(III) Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in t

(III) Estimate how many molecules of air are in each 2.0-L...

(III) Estimate how many molecules of air are in each $2.0-\mathrm{L}$ breath you inhale that were also in the last breath Galileo took. $[$ Hint: Assume the atmosphere is about 10 $\mathrm{km}$ high
and of constant density.

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental constant that defines the number of molecules in one mole of a substance (approximately 6.022 x 10^23). It allows us to convert between the macroscopic amount of a substance (measured in moles) and the microscopic number of molecules, which is crucial for estimating the number of air molecules in a given volume, such as a breath.

Mole Concept and Gas Estimation

The mole concept is central in chemistry for quantifying the amount of a substance. By determining the number of moles of gas in a particular volume using principles like the ideal gas law, one can then calculate the number of individual molecules present. This is particularly useful when assessing how many molecules are shared between different volumes of air over time.

Constant Density Atmosphere

The assumption of a constant density atmosphere simplifies calculations by treating the atmospheric air as uniformly distributed with height. This idealization lets us estimate the total volume of the atmosphere and, by extension, the fraction of molecules present in any given volume of air, such as a 2.0-L breath, exposing how even historically separated samples of air are mixed.

Volume Estimation and Mixing

Estimating the volume of the Earth’s atmosphere involves using the Earth's surface area and the assumed height of the atmosphere. This calculation is key for understanding the proportion of a single breath relative to the entire atmosphere. The concept also underscores the extensive mixing of air molecules over time, suggesting that molecules from a breath taken long ago can still be present in a current breath.

Transcript

00:01 Estimate how many molecules of air are in each two -liter breath you inhale that were also in the last breath the galileo took.

00:08 Hint, assuming the atmosphere is about 10 kilometres high and of constant density.

00:13 So, we're going to work backwards from this.

00:16 So the number of molecules, so n number of molecules is equal to the number of molecules in the atmosphere, so n at, multiplied by the probability of any molecule, so any molecule being in any two random breaths, in any two breaths.

01:02 So we're assuming the two events are linearly independent.

01:05 So that's simply the probability of any molecule in any two breaths is simply the probability of any molecule in any one breath squared.

01:14 And the probability of any molecule in one given breath is, so we're just going to call p for shortcut.

01:26 So p total is going to be the square of the probability of it being in any one breath.

01:32 And we give that probability is given by v, the volume of a breath over the volume of the atmosphere.

01:45 And that's squared.

01:47 So we're going to calculate both of those.

01:50 So v of breath, v breath is equal to 2 litres.

01:59 So in s .i.

02:00 Units, that's 2 times 10 to the minus 3.

02:04 Volume of atmosphere, the atmos is equal to.

02:13 And so we're using standard four -thirds pi are cubed minus four -thirds pi a cube...

Please give Ace some feedback