Unit 1 Quick Reference: Measurements and SI Units

Scientific Method

• Scientific Method: approach to acquire knowledge through observation of phenomena.

• Experiment: observation tested in a controlled, repeatable process; leads to a rational conclusion.

• Hypothesis: tentative, testable explanation for observations.

• Theory: tested explanation of basic natural phenomena.

• Law: summarizes a vast number of observations and describes/predicts aspects of the natural world.

• Process (summary): Observe phenomena

  form hypothesis

  make predictions

  test with experiment

  analyze results

  evaluate hypothesis

  refine or reject

  establish theory

  report findings

  reproduce results.

Measurements and Uncertainty

• Measurements are essential for characterizing properties; two key parts: standardization of units and precision/accuracy.

• Precision: agreement among repeated measurements.

• Accuracy: agreement between measured value and true/accepted value.

• Uncertainty: most measurements have some uncertainty; last digit is often estimated.

• Digital instruments: output a certain value plus one uncertain digit.

• Analog instruments: readings based on markings; uncertainty estimated by user.

• Significant figures: reflect certain digits plus one estimated digit.

Significant Figures (SF)

• Rules:

  • Nonzero integers are always significant: 7256

  4 sig figs; 8.29

  3 sig figs.

  • Leading zeros are not significant: 0.0392

  3 sig figs.

  • Trailing zeros are not significant unless after a decimal point: 8200

  2 sig figs; 6230.00

  6 sig figs.

  • Captive zeros are significant: 43.07

  4 sig figs.

  • Exact numbers have infinite sig figs: 1 penny, 1 L = 1000 mL, 1 in = 2.54 cm.

• Exact numbers have unlimited sig figs; use them accordingly in calculations.

Scientific Notation

• Representation: A.XX \times 10^{n} with 1 \le A < 10 and n integer; all digits in A are significant.

• Examples: 437000

  4.37\times 10^{5}; 0.009740

  9.740\times 10^{-3}

Significant Figures in Mathematical Operations

• Multiplication/Division: sig figs in result = least number of sig figs among inputs.

  • Example: 16.84 / 2.54 = 6.6299 has 3 sig figs

  use 6.63.

• Addition/Subtraction: number of decimal places in result = least precise decimal place among inputs.

  • Example: 68 + 1190 = 1258

  3 sig figs shown as appropriate by decimal places; effectively align decimal places.

• Practice: carry all digits through calculations, then round only at the end; track SFs for each step if multiple steps (PEMDAS).

Rounding Practice

• Round (1.23 g - 0.567 g) to appropriate SFs: 0.34442 \text{ cm}^3

  rounded value according to SF rule.

SI Units (Base Units)

• Length: \text{m} \text{(meter)}, Symbol: m

• Mass: \text{kg} \text{(kilogram)}, Symbol: kg

• Time: \text{s} \text{(second)}, Symbol: s

• Temperature: \text{K} \text{(Kelvin)}, Symbol: K

• Amount of substance: \text{mol} \text{(mole)}, Symbol: mol

• Electric current: \text{A} \text{(ampere)}, Symbol: A

SI Derived Units

• Area: m^2

• Volume: m^3

• Density: \frac{kg}{m^3}

• Speed: \frac{m}{s}

• Acceleration: \frac{m}{s^2}

• Force: kg\cdot \frac{m}{s^2} (newton, N)

• Pressure: \frac{kg}{m\cdot s^2}

• Energy: kg\cdot \frac{m^2}{s^2}

SI Prefixes

• Tera (T): 10^{12}

• Giga (G): 10^{9}

• Mega (M): 10^{6}

• Kilo (k): 10^{3}

• Hecto (h): 10^{2}

• Deca (da): 10^{1}

• Deci (d): 10^{-1}

• Centi (c): 10^{-2}

• Milli (m): 10^{-3}

• Micro (µ): 10^{-6}

• Nano (n): 10^{-9}

• Pico (p): 10^{-12}

• Example conversions shown on slide (e.g., 1 m = 1 \times 10^{3} mm, 1 m = 10^{2} cm, etc.).

Conversion Factors (selected)

• Length: 1\text{ in} = 2.54\text{ cm} (exact); 1\text{ cm} = 0.39370\text{ in}; 1\text{ m} = 39.37\text{ in}

• Mass: 1\text{ kg} = 2.2046\text{ lb}; 1\text{ lb} = 453.59\text{ g}

• Volume: 1\text{ L} = 10^{-3}\text{ m}^3 = 1\text{ dm}^3 = 1000\text{ mL}

• Density/Volume relationships and common equivalences are listed where needed.

• Note: 1 in^3 = 16.39 cm^3 (approx 16.4 cm^3).

Temperature Scales and Conversions

• Fahrenheit to Celsius and Kelvin:

  • K = {}^\circ C + 273.15

  • ^\circ C = K - 273.15

  • ^\circ C = \frac{5}{9}\left(^\circ F - 32\right)

  • ^\circ F = {}^\circ C \left(\frac{9}{5}\right) + 32

• Use same precision as the measured temperature during conversion.

Dimensional Analysis (Unit Analysis)

• Method: carry units through calculations to convert from starting units to desired units.

• Example concept: convert mass/weight and volume through unit factors to compute results; track units at every step.

Big Problem (Unit Conversions & Dosing Concept)

• Problem setup (from slides): dose = 3\ \text{mg per kg per day}; weight in pounds (lbs); concentration = 4.0000\ \text{mg/mL}; bottle price = 27.99\; price per volume = 0.89\\,/\text{in}^3.

• Key formulas:

  • Dose per administration (twice daily):

    \text{mg per dose} = \left(3\ \text{mg/kg/day}\right) \times \text{(weight in kg)} \div 2

  • Convert weight to kg: 1\ \text{kg} = 2.2046\ \text{lb};

    weight(kg) = \frac{\text{weight(lb)}}{2.2046}

  • Bottle volume from price: V_{\text{bottle}} = \frac{27.99}{0.89}\ \text{in}^3

  • Convert bottle volume to mL: V{\text{bottle}}(\text{mL}) = V{\text{bottle}} \times 16.387

  • Total mg in bottle: \text{Total mg} = V_{\text{bottle}}(\text{mL}) \times 4.0000\ \text{mg/mL}

  • Doses per bottle: N = \left\lfloor \dfrac{\text{Total mg}}{\text{mg per dose}} \right\rfloor

• Note: Final numeric value depends on the patient

's weight in pounds; the framework above shows how to compute.

Summary Concepts and Equations

• Core ideas: Scientific Method; Measurements & Units; Precision & Accuracy; Sig figs; Unit Conversions; Temperature Conversions; Dimensional Analysis.

• Key equations:

  • K = {}^\circ C + 273.15

  • ^\circ C = K - 273.15

  • ^\circ C = \frac{5}{9}\left(^\circ F - 32\right)

  • ^\circ F = {}^\circ C \left(\frac{9}{5}\right) + 32