# ‘Month Of Equations’: What Does Half Life Really Mean?

### October 21: Half Life equation for the 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 EffectChandrasekhar LimitDebye 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 does it really means?

Previous in Series:What Does Relativistic Doppler Effect Really Mean?

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 a particular nucleus 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 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. Half life vs concentration graph for radioactive decay of an element

What’s important to note is the trend of decay. As time progresses, the decay decreases. The most stunning feature of 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.

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. “It 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 it is also a measure of the stability of an atom.Hence, it would not be wrong if I say that the half life of a radioactive element is indeed a very important and fundamental parameter to describe and understand its anatomy.

Now, to make this equation on the Half life of elements even more interesting, I would like to add a fun fact in this article before leaving. And, here it is , “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! Mind boggling ! Isn’t it ?

Next in Series:What Does The KdV Equation In Plasma Really Mean?

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