Key Math Skills for Chemistry and Forensic Applications

Key Math Skills for Chemistry

• Place Value Identification

  • Determining the place value of specific digits within numbers requires identifying position relative to the decimal point.
  • For the number 97.5689, the digit 5 is in the tenths place.
  • For the number 375.88, the digit 7 is in the tens place.
  • For the number 46.1000, the second 0 after the decimal point is in the hundredths place.

• Evaluating Signed Numbers and Operations

  • Subtraction of a negative number is equivalent to adding its positive value:
    • $15 - (-8) = 15 + 8 = 23$
  • Addition of two negative numbers yields a negative total:
    • $-8 + (-22) = -30$
  • Order of operations requires performing multiplication prior to addition:
    • $4 \times (-2) + 6 = -8 + 6 = -2$
  • Absolute value and signed operations:
    • $-|1| - (-9) = -1 + 9 = 8$
    • Alt notation solution: $-11 - (-9) = -2$
    • $34 + (-55) = -21$
  • Division involving a negative signed numerator:
    • $\frac{-56}{8} = -7$

• Solving Algebraic Equations for Variables

  • Solving for $b$ in linear single-variable equations:
    • Equation: $2b + 7 = b + 10$
      • Subtract $b$ from both sides: $b + 7 = 10$
      • Subtract 7 from both sides: $b = 3$
    • Equation: $3b - 4 = 24 - b$
      • Add $b$ to both sides: $4b - 4 = 24$
      • Add 4 to both sides: $4b = 28$
      • Divide by 4: $b = 7$
  • Solving for $a$ in linear single-variable equations:
    • Equation: $4a + 4 = 40$
      • Subtract 4 from both sides: $4a = 36$
      • Divide by 4: $a = 9$

• Clinical and Practical Percentage Calculations

  • Calculating percentage of patient flu shots:
    • A clinic has 25 patients on Friday morning, and 21 are given flu shots.
    • $\text{Percentage} = \left(\frac{21}{25}\right) \times 100 = 84\%$
    • Expressed to the ones place: $84\%$
  • Calculating percentage of silver in an alloy:
    • An alloy contains $56\,\text{g}$ of pure silver and $22\,\text{g}$ of pure copper.
    • $\text{Total mass} = 56\,\text{g} + 22\,\text{g} = 78\,\text{g}$
    • $\text{Percentage silver} = \left(\frac{56\,\text{g}}{78\,\text{g}}\right) \times 100 = 71.79\% \approx 72\%$
    • Expressed to the ones place: $72\%$
  • Calculating percentage of dimes in a coin collection:
    • A coin collection contains 11 nickels, 5 quarters, and 7 dimes.
    • $\text{Total coins} = 11 + 5 + 7 = 23$
    • $\text{Percentage dimes} = \left(\frac{7}{23}\right) \times 100 = 30.43\% \approx 30\%$
    • Expressed to the ones place: $30\%$
  • Calculating percentage of male newborn babies:
    • 35 babies were born in May at a local hospital; 22 were boys.
    • $\text{Percentage boys} = \left(\frac{22}{35}\right) \times 100 = 62.86\% \approx 63\%$
    • Expressed to the ones place: $63\%$
  • Calculating percentage of zinc in a metal alloy:
    • An alloy contains $67\,\text{g}$ of pure gold and $35\,\text{g}$ of pure zinc.
    • $\text{Total mass} = 67\,\text{g} + 35\,\text{g} = 102\,\text{g}$
    • $\text{Percentage zinc} = \left(\frac{35\,\text{g}}{102\,\text{g}}\right) \times 100 = 34.31\% \approx 34\%$
    • Expressed to the ones place: $34\%$
  • Calculating percentage of pennies in a collection:
    • A collection contains 15 pennies, 14 dimes, and 6 quarters.
    • $\text{Total coins} = 15 + 14 + 6 = 35$
    • $\text{Percentage pennies} = \left(\frac{15}{35}\right) \times 100 = 42.86\% \approx 43\%$
    • Expressed to the ones place: $43\%$

Graphical Analysis of Cooling and Post-Mortem Data

• Graph Analysis: Temperature of Tea versus Time for Cooling

  • Title Significance: Indicates the relationship between the cooling temperature of tea and elapsed time.
  • Vertical Axis (Y-axis):
    • Measures: Temperature ($^\circ\text{C}$).
    • Range of values: $0\,\text{to } 80\,^\circ\text{C}$.
  • Horizontal Axis (X-axis):
    • Measures: Time (min).
    • Range of values: $0\,\text{to } 100\,\text{min}$.
  • Observed Trend: Temperature decreases as time increases.
  • Data Interpolation:
    • Temperature of tea after $20\,\text{min}$: Approximately $50\,^\circ\text{C}$.
    • Time required to reach $45\,^\circ\text{C}$: Approximately $38\,\text{min}$.

• Graph Analysis: Post-Mortem Body Temperature versus Time

  • Title Significance: Displays post-mortem body temperature decline as time progresses following death.
  • Vertical Axis (Y-axis):
    • Measures: Post-Mortem Body Temperature ($^\circ\text{C}$).
    • Range of values: $20\,\text{to } 40\,^\circ\text{C}$.
  • Horizontal Axis (X-axis):
    • Measures: Time (hours) Since Death.
    • Range of values: $0\,\text{to } 25\,\text{hours}$.
  • Observed Trend: Body temperature decreases over time following death.
  • Data Interpolation:
    • Time needed to reach a body temperature of $28\,^\circ\text{C}$: $11.5\,\text{hours}$.

• Forensic Application: Estimating Time of Death

  • Case Scenario: Gloria's body temperature was measured by the coroner at 9 P.M. as $34\,^\circ\text{C}$.
  • Determination using graph:
    • A body temperature of $34\,^\circ\text{C}$ corresponds to $5\,\text{hours}$ elapsed since death.
    • Calculation of time of death: $9\,\text{P.M.} - 5\,\text{hours} = 4\,\text{P.M.}$

Writing and Comparing Numbers in Scientific Notation

• Converting Standard Numbers to Scientific Notation

  • Numbers greater than 1 (positive exponent):
    • $55\,000 = 5.5 \times 10^4$
    • $480 = 4.8 \times 10^2$
    • $670\,000 = 6.7 \times 10^5$
    • $180\,000\,000 = 1.8 \times 10^8$
    • $750 = 7.5 \times 10^2$
    • $1500 = 1.5 \times 10^3$
  • Numbers less than 1 (negative exponent):
    • $0.000\,005 = 5 \times 10^{-6}$
    • $0.000\,14 = 1.4 \times 10^{-4}$
    • $0.0072 = 7.2 \times 10^{-3}$
    • $0.000\,06 = 6 \times 10^{-5}$
    • $0.15 = 1.5 \times 10^{-1}$
    • $0.024 = 2.4 \times 10^{-2}$

• Comparing Magnitudes in Pairs of Numbers

  • Identifying the larger number in a pair:
    • Between $7.2 \times 10^3$ and $8.2 \times 10^3$: The larger number is $8.2 \times 10^3$.
    • Between $4.5 \times 10^{-4}$ and $3.2 \times 10^{-2}$: The larger number is $3.2 \times 10^{-2}$.
    • Between $1 \times 10^4$ and $1 \times 10^{-4}$: The larger number is $1 \times 10^4$.
    • Between $0.000\,552$ ($5.52 \times 10^{-4}$) and $6.8 \times 10^{-2}$: The larger number is $6.8 \times 10^{-2}$.
  • Identifying the smaller number in a pair:
    • Between $4.9 \times 10^{-3}$ and $5.5 \times 10^{-4}$: The smaller number is $5.5 \times 10^{-4}$.
    • Between $1250$ ($1.25 \times 10^3$) and $3.4 \times 10^2$ ($340$): The smaller number is $3.4 \times 10^2$.
    • Between $0.000\,000\,4$ ($4.0 \times 10^{-7}$) and $5.0 \times 10^2$: The smaller number is $0.000\,000\,4$.
    • Between $2.50 \times 10^{-2}$ and $4 \times 10^5$: The smaller number is $2.50 \times 10^{-2}$.

Clinical Update and Forensic Toxicology

• Forensic Investigation Summary: Ethylene Glycol Poisoning

  • Forensic scientist Sarah conducted laboratory tests on victim blood samples and identified ethylene glycol.
  • Quantitative blood tests revealed that the victim had ingested $125\,\text{g}$ of ethylene glycol.
  • Analysis of liquid from a glass at the crime scene confirmed ethylene glycol had been added to an alcoholic beverage.
  • Fingerprint evidence on the glass containing ethylene glycol matched the victim's husband.
  • A container of antifreeze found in the pantry/home matched the color of the liquid in the glass.
  • Combined physical and chemical evidence led to the arrest and conviction of the husband for poisoning his wife.

• Chemical and Physical Properties of Ethylene Glycol

  • Appearance: Clear, thick liquid.
  • Taste: Sweet-tasting.
  • Odor: Odorless.
  • Solubility: Readily mixes with water.
  • Availability: Readily accessible due to common application as automobile antifreeze and brake fluid component.

• Toxicology, Symptoms, and Treatment

  • Initial Symptoms: Mimic alcohol intoxication, leaving victims unaware of ingestion.
  • Systemic Toxicity: Causes central nervous system depression, cardiovascular damage, and renal (kidney) failure.
  • Medical Treatment: Hemodialysis can be performed to extract ethylene glycol from the bloodstream if detected early.
  • Toxicity Thresholds:
    • Toxic Threshold Ratio: $1.5\,\text{g}$ of ethylene glycol per $1\,\text{kg}$ of body mass ($1.5\,\text{g/kg}$).
    • Fatal Dose Example: A dose of $75\,\text{g}$ can be fatal for a $50\text{-kg}$ ($110\text{-lb}$) individual.

• Categorization of Forensic Police Report Statements

  • Statement 1: "Gloria may have had a heart attack."
    • Classification: Hypothesis
  • Statement 2: "Test results indicate that Gloria was poisoned."
    • Classification: Conclusion
  • Statement 3: "The liquid in the glass was analyzed."
    • Classification: Experiment
  • Statement 4: "The antifreeze in the pantry was the same color as the liquid in the glass."
    • Classification: Observation
  • Statement 5: "Gloria may have committed suicide."
    • Classification: Hypothesis
  • Statement 6: "Sarah ran blood tests to identify any toxic substances."
    • Classification: Experiment
  • Statement 7: "The temperature of Gloria's body was $34\,^\circ\text{C}$."
    • Classification: Observation
  • Statement 8: "The fingerprints found on the glass were determined to be her husband's."
    • Classification: Conclusion

• Quantitative Forensic Calculations

  • Calculating Ethylene Glycol Percentage in Crime Scene Liquid:
    • A container found in the victim's home contained $120\,\text{g}$ of ethylene glycol in $450\,\text{g}$ of liquid.
    • $\text{Percentage} = \left(\frac{120\,\text{g}}{450\,\text{g}}\right) \times 100 = 26.67\% \approx 27\%$
    • Expressed to the ones place: $27\%$
  • Calculating Fatal Percentage Threshold of Body Mass:
    • Toxic quantity is defined as $1.5\,\text{g}$ of ethylene glycol per $1000\,\text{g}$ of body mass.
    • $\text{Fatal Percentage} = \left(\frac{1.5\,\text{g}}{1000\,\text{g}}\right) \times 100 = 0.15\%$