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?

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.

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}.

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.