; volume is derived in SI, commonly used as L and mL in practice.
• Master the conversion factors and how to set up dimensional analysis problems so that unwanted units cancel cleanly.
• Be comfortable with both the conceptual understanding (what SF, accuracy, precision mean) and practical computation (rounding, multi-step calculations, and notation).
• In health/medicine contexts, pay attention to concentrations, mg/kg dosing, and practical conversion to dose or tablet counts, using the given conversion factors carefully.

Quick Practice Prompts (to test yourself)

• Convert 2.50 kg to g with correct SF; express as 2.50×10^3 g if needed.
• A 3000 mg dose is prescribed for a patient who weighs 70 kg. What is the dose in mg/kg? Is this ≤1 mg/kg, etc. (conceptual exercise).
• If density of a substance is 0.791 g/mL, what is its volume if mass is 65.0 g? Provide answer in mL with proper SF.
• Express 0.036 in scientific notation; express 2500 in scientific notation with appropriate SF.
• A drug comes as 80 mg per 2.5 mL. How many mL are required for a 240 mg dose?