Today Alan Mason overhears a conversation all about half life.

Curio: What exactly is a “half life?”

Alfredo: It’s a general term, to describe almost anything which declines regularly over a period of time.

Curio: Anything?

Alfredo: Anything that shows regular decline, like the charge in a capacitor, or the mass of an evaporating liquid.

Curio: Isn’t it something to do with radioactivity?

Alfredo: That is the most well-known and most widespread use of the term “half-life” these days.

Curio: So what is it exactly?

Alfredo: Formally, it is, “The time in which the amount of a radioactive nuclide decays to half its original value.”

(References A)

Curio: Radioactive nuclide?

Alfredo: A radioactive isotope of an element. Let’s take a well-known example.

Carbon (**2**) is a common element whose Atomic Number is 6, and whose Atomic Weight is 12.011. As you know, the Atomic Number tells us there are six protons in the nucleus, which determine its chemical properties. The nucleus also contains six neutrons; hence the Atomic Weight is 12.

Curio: Didn’t you say the Atomic Weight was 12.011?

Alfredo: Correct, and it is that odd 0.011 which is important. The early workers on atomic weights were puzzled that the results did not come out to nice round numbers, which led to problems in trying to arrange the known elements logically.

It was Dmitri Mendeleev (**3**), in 1869, who saw that the chemical properties of elements were of over-riding importance, not the atomic weights. He was criticised, at the time, because his approach was purely qualitative, and not mathematical as the atomic weights were, but in 1914 Henry Moseley (**4**) was able to measure atomic numbers directly. He proved that Mendeleev was correct.

Now back to that strange, 0.011. It tells us that any sample of pure carbon must contain a small proportion of heavier isotopes. If it was all Carbon 12, as in Figure 2, its Atomic Weight would be exactly 12 as the early nineteenth century researchers hoped it would be.

We now know that there are three naturally-occurring isotopes of carbon, known as “Carbon 12”, “Carbon 13” (**5**) and “Carbon 14” (**6**). While Carbon 12 and Carbon 13 are stable, and do not alter over long periods of time, Carbon 14 is unstable. A Carbon 14 atom transmutes into a Nitrogen 14 atom. At any moment in time, a proportion of the millions of C14 atoms changes into N14 atoms, while the vast majority remain unchanged. Why some change, and some do not, is one of the mysteries of physics. In a mass of carbon the amount of C14 slowly decreases over many centuries. Its half –life is 5, 730 years.

Curio: Why choose the “half-life”; why not the whole life?

Alfredo: The process is one of exponential decline; the graph drops away steeply at first and then falls much more gently. Mathematically, we say the graph becomes nearly asymptotic as it runs close the x-axis, but takes a long time to meet it. To calculate the full life would give us needlessly large figures. The half-life is much more convenient as a figure and is relatively easy to measure and calculate. Remember it is essentially a statistical concept; it deals with the properties of a large population of atoms.

Curio: I have friend (**8**) who says that the term half life can be applied to an atom or a particle.

Alfredo: He’s wrong. Just think about it. A single atom of C14 is either intact or it is not, and is then changed to N14;

there is no halfway house. “Half life” is applied to POPULATIONS, not to individual atoms. Take a parallel illustration. If the census tells us that 55% of the 60 million population of Britain is female, will 55% of Birmingham, with a population of over a million also be female? Yes. The village of Newham has a population of only 300 people, so what percentage will be women? It will probably be about 55% plus or minus 10%. Now take a single individual, Charlie Brown,

is he 55% female? The idea is nonsense.

Curio: I see, Sir, you argue the *reductio ad absurdum*.

Alfredo: Exactly.

Curio: Can we follow the half-life process more closely?

Alfredo: Yes, I would like to do this by returning to your interesting question, “Why not full life?”

Let us assume that we have carbon in the form or carbon dioxide gas at STP (standard temperature and pressure, that is, 0 degrees C and 760 mm mercury). Following, there are three key pieces of data.

Three Important Facts

(i) A mole of this gas, that is the gram-molecular weight, or 44 grams, contains 6.022 X 10^{23} molecules, the Avogadro number. (References C)

(ii) The ratio of C14 to C12 is 1.5: 10^{12} (References D)

(iii) The half –life of C14 is 5, 730 years (References D)

How Many C14 Molecules in a Mole?

The curiously expressed ratio 1.5: 10^{12} can be better expressed as 1: 2/3 X 10^{20} = 1: 6.667 X 10^{19}

The number of C14 molecules in a mole of CO_{2} will be 6.022 X 10^{23}/6.667 X 10^{19} = 6.022/6.667 X 10^{23}/10^{19} = 0.9033 X 10^{4} or 9.033 X 10^{3}. In plain English, there are 9, 033 molecules of C14O_{2} in a mole of CO_{2}.

Calculating a “Full Life”

It will take 5,730 years for the number of C14O_{2} molecules to drop to 4,500.

Table 1: Decay of C14

Half life periods |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |

Number of molecules |
9033 |
4516 |
2258 |
1129 |
564 |
282 |
141 |
71 |
35 |
18 |
9 |
4 |
2 |
1 |

Thus in 13 steps the number of unstable carbon atoms will drop to one, which could drop to zero at any moment, or hang on for another millennium.

The time period for this change, or “Full Life”, is 5,730 X 13 years = 75, 490 years.

Curio: The table is very revealing. I can see that when you have an odd number like 35 at stage 8 you have to decide on whether stage 9 has 17 or 18. The more stages you have, the greater this inaccuracy. I can see why the half life is really the most satisfactory measure.

Alfredo: Yes, when you are dealing with a statistical parameter, the larger the population the better.

Curio: Isn’t Carbon 14 used to date fossil material?

Alfredo: Yes, but that is a longer story.

Curio: Shall we go and have a glass of Marsala at the “Albergo di Sicilia”?

Alfredo: A great idea.

(On a personal note, in the 1950s, when I was a teenager living in London, I tried to get hold of some Marsala wine. At this remove in time, I can’t remember what promoted this curiosity; perhaps I had read about it somewhere. The local off-licences (shops selling bottled beers, wines and spirits) had never heard of Marsala, so I decided to try Soho.

I often went there on a Saturday morning, to visit the bookshops. It was rather different from the way it is today; a much poorer and more workaday district, full of small businesses. There were lots of nice ladies there, who said, “Hello” to me in a friendly fashion. Enquiring in a shop about Marsala wine, I was directed to, “Parmeejarnofeeleeo” in Old Compton Street.

As I had the street name I was sure to find this mysterious address. It turned out to be “Parmigiano Figlio”, (Parmigiano Brothers), a wine-importing business. I bought a bottle of Marsala, which is a fortified Sicilian wine, like a sweet sherry, ideal for an aperitif before a meal, as Curio and Alfredo probably intended.)

REFERENCES

A. Dictionary of Physics, edited by Valerie Pitt, Penguin, 1979

B. “Michelangelo” by Gilles Néret, Taschen, 1998

C. Dictionary of Science, edited by E V Uvarov, and D R Chapman, Penguin, 1979

D. Wikipedia article, “Radiocarbon dating”

E. Figures 7 and 8 are details from “The Hemicycle of the Ecole des Beaux-Arts”, a mural by the French painter, Paul Delaroche, (1797 – 1856)

ILLUSTRATIONS

1. “David” a detail from a statue by Michelangelo Buonarroti (References B)

2. The Carbon 12 atom (Author)

3. Dmitri Mendeleev, chemist 1834 – 1907 (Google image)

4. Henry Moseley, chemist, 1887 – 1915 (Google image)

5. The Carbon 13 atom (Author)

6. The Carbon 14 atom (Author)

7. Alfredo and Curio in conversation (detail from Ref E)

8. A heated debate (detail from Ref E)

9. A glass of Marsala (Google image)

10. A street of cafes and restaurants in the town of Marsala, Sicily (Google image)

Exactly! QED. So what’s wrong with Hydronium having a half life?

Hydronium has an incredibly short half life, the proton jumping from one water molecule to another. Not sure if Marsala has a half life though?

I think Alfredo must be a chemist and not a physicist, on a quick look around the internet I have found many references to particle half-life or decay:

Wiki:

While a free neutron has a half life of about 10.2 min, most neutrons within nuclei are stable. According to the nuclear shell model, the protons and neutrons of a nuclide are a quantum mechanical system organized into discrete energy levels with unique quantum numbers.

Hyperphysics:

Along with protons, neutrons make up the nucleus, held together by the strong force. The neutron is a baryon and is considered to be composed of two down quarks and one up quark. A free neutron will decay with a half-life of about 10.3 minutes but it is stable if combined into a nucleus.

So perhaps not as absurdum as first thought!

BTW – the half life of Marsala is a great deal less once the cork has been removed.

This essay gives a clear definition of what “half-life” means with reference to radio-active decay. The critics need to give me a definition of what “half life” means with reference to a single particle. I wait with expectation.

Like Alfredo, I am no physicist but this seems to be a good definition of the half life of a Neutron:

‘Outside the nucleus, free neutrons are unstable and have a mean lifetime of 881.5±1.5 s (about 14 minutes, 42 seconds); therefore the half-life for this process (which differs from the mean lifetime by a factor of ln(2) = 0.693) is 611.0±1.0 s (about 10 minutes, 11 seconds). Beta decay of the neutron, described above, can be denoted by the radioactive decay:

n^0 → p+ + e− + νe-

where p+, e−, and νe- denote the proton, electron and electron anti-neutrino, respectively.’

Alfy, why do you keep banging on about particles? Hydronium is not a particle, its a molecule. So I ask again, whats wrong with Hydronium having a half life?