Half Life Formula

Half life is a particular phenomenon that takes place every day in various chemical reactions as well as nuclear reactions. One can get an idea about half life by imagining a situation in which an individual watches a movie in a theatre. This individual is eating from a tub of popcorn. After about 15 minutes, half the popcorn is over. The rest of the popcorn continues until the rest of the movie. Most noteworthy, this shows that the rate of popcorn eating was not at a steady pace and that the half-life of popcorn is of 15 minutes. Learn the half life formula here.

What is Half Life?

Half-life refers to the amount of time it takes for half of a particular sample to react. Furthermore, it refers to the time that a particular quantity requires to reduce its initial value to half.

This concept is quite common in nuclear physics and it describes how quickly atoms would undergo radioactive decay. Moreover, it could also mean how long atom would survive radioactive decay. Also, the half-life can facilitate in characterizing any type of decay whether exponential or non-exponential. A good example can be that the medical sciences refer to the half-life of drugs in the human body which of biological nature.

Half Life Formula

One can describe exponential decay by any of the three formulas

N(t) = N0\(\frac{1^{\frac{t}{t^{\frac{1}{2}}}}}{2}\)

N(t) = N0\(e^{-\frac{t}{\tau}}\)

N(t) = N0 \(e^{-\lambda t}\)

Where,

N0 refers to the initial quantity of the substance that will decay. The measurement of this quantity may take place in grams, moles, number of atoms, etc.

N(t) is the quantity that still remains and its decay has not taken place after a time t,

\(t\frac{1}{2}\) represents the half-life of the decaying quantity,

τ is a positive number and is the mean lifetime of the decaying quantity,

λ is a positive number and is certainly the decay constant of the decaying quantity.

There is a direct relation between the three parameters, \(t\frac{1}{2}\), τ, and λ which is follows:

\(t\frac{1}{2}\)= \(\frac{In(2))}{\lambda }\) = \(\tau In(2))\)

where ln(2) happens to be the natural logarithm of 2 (approximately 0.693).

Half Life Formula Derivation

First of all, we start from the exponential decay law which is as follows:

N(t) = N0\(e^{-\lambda t}\)

Furthermore, one must set t = \(T\frac{1}{2}\) and N(\(T\frac{1}{2}\)) = ½ N0.

N(\(T\frac{1}{2}\) ) = \(\frac{1}{2}N0\) = N0\( e^{-\lambda T\frac{1}{2}}\)

Now divide through by N0 and take the logarithm,

½ = \(e^{-\lambda t}\), this leads to In(1/2) = \(-\lambda T^{\frac{1}{2}}\)

Now solving for \(T\frac{1}{2}\),

\(T\frac{1}{2}\) = \(-\frac{1}{\lambda }In(\frac{1}{2})\)

Following the laws of logarithms, one can take the “-1” up s an exponent of the logarithm. Finally, this gives

\(T\frac{1}{2}\) = \(\frac{In(2))}{\lambda }\)

Solved Examples on Half Life Formula

Q1. Calculate the half-life of a radioactive substance whose disintegration constant happens to be 0.002 1/years?

Answer:  the quantities available here are,
λ = 0.002 1/years
Consequently, the half life equation becomes:
\(t\frac{1}{2}\) = 0.693/ λ
\(t\frac{1}{2}\) = 0.693/0.002 = 346.5

Hence, the half-life of this particular radioactive substance is 346.5 years.