'Month of Equations': What Does Half Life Really Mean?

                                       October 21: Half Life

21 Half Life

Meaning of Equation:

The half life period is the time required for a quantity to reduce to half its initial value. Within a half life, the probability of decaying is 50%.

Today's equation is easy. After the previous equations on Relativistic Doppler Effect, Chandrasekhar Limit, Debye Length and Stefan's Law, I decided to include at least one equation of nuclear physics in my series and chose half life, which is derived from the natural law of radioactivity. Let us now try to understand what half life really means?

In physics, half life applies to the field of radioactivity. If the nucleus of an atom is too heavy, it disintegrates into smaller, stable nuclei by emitting radiation. The radiation is of three types: alpha, beta and gamma. But that's not important to today's equation. There is only one important point we must know: Radioactivity is a spontaneous phenomenon. This means that we don't know when will a particular nuclei will disintegrate. All we can do is to study that nuclei for some time and know the "trend" of its decay. By trend, I mean that how much time it takes to decay into some fraction of its initial quantity. The equation of radioactive decay tells us that this process is an exponentially decaying one. What does that mean? This means that the quantity of the decaying matter will fall of exponentially with time as shown below in the graph.


This curve shows the decay of a radioactive atom (Na-24 in this case). What's important to note is the trend of decay. As time progresses, the decay decreases. The most stunning feature of an exponential decay is that it never reaches zero. Mathematically speaking, the curve will only touch the x axis at infinity. It asymptotically approaches the axis. This is the reason why traces of radioactivity are still found at the bombing sites of Hiroshima and Nagasaki.

Now half life is the time period required for a radioactive quantity to decay to half of its initial value. Suppose you have 10 g of a radioactive element say X. Let us assume its half life to be 2 minutes. Then after 2 minutes, you'll be left with 5 grams of X. After 4 minutes, 2.5 grams of X, after 6 minutes 1.25 grams of X and so on and so forth. Did you notice something? This process will never stop. You keep on dividing but you'll never reach 0. This explains the shape of the curve above.

Also See: What Does The Schrodinger Wave Equation Really Mean?

One last important concept of today's equation is that half life is better explained in terms of probability. Suppose I have one radioactive atom whose half life is 2 minutes. It does not mean that after 2 minutes, half of the atom will be left. Thus half life is defined in terms of probability. "Half life is the time required for exactly half of the entities to decay on average." In other words, the probability of a radioactive atom decaying within its half life is 50%. Thus half life is also a measure of the stability of an atom.

Before leaving, I will add a fun fact. Tellurium-128 is the most stable radioactive isotope. Its half life is of the order of 10^24 years. That is 1,000 trillion times the age of the universe!

Previous in series: What Does The Relativistic Doppler Effect Really Mean?