How to model radioactive decay with Beer?
A Brief Introduction
The very first thing that comes to someone’s mind when they hear ‘radioactivity’ is danger or the Chernobyl disaster. However, radioactivity is much more than danger, and is quintessential to today’s life. It is essential to us, and can be found in the most simple devices such as a smoke detector to the most complex fields in science such as nuclear medicine.
Radioactive decay of an atom is the spontaneous transformation of an unstable atom’s nucleus into a lighter one and also includes the release of radiation in the form of alpha and beta minus particles and gamma rays. In some rare situations, some other particles such as beta plus particles and others are also released. Large nuclei have more protons stuffed in the nucleus, and because of this the protons repel more. Near the center, the protons are held together by the strong nuclear force, however, as distance increases the magnitude of this force decreases, and radioactive decay happens. The binding energy, i.e the amount of energy required to break apart an atom, is less in these large nuclei, hence radioactive decay is more evident.
The graph below shows the binding energy per nucleon for different elements. As we can see, Uranium-238 is on the far right end of the graph, and hence when it decays radioactively, it releases immense energy. With today’s technology, it is not possible to harness the energy released by the transformation of hydrogen to helium, hence for now, it is only good for making weapons such as the hydrogen bomb. However, radioactivity is used in many scientific and medical fields as well. Gamma radiation can easily penetrate through human tissue, it has many medical applications such as X-rays and CT scans. Nuclear medicine is a new field in science, which builds upon the principles of radioactivity and nuclear physics. PET scans and SPECT scans are some examples.
Nucleon number vs BE per nucleon graph
It is extremely difficult to model radioactive decay because sometimes it can be extremely slow (some elements have half-life of more than 1000 years), whereas sometimes it can be extremely dangerous as well. For example, when Radium decays to Radon releasing an alpha particles, 4.84 MeV of energy is released. Only a few eV are required to ionize air and many other substances, this can be harmful. In addition to this, the particles in nuclei (nucleons) can be modelled by a wave function, giving the probability of them being in or out of the nucleus. Hence, this makes radioactive decay an extremely random process as well. In simple words, we cannot predict the moment in time when a nucleus will decay.
The law of radioactivity says that dN/dt = -λN, i.e the no. nuclei that will decay per unit time is proportional to the remaining nuclei. Beer foam does something similar to this phenomenon. Bubbles of beer foam burst randomly just like the random decay of radioactive nuclei, and the rate of decay of bubbles is almost proportional to the number of bubbles. So, can beer foam model radioactive decay?
A perfect beer foam (beer head)
Some equations you can graph to see the results
Law of radioactivity says dN/dt = -λN
dN/N = -λdt
integrating both sides, we get N = Ae^(-λt)
where N is the no. bubbles at time t, A is the initial no. bubbles and λ is the decay constant.
However, it is not possible to precisely count the no. bubbles per unit time, but as the bubbles burst the level of the beer goes up, so we are assuming that when bubbles burst beer is made. So bubbles burst ∝ height of beer.
this gives, H = he^(-λt)
taking, ln of both sides, we get lnH = lnh -λt
this would give a straight line with the slope = -λ
An exponential decay curve for N = Ae^(-λt) indicates to the fact that beer can indeed model radioactive decay
If beer accurately models radioactive decay, then it can be used to find the half life of this sample of beer. If we graph dH/dt vs t, where dt is 10s, the value corresponding to halfway through the y-axis value would be our half-life value.Multiple pairs of points can be used to average the half-life value, and also confirm if the curve is an exponential decay curve or not. The graph below gives an example of the same
An Extremely Simple but Useful Derivation
A formula for half-life can also be derived as well.
Half-life is when half of the substance has decayed, hence H=h/2
h/2 = he-λt
h cancel out, and ln of both sides gives-ln1/2 = λt
this gives ln2/λ = t
half life (T1/2) = ln2/λ
An Interesting Use of Beer
If beer can model radioactivity, then it might have some very interesting properties as well. It has been seen that beer can help to stop tritium contamination. Tritium, an isotope of hydrogen with 2 extra neutrons, if enters in the bloodstream can do a lot of severe damage by changing the concentrations of fluids as well as altering DNA and causing cancer. Half-life of tritium in human body is about 7-9 days. Beer has proved to be a substance which can reduce the tritium concentration in human blood.
Sample graph for height of beer vs ln(height of beer)
Sample graph for activity vs time with T1/2 calculations
The Thunderbolt Project
If you could spend one day with one of these, who would it be?
An elusive particle which is the antiparticle of itself is seen after 80 years
Link to above mentioned discovery