Problem 67 - Chemistry: Principles and Reactions

Question

Each time you inhale, you take in about \(500 \mathrm{~mL}\) (two significant figures) of air, each milliliter of which contains \(2.5 \times 10^{19}\) molecules. In delivering the Gettysburg Address, Abraham Lincoln is estimated to have inhaled about 200 times. (a) How many molecules did Lincoln take in? (b) In the entire atmosphere, there are about \(1.1 \times 10^{44}\) molecules. What fraction of the molecules in the earth's atmosphere was inhaled by Lincoln at Gettysburg? (c) In the next breath that you take, how many molecules were inhaled by Lincoln at Gettysburg?

Step-by-Step Solution

Answer

Answer: In your next breath, about 284 molecules will be from the ones that were inhaled by Lincoln at Gettysburg.

Step 1: a) Calculate the total number of molecules Lincoln took in during his speech

To find out the total number of molecules Lincoln took in during his speech, we have to multiply the number of inhales (200) by the volume of air in each inhale (500mL) and by the number of molecules in each milliliter of air. Total number of molecules = Inhales × Volume per inhale × Molecules per mL Total number of molecules = 200 × 500mL × 2.5 × 10^{19} Total number of molecules = 2.5 × 10^{25} So, Lincoln took in a total of 2.5 × 10^{25} molecules during his speech.

Step 2: b) Calculate the fraction of the total molecules in the Earth's atmosphere inhaled by Lincoln

To find out the fraction of the total molecules in the Earth's atmosphere that were inhaled by Lincoln at Gettysburg, we have to divide the number of molecules inhaled by Lincoln by the total number of molecules in the Earth's atmosphere. Fraction = (Molecules inhaled by Lincoln) / (Total molecules in the atmosphere) Fraction = (2.5 × 10^{25}) / (1.1 × 10^{44}) Fraction = 2.273 × 10^{-19} Therefore, the fraction of the total molecules in the Earth's atmosphere that were inhaled by Lincoln at Gettysburg is 2.273 × 10^{-19}.

Step 3: c) Calculate how many molecules in the next breath that you take were inhaled by Lincoln

To find out how many molecules in the next breath you take were inhaled by Lincoln at Gettysburg, we have to multiply the volume of one inhale (500mL) by the number of molecules per mL of air, and then multiply it by the fraction that we calculated previously. Molecules in next breath = (Volume per inhale × Molecules per mL) × (Fraction of molecules inhaled by Lincoln) Molecules in next breath = (500mL × 2.5 × 10^{19}) × (2.273 × 10^{-19}) Molecules in next breath ≈ 284.125 Since the number of molecules cannot be a fraction, we can round it up to the nearest whole number. Molecules in next breath ≈ 284 So, in your next inhalation, about 284 molecules will be from the ones that were inhaled by Lincoln at Gettysburg.

Key Concepts

Significant Figures

When performing calculations in chemistry and physics, it's important to consider significant figures. Significant figures are the digits in a number that contribute to its precision. They include all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion.

• In this exercise, the amount of air inhaled with each breath (500 mL) is specified with two significant figures.
• This indicates that only the first two digits of this number are reliable or significant when making calculations.
• Ensuring you maintain the correct number of significant figures is crucial for keeping your answers accurate without implying false precision.

Remember, the accuracy of your final result is limited by the measurement with the fewest significant figures. In our calculations, we'd report 2.5 × 10²⁵ molecules, respecting the significant figures provided.

Molecule Counting

Molecule counting in this scenario involves understanding the sheer number of molecules you inhale with every breath. To calculate the total molecules inhaled by Lincoln, we followed these steps:

• First, identify the number of inhales: Lincoln inhaled 200 times during his speech.
• Next, calculate the total volume of air inhaled: 200 inhales × 500 mL per inhale = 100,000 mL.
• Multiply the total volume by the number of molecules per mL: 100,000 mL × 2.5 × 10¹⁹ molecules/mL = 2.5 × 10²⁵ molecules.

Molecule counting in chemistry often deals with very large numbers, underscoring why it's essential to understand how to mathematically handle these quantities. Using scientific notation simplifies dealing with such large numbers.

Avogadro's Number

Avogadro's number, which is approximately 6.022 × 10²³, is a fundamental concept in molecular chemistry. It represents the number of atoms, ions, or molecules in one mole of a substance.

• Although Avogadro's number isn't directly used in this problem, it's related as it helps in understanding the vastness of molecular numbers we encounter.
• Considering that even a tiny volume of air contains a huge number of molecules highlights the importance of Avogadro's number in making chemical calculations feasible.
• This principle underlines calculations like those in our exercise, where we discern how many molecules are shared between different inhalations.

Using Avogadro's number allows chemists to transition between macroscopic measurements we can see and microscopic information about atoms and molecules, helping bridge the gap and making practical, real-world chemistry possible.