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# Need help with a Fermi problem of "Caeser's Last Breath"?

A famous example of a Fermi problem is "Caeser's last breath" which estimates that you, right now, are breathing some of the molecules exhaled by Julius Caesar just before he died.

Assumptions: 1. The gas molecules from Caesar's last breath are now evenly dispersed in the atmosphere. 2. The atmosphere is 50 km thick, has an average temperature of 15 °C, and an average pressure of 0.20 atm. 3. The radius of the Earth is about 6400 km. 4. The volume of a single human breath is roughly 500 mL.

Perform the following calculations, reporting all answers to two significant figures.

1. Calculate the total volume of the atmosphere in cubed meters.

2. Calculate the total number of gas molecules in the atmosphere.

3. Calculate the number of gas molecules in Caesar\'s last breath (37 °C and 1.0 atm).

4. What fraction of all air molecules came from Caesar\'s last breath?

5. About how many molecules from Caesar's last breath do you inhale each time you breathe?

Please help, I've gotten this question wrong fire times already!!!!

### 3 Answers

- Dr WLv 78 years agoFavorite Answer
*** 1 ***

volume of a sphere = 4/3 x pi x r³

and we want volume of sphere with diameter 6450km - volume of sphere with diameter = 6400km. this gives volume of the atmosphere..

Vatm = [(4/3 x pi x (6450km)³) - (4/3 x pi x (6400km)³)] x (1000m / 1km)³ = 2.6x10^19 m³

*** 2 ***

PV = nRT

n = PV/(RT) = (0.20atm) x (2.59x10^19 m³ x 1000L/1m³) / ((0.08206 Latm/molK) x (288.15K))

n = 2.19x10^20 moles

and number of molecules...

2.19x10^20 mol x (6.022x10^23 molecules / mol) = 1.3x10^44

*** 3 ***

PV = nRT

n = PV/(RT) = (1.0atm) x (0.500L) / ((0.08206 Latm/moLK) x (310.15K)) = 0.0196 mol

number of molecules..

0.0196 mol x (6.022x10^23 molecules / mol) = 1.2x10^22 molecules

*** 4 ***

volume fraction = (1.2x10^22 molecules Caesar exhaust / 1.3x10^44 air) = 9.0x10^-23 molecules / molecule

*** 5 ***

assuming your lung capacity is also 500mL at 37°C and 1.0atm

you breathe in..

(1.2x10^22 molecules air / breath) x (9.0x10^-23 molecules Caesar molecules / 1 molecule air) = 1.0 Caesar molecules per breath.