Enrico Fermi (1901-1954) was a famous physicist who liked to pose what are now k
ID: 1065220 • Letter: E
Question
Enrico Fermi (1901-1954) was a famous physicist who liked to pose what are now known as Fermi problems in which several assumptions are made in order to make a seemingly impossible estimate. Probably the most famous example is the estimate of the number of piano tuners in Chicago using the approximate population of the city and assumptions about how many households have pianos, how often pianos need tuning, and how many hours a given tuner works in a year. Another famous example of a Fermi problem is "Caesars 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. Calculate the total volume of the atmosphere Number In Calculate the total number of gas molecules in the atmosphere. Number molecules continued below. Calculate the number of gas molecules in Caesar's last breath (37 °C and 1.0 atm)Explanation / 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.
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