Half-Life

What is Half-Life?

The time required for a quantity to reduce to half of its initial value is known as the half-life. This term is usually used in nuclear physics to describe how fast the unstable atoms undergo radioactive decay and for how long the stable one survives. 

It characterizes any type of exponential or non-exponential decay. For your better understanding, let’s take an example; medical science mostly refers to the biological half-life of chemicals and drugs in the human body. The opposite of half-life is doubling time.

In 1907, Ernest Rutherford discovered the principle of the half-life period, which was shortened to half-life. He applied the principle of radioactive element’s half-life to the studies of age determination of rocks by calculating the decay period of radium to lead-206.

Half-life is the characteristic unit for the exponential decay equation and is constant over the lifetime of an exponentially decaying quantity. 

The time required for exactly half of the entities to decay is called a half-life. If there is just one radioactive atom and the half-life is only one second, then there will not be half of an atom left after one second. It is mostly defined in terms of probability, and the probability of decay of radioactive atoms within its half-life is 50%. After the half-life, the process changes randomly, so not only half of the atoms are left, but only approximate. However, if many identical atoms collapse, the law of large numbers suggests that it is a very good approximation to say that half of the atoms remain after half-life. 

What are the formulas of half-life in exponential decay?

The following three formula are used in an exponential decay:

N(t) = N0 (1/2)t/t1/2

N(t) = N0 e-t/T

N(t) = N0 e-λt

Where, 

• N0 is the initial quantity of the decaying substance, measured in grams, moles, number of atoms etc.
• N(t) is the quantity that remains constant and has not decayed after a time t.
• t1/2 is the half time of the quantity which is decaying.
• λ is a positive number known as the decay constant of the decaying quantity.
• τ is also a positive quantity which is known as the mean lifetime of the decaying quantity

All the three parameters t1/2, τ, λ  are related in the following way:

t1/2 = ln (2)/ λ = τ ln (2), here ln (2) is the natural logarithm of 2 (approx. 0.693)

What is the first-order reaction of half-life?

In this first-order reaction, the concentration of the reaction will continue to decrease as time progresses until it reaches zero and the length of and half-life becomes constant, being independent of the concentration.

The time for [A] to decrease from [A]0 to ½ [A]0 in the first-order reaction by the following equation:

kt1/2 = -ln (1/2 [A]0/ [A]0) = -ln ½ = ln (2)

In a primary reaction, the half-life of the reactants is independent of their initial concentration. Therefore, if the concentration of A is [A] at any stage of the reaction, it is reduced to 1/2 [A]0  after another interval of (ln 2)/k. Therefore, the half-life of the primary reaction is given as follows:

t1/2 = ln 2/k

The half-life of a first-order reaction depends on the reaction rate constant and is independent of its initial concentration.

Conclusion: 

Now that we’ve learned everything about half-life, we know that half-life is also very important in our daily lives. It shows that we humans tend to have the same qualities as everything else in this world collapses. Spending time researching something that changes from time to time can be a waste of time. So, we need to keep pace in order to catch up with where we are, which Half-life represents.