General, Organic, and Biochemistry - Lecture 1.4 Practice Flashcards

Significant Figures in Calculations

Addition and Subtraction Rule for Significant Figures

• When adding or subtracting numbers, the final result must have the same number of digits after the decimal point (decimal places) as the input value with the fewest decimal places.
• For numbers expressed in scientific notation, adjust the exponent of one of the numbers (typically the one with the smaller power of ten) so that both numbers share identical powers of ten before adding or subtracting.

Example Problem: Addition in Scientific Notation

• Calculate:   $9.47 \times 10^{-6} + 9.3 \times 10^{-5}$
• Step 1: Adjust $9.47 \times 10^{-6}$ to have an exponent of $-5$:   $9.47 \times 10^{-6} = 0.947 \times 10^{-5}$
• Step 2: Align the decimal points and add the coefficients:   $0.947 \times 10^{-5} + 9.3 \times 10^{-5} = 10.247 \times 10^{-5}$
• Step 3: Identify the limiting decimal places:
  • $0.947$ contains 3 decimal places.
  • $9.3$ contains 1 decimal place (limiting term).
  • The sum aligned at 1 decimal place is $10.2 | 47 \times 10^{-5}$.
• Step 4: Convert to standard scientific notation and round to 1 decimal place (3 significant figures overall):   $10.247 \times 10^{-5} = 1.0247 \times 10^{-4} \rightarrow 1.02 \times 10^{-4}$

Multiplication and Division Rule for Significant Figures

• The result of a multiplication or division calculation can contain no more significant figures than the input value with the fewest significant figures.
• The least precise input number dictates the overall precision of the calculated result.

Example Problem: Multiplication and Division Chain

• Calculate:   $\frac{4.2 \times 1.594}{2.255}$
• Calculator output:   $2.9688692$
• Significant figure evaluation of input values:
  • $4.2$ contains 2 significant figures.
  • $1.594$ contains 4 significant figures.
  • $2.255$ contains 4 significant figures.
• Because $4.2$ is the least precise number with only 2 significant figures, the final result must be rounded to 2 significant figures:   $\text{Result} = 3.0$

Division of Numbers Written in Scientific Notation

• To divide numbers written in scientific notation, group the mantissas (coefficients) together and the powers of ten together using the associative property. Divide the mantissas, and subtract the exponent in the denominator from the exponent in the numerator using the quotient rule of exponents.

Example Problem: Division in Scientific Notation

• Calculate:   $\frac{3.9 \times 10^1}{-3.0 \times 10^{-4}}$
• Step 1: Separate mantissas and powers of ten:   $\left(\frac{3.9}{-3.0}\right) \times \left(\frac{10^1}{10^{-4}}\right)$
• Step 2: Divide the mantissas:   $\frac{3.9}{-3.0} = -1.3$
• Step 3: Apply the quotient rule for exponents ($\frac{10^a}{10^b} = 10^{a-b}$):   $\frac{10^1}{10^{-4}} = 10^{1 - (-4)} = 10^5$
• Step 4: Combine the simplified terms:   $-1.3 \times 10^5$

Significant Figures in Mixed Calculations

• For calculations involving a sequence of operations (such as subtraction followed by division), track significant figures at each step based on the operational rules.

Case Study: Vaccine Efficacy Calculation

• Scenario: A clinical trial randomly selects $43,000$ individuals, split evenly into Group A (vaccine) and Group B (placebo).
  • Group A recorded $8$ infections.
  • Group B recorded $162$ infections.
• Efficacy Definition: The percentage of individuals who avoided infection:   $\text{Efficacy} = \left(\frac{162 - 8}{162}\right) \times 100\%$
• Step-by-Step Calculation:
  • Subtraction in numerator:     $162 - 8 = 154$     (Since $162$ and $8$ are exact whole counts precise to the ones place, $154$ maintains precision to the ones place, giving $3$ significant figures).
  • Division and percentage scaling:     $\left(\frac{154}{162}\right) \times 100\% = 95.0617...\%$
  • Rounding: Both $154$ and $162$ have $3$ significant figures, so the final calculated result maintains $3$ significant figures:     $\text{Efficacy} = 95.1\%$
• Practical Note: News and public reports frequently state this efficacy as $95\%$ to simplify communication and remove uncertainty. Although one group had $8$ more individuals than the other, this minor difference does not alter the precision of the final percentage.

Fundamentals of Measurement and Units

The Metric and English Systems

• A measurement consists of both a numeric value and a unit indicating the physical quantity being measured. A numeric measurement without units is completely meaningless.
• English System:
  • A collection of functionally unrelated units with non-uniform conversion factors, making unit conversion difficult.
  • Example relationship:     $1\,\text{ft} = 12\,\text{in} = 0.33\,\text{yd} = \frac{1}{5280}\,\text{mi}$
• Metric System:
  • A systematic decimal system composed of units related to one another by powers of ten.

Core Physical Quantities and Standard Units

• Mass:
  • Definition: The quantity of matter in an object. Mass is distinct from weight.
  • Distinction from Weight: Weight depends on gravity:     $\text{Weight} = \text{mass} \times \text{acceleration due to gravity}$
  • Standard Metric Unit: Gram ($\text{g}$).
  • Common English Unit: Pound ($\text{lb}$).
  • Metric-English Equivalence:     $1\,\text{lb} = 454\,\text{g}$
  • Laboratory Measurement: Mass must be measured on a balance rather than a scale.
• Length:
  • Definition: The distance between two spatial points.
  • Standard Metric Unit: Meter ($\text{m}$).
  • Common English Unit: Yard ($\text{yd}$).
  • Metric-English Equivalence:     $1\,\text{yd} = 0.914\,\text{m}$
• Volume:
  • Definition: The three-dimensional space occupied by an object.
  • Standard Metric Unit: Liter ($\text{L}$).
  • Common English Unit: Quart ($\text{qt}$).
  • Metric-English Equivalence:     $1\,\text{qt} = 0.946\,\text{L}$
• Time:
  • Metric Unit: Second ($\text{s}$).

Metric Prefixes

• Metric prefixes modify base units (such as $\text{g}$, $\text{m}$, or $\text{L}$, represented generically by $x$) by specific powers of ten.
• Basic units are standard units of a quantity without any metric prefix attached.

Unit Ranking Example

• Task: Rank the mass units $\text{g}$, $\text{cg}$, and $\text{Mg}$ from greatest to least mass.
  • $\text{Mg} = 10^6\,\text{g}$
  • $\text{g} = 10^0\,\text{g}$
  • $\text{cg} = 10^{-2}\,\text{g}$
• Correct Ranking:   $\text{Mg} > \text{g} > \text{cg}$

Dimensional Analysis and Unit Conversions

The Factor-Label Method

• Unit conversions within the metric system or between English and metric systems are performed using the Factor-Label Method (also called Dimensional Analysis).
• Conversion factors are fractions derived from exact equivalence relationships between two units.
• Exact Numbers Rule: All conversion numbers in standard conversion tables (such as metric prefixes and English-metric conversion factors) and temperature formulas are exact numbers. Exact numbers have infinitely many significant figures and do not restrict the significant figures of calculated results.

Constructing and Applying Conversion Factors

• From an equivalence relationship such as $1\,\text{gal} = 4\,\text{qt}$, two reciprocal conversion factors can be written:   $\frac{1\,\text{gal}}{4\,\text{qt}} \quad \text{or} \quad \frac{4\,\text{qt}}{1\,\text{gal}}$
• To convert a value, multiply by the conversion factor that places the original unit in the denominator (ensuring it cancels out) and the desired unit in the numerator.

Example 1: Volume Conversion

• Task: Convert $12\,\text{gal}$ to $\text{qt}$.
• Given Data: $12\,\text{gal}$.
• Setup:   $12\,\text{gal} \times \frac{4\,\text{qt}}{1\,\text{gal}} = 48\,\text{qt}$
• Diagnostic Unit Check:
  • If the inverted fraction were accidentally used:     $12\,\text{gal} \times \frac{1\,\text{gal}}{4\,\text{qt}} = 3\,\frac{\text{gal}^2}{\text{qt}}$
  • The resulting units $\frac{\text{gal}^2}{\text{qt}}$ fail to cancel, indicating an incorrect arrangement.

Example 2: Metric Time Conversion

• Task: Convert $2831.4\,\text{ms}$ to $\text{s}$.
• Equivalence: $1\,\text{ms} = 10^{-3}\,\text{s}$.
• Setup:   $2831.4\,\text{ms} \times \frac{10^{-3}\,\text{s}}{1\,\text{ms}} = 2.8314\,\text{s}$
• Significant Figures: The initial value $2831.4$ contains $5$ significant figures; since the metric conversion constant is exact, the result retains $5$ significant figures:   $2.8314\,\text{s}$

Example 3: Multi-Step Mass Conversion

• Task: Convert $8.7\,\text{lb}$ to $\text{mg}$, reporting the answer with the correct number of significant figures.
• Required Conversion Equivalences:
  • $1\,\text{lb} = 453.59237\,\text{g}$
  • $1\,\text{mg} = 10^{-3}\,\text{g}$ (or $1\,\text{g} = 10^3\,\text{mg}$)
• Intermediate Calculation Rule: Carry at least two extra digits during intermediate steps to prevent rounding errors, rounding only the final result to the correct number of significant figures.
• Step-by-Step Procedure:
  1. Convert pounds to grams:      $8.7\,\text{lb} \times \frac{453.59237\,\text{g}}{1\,\text{lb}} = 3.946... \times 10^3\,\text{g}$      (If the incorrect factor $\frac{1\,\text{lb}}{453.59237\,\text{g}}$ were used, units would become $\frac{\text{lb}^2}{\text{g}}$).
  2. Convert grams to milligrams:      $3.946... \times 10^3\,\text{g} \times \frac{10^3\,\text{mg}}{1\,\text{g}} = 3.946... \times 10^6\,\text{mg}$
• Single-Chain Setup:   $8.7\,\text{lb} \times \frac{453.59237\,\text{g}}{1\,\text{lb}} \times \frac{10^3\,\text{mg}}{1\,\text{g}} = 3.9462536 \times 10^6\,\text{mg}$
• Final Rounding: Since $8.7\,\text{lb}$ contains $2$ significant figures, round the answer to $2$ significant figures:   $3.9 \times 10^6\,\text{mg}$

Temperature Scales and Conversions

Physical Nature of Temperature

• Temperature measures the degree of "hotness" or thermal energy of an object.
• The Kelvin Scale ($\text{K}$) is directly proportional to molecular motion. As molecular speed increases, the Kelvin temperature increases proportionally.

Comparison of Temperature Scales

• Key physical benchmarks across Kelvin ($\text{K}$), Celsius ($^\circ\text{C}$), and Fahrenheit ($^\circ\text{F}$):

• Benchmark Reference Values:
  • Boiling Point of Water:
  • Kelvin: $373\,\text{K}$
  • Celsius: $100\,^\circ\text{C}$
  • Fahrenheit: $212\,^\circ\text{F}$
  • Body Temperature:
  • Kelvin: $310\,\text{K}$
  • Celsius: $37\,^\circ\text{C}$
  • Fahrenheit: $98.6\,^\circ\text{F}$
  • Room Temperature:
  • Kelvin: $298\,\text{K}$
  • Celsius: $25\,^\circ\text{C}$
  • Fahrenheit: $77\,^\circ\text{F}$
  • Freezing Point of Water:
  • Kelvin: $273\,\text{K}$
  • Celsius: $0\,^\circ\text{C}$
  • Fahrenheit: $32\,^\circ\text{F}$

Temperature Transformation Equations

• Fahrenheit ($T_{^\circ\text{F}}$) to Celsius ($T_{^\circ\text{C}}$):   $T_{^\circ\text{C}} = \frac{T_{^\circ\text{F}} - 32}{1.8}$
• Celsius ($T_{^\circ\text{C}}$) to Fahrenheit ($T_{^\circ\text{F}}$):   $T_{^\circ\text{F}} = T_{^\circ\text{C}} \times 1.8 + 32$
• Celsius ($T_{^\circ\text{C}}$) to Kelvin ($T_{\text{K}}$):   $T_{\text{K}} = T_{^\circ\text{C}} + 273.15$

Temperature Conversion Examples

Example 1: Celsius to Fahrenheit

• Task: Convert $75\,^\circ\text{C}$ to $^\circ\text{F}$.
• Calculation:   $T_{^\circ\text{F}} = 1.8 \times 75 + 32$$1.8 \times 75 = 135$$135 + 32 = 167\,^\circ\text{F}$
• Significant Figures: Since $75\,^\circ\text{C}$ contains $2$ significant figures, express the answer in scientific notation rounded to $2$ significant figures:   $1.7 \times 10^2\,^\circ\text{F}$

Example 2: Fahrenheit to Kelvin

• Task: Convert $-10.\,^\circ\text{F}$ to $\text{K}$.
• Solution Road Map:
  1. Calculate $T_{^\circ\text{C}}$ from $T_{^\circ\text{F}}$.
  2. Calculate $T_{\text{K}}$ from $T_{^\circ\text{C}}$.
• Step 1: Calculate $T_{^\circ\text{C}}$:   $T_{^\circ\text{C}} = \frac{-10. - 32}{1.8} = \frac{-42}{1.8} = -23.333...\,^\circ\text{C}$   (Rounding to ones place yields $-23\,^\circ\text{C}$).
• Step 2: Calculate $T_{\text{K}}$:   $T_{\text{K}} = -23.333... + 273.15 = 249.816...\,\text{K}$$-23. + 273.15 = 250.15\,\text{K}$
• Significant Figures: Expressed in scientific notation rounded to $3$ significant figures:   $2.50 \times 10^2\,\text{K}$