Further physics - The last breath of Caesar
Tong Shiu-sing & Hui Pak-ming
(Translation by Tong Shiu-sing & Janny Leung)

Introduction

Enrico Fermi
Enrico Fermi

Enrico Fermi (1901-1954) is a very famous modern physicist. Under his guidance, human have created the first atomic bomb and successfully controlled thermonuclear reactions, leading the world into an atomic era. His life was very interesting, and the most well-known events concerned his ability to make order of magnitude estimation in physical problems. It is said that when first atomic bomb in the world experimentally exploded, Fermi shed paper debris at a great distance from the centre of explosion, and carefully observed the drifting distance of the debris under air pressure; after a few minutes of mental arithmetic, he estimated that the power of explosion was about several ten thousand tons of TNT. The estimation corresponds to the correct order produced by precise instrument available a few weeks later, and this surprised his colleagues a lot.

Question

Fermi liked using simplified problems to stimulate his students to make order of estimation in physical problems. One of the interesting problems is: "When you take a single breath, how many molecules of gas you intake would have come from the dying breath of Caesar?" For the sake of simplicity, we can assume that the molecules which Caesar exhaled in his last breath have diffused evenly to the whole atmosphere, and these molecules were not absorbed by the ocean or plants for thousands of years. Although these are not valid assumptions, they can help us forget about the complexity of the real world, and to make elementary estimations in the simplest way. For the convenience of your calculation, we have given hints and information as follows:

Hints

We shall at first estimate the ratio of the gas volume exhaled in a single breath to the volume of the whole atmosphere (for simplicity, you may assume that all gases are evenly distributed in a layer which has a thickness of 50 km on the surface of the earth). Then you can estimate, in your single breath, how much volume of gas would have come from Caesar's last exhalation, and finally, with data on the density of the atmosphere and the average mass of an air molecule, you can estimate the no of molecules that the volume contains.

Assume that Radius of the earth R = 6,400 km Volume of gas in a single breath ~ 1 litre Thickness of the atmosphere ~ 50 km Mass of a proton mass of a neutron Density of the atmosphere on the surface of the earth =

Reference: The Fermi Solution, Hans Christian von Baeyer

Answer

First we shall estimate the volume of the earth's atmosphere V. Since the thickness of the atmosphere is much less than the radius of the earth, we have


The air intake in a single breath is about 1 litre, i.e., . Assuming that the gas exhaled from Caesar's last breath is evenly distributed in the atmosphere, we can deduce that, in a single breath, the volume of gas that one intake from Caesar's last breath is


Nitrogen is the major component of the atmosphere of the earth, and oxygen the second. Since the molecular masses of nitrogen and oxygen do not differ much, we will simply use nitrogen in our estimation. A nitrogen molecule has two atoms, each with 7 protons and 7 neutrons. Neglecting the mass of an electron, a nitrogen molecule would have a mass of



Hence in a single breath, the number of the molecules that comes Caesar's last exhalation would roughly be


i.e., when we take a single breath, we would have intake a single molecule which comes from Caesar's last exhalation.

Note

Some students may notice that I have only taken one significant figure throughout my calculations. Frankly speaking, I have not used a calculator in any of the calculations above. From your letters, I notice that some students have tried to perform very accurate calculations, and some of them have even made an effort to analyze the composition of air. In fact, these are not necessary, because under the many assumptions that we have used to simplify our problem (e.g. we have assumed that the density of air is uniform, which is obviously not true), it is virtually impossible to obtain an accurate answer. As a matter of fact, the spirit of the "Fermi problem" lies on training us to deal with a problem in which detailed information and calculation techniques are not available, and yet we can still make a very rough, but barely reliable estimation to get the right order of magnitude. Just think about it, when the first atomic bomb exploded in a trial, Fermi did not have a calculator or precise instrument in hand, his great charm came only from a few pieces of paper debris and the immense power of imagination!