## What is Half-Life and its Formula?

Half-life is the time required for half of a quantity of a radioactive substance to undergo decay or transformation. It is a characteristic property of each radioactive isotope and is used to describe the rate of decay, providing a measure of the stability or persistence of a radioactive material. The half-life formula is an equation we use to calculate the rate of disintegration of unstable atomic nuclei which leads to the emission of alpha (α), beta (β), or gamma (γ) particles.

The half-life formula is written as **T _{1/2} = (0.693) / λ**, or

**T**, or

_{1/2}= ln(2) / λ**T**

_{1/2}= (Log_{e}

^{2}) / λWhere

**T _{1/2} = Half-life**

**λ = decay constant**

**Note:** We can use t_{1/2} or T_{1/2} to indicate the half-life of a radioactive element

Therefore, we can use the above formulae to solve half-life problems. We need to understand that the half-life of a radioactive element is the time taken for half the atoms of the element to decay. We can also define half-life as the time taken for a given mass of a radioactive substance to disintegrate to half its initial mass. The number of half-life formula are 3 and we can apply any of the formulae to solve a problem.

The relation of N and T_{1/2} is in a graph of N versus T_{1/2} below. The graph below describes a decay curve.

## Half-life Equations

We can use the following new equations to calculate the half-life of a radioactive element:

- Half-life,
**T**[ Where R = 2_{1/2}= t / (log_{2}R) = (t x log_{2}R) / logR^{n }= N_{1}/ N_{2}; and n = t / T_{1/2}] - The second formula we can use for the half-life is
**T**_{1/2}= (0.693) / λ - We also have another formula for calculating half-life which is
**T**_{1/2}= ln(2) / λ - The last formula for calculating half-life is
**T**_{1/2}= (Log_{e}^{2}) / λ - The formula for calculating the number of atoms that decay,
**N**_{d}= N_{1}– N_{2}= N_{1}((R-1) / R) = N_{2}(R – 1) - Fraction remaining of undecayed radioactive elements,
**f**_{r}= N_{1}/ N_{2}= 1 / R - The fraction of decayed atoms,
**f**_{d}= N_{d}/ N_{1}= (R – 1) / R

Where

T_{1/2} = Half-life of a radioactive element

t = time it takes a radioactive element to decay or disintegrate

n = number of half-lives

N_{1} = Initial mass or the initial number of atoms present/initial count rate.

And N_{2} = final mass of the final number of atoms remaining undecayed/final count rate.

N_{d} = Number of atoms or mass of atom that has decayed or disintegrated.

R = Disintegrating ratio

f_{r} = fraction of initial number of atoms remaining undecayed

We also have f_{d} = fraction of the initial number of atoms that have decayed.

**Note:**

From the above half-life formulae, the first formula is called **Zhepwo radioactive equation** while the remaining formulae are called the **Zhepwo derivative(s)**. Hence, we can be able to differentiate between the two groups of equations.

## Derivation of Half-Life Formula

Here is how to derive the formula:

Since the rate of disintegration is proportional to the number of atoms present at a given time, we can say that

-(dN/dt) **∝** N

or dN/dt = -λN

Where

λ = constant of proportionality which is referred to us as a **decay constant** of the element. We can write the above equation (dN/dt = -λN) as** λ = – 1/N (dN/dt)**.

Hence the formula for decay constant is **λ = – 1/N (dN/dt)**

After integrating the above equation, we will have

**N = N _{0}e^{-λt}**

Where **N _{0}** is the number of atoms present at a time t = 0 (i.e at the time when observations of decay were begun). N = the number of atoms present at time t.

We can now change **N = N _{0}/2** into

**N = N**to obtain the time required for half of the atoms to disintegrate (half-life)

_{0}e^{-λt}**N _{0} / 2 = N_{0}e^{-λt}**

And **N _{0}** will cancel each other from both sides to obtain

**1 / 2 = e ^{-λt1/2}**

We will take the natural or Naperian logarithm of both sides to get

**log _{e}(1/2) = -λt**

_{1/2}

We need to remember that log_{e}e^{n} = n

Therefore, from the left-hand side of the equation (**log _{e}(1/2) = -λt**

_{1/2}). We can see that

**log _{e}(1/2) = log_{e}1 – log_{e}2 = 0 – log_{e}2 = – log_{e}2 = – 0.693 **

Thus, **-λt**_{1/2} = **– 0.693**

and **t**_{1/2} = 0.693 / λ

## What is Half-Life?

The half-life of a radioactive element is the time it takes half of the atoms initially present in the element to disintegrate or decay.

## Knowledge of Logarithm for Calculating Half-life

The knowledge of the theory of logarithms will help us to understand how to calculate half-life. Here are logarithmic terms in a tabular form to help you understand the topic better.

R = 2^{n} | log_{2} R = n |

2 = 2^{1} | log_{2} 2 = 1 |

4 = 2^{2} | log_{2} 4 = 2 |

8 = 2^{3} | log_{2} 8 = 3 |

16 = 2^{4} | log_{2} 16 = 4 |

32 = 2^{5} | log_{2} 32 = 5 |

64 = 2^{6} | log_{2} 64 = 6 |

128 = 2^{7} | log_{2} 128 = 7 |

256 = 2^{8} | log_{2} 256 = 8 |

512 = 2^{9} | log_{2} 512 = 9 |

1024 = 2^{10} | log_{2} 1024 = 10 |

## What is the Formula for the Disintegration Ratio?

The disintegration ratio is a newly coined expression, it is NOT a new or additional concept in physics. It does not contradict any term or concept in radioactivity. Therefore, It is simply coined to name for an established relationship (N_{o} / N = 2^{n}). Thus, it is modification R = N_{1} / N_{2} = 2^{n} which is a simplified application in solving radioactive decay problems.

## Half-life Formula: Conventional Method of Calculating Half-Life

Assume that a radioactive element with a half-life of 5 seconds contain 192 atoms initially.

After the first 5 seconds (1 half-life), 96 atoms would decay and 96 atoms would remain.

In another 10 seconds (2 half-lives), 144 atoms would decay and 48 atoms would remain.

When we move to the next 15 seconds (3 half-lives), 168 atoms would decay and 24 atoms would remain.

After 20 seconds (4 half-lives), 180 atoms would decay and 12 atoms would remain.

In another 25 seconds (5 half-lives), 186 atoms would disintegrate and 6 atoms would remain.

## Half-Life Formula: How to Calculate Half-life in Physics

Here is a solved problem to help you understand how to apply half-life formula

### Problem

A radioactive element has a decay constant of 0.077 per second. Calculate its half-life.

#### Solution

The decay constant, λ = 0.077 s^{-1}

We will use the formula that says

T_{1/2} = (0.693) / λ = 0.693 / 0.077 = 9 s

**Therefore, the half-life of the radioactive element is 9 seconds.**

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