# Half Life Calculator

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## How to use this calculator

Our half-life calculator gives accurate answers to the calculations involving various parameters pertaining radioactive decay. It also displays the formula used in the computation and steps followed in arriving at the answer. You need to follow key steps when using this calculator

i) Steps 1 will answer the question, what do you want? Here you click drop down arrow and select the parameter or the quantity you want the calculator to compute

ii)  Step 2, identify the parameters or the quantities you already have, you can select from the many in the next dropdown arrow

iii) Key in the calculator the values of the parameters you already have

iv) Click calculate and the answer will be displayed just below the “calculate”  button

v)  View the steps followed and the formula used by simply clicking “click to see steps below” just below the answer.

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## What is half-life?

Radioactive materials are generally unstable and therefore undergo spontaneous disintegration (radioactive decay) to form a stable nuclide, radiations and energy.

Half-life (denoted as ${t}_{1/2}$) is the time it takes for a given radioactive substance to decay to half its original/initial amount. The amount in this case refers to mass or number of atoms of a radioactive sample. In terms of activity, half-life can be defined as the time it takes for the activity (disintegration per second) of an unstable nuclide to decrease by one-half as it undergoes radioactive decay. Radioactive decay yields stable nuclides, particles (alpha, beta, gamma rays) and energy.

Change in the mass of nuclide, number of atoms as well as the time it takes for the decay to occur are useful in the calculation of half-life. Radioactive nuclides have different (unique) half-lives that are independent of concentration and environmental factors such as change in temperature and pressure. The half-lives of these radioactive isotopes range from fractions of a second to billions of years. For example, Nobelium-254 has a half-life of 3 seconds while uranium-238 has a very long half-life of 4.5 billion years.

Half-live has been used to describe other decays which may or not be exponential. In medical sciences for example, half-life describes the change in the concentration of a drug in the body. In the medical field, half-life is classified as a pharmacokinetic parameter that refers to the time it takes for the concentration of a drug in the body to decrease by one-half.

Half-life finds its application in dating or estimating ages of archaeological materials. Carbon-14 which has a half-life of 5730 years has widely been used to estimate ages of materials of up to 50,000 years old.

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## What is the half-life formula equation?

By denoting the mass radioactive nuclide remaining or the number of atoms/nuclei remaining after time, t, as ${N}_{t}$ and the initial mass/number of atoms or nuclei as ${N}_{o}$, the equation connecting these quantities and half-life (${t}_{1/2}$) is;

${N}_{t}$${\left(\frac{1}{2}\right)}^{\frac{t}{{t}_{1/2}}}$${N}_{o}$                                                                                      1

This equation can be rearranged such ${t}_{1/2}$ is given in terms of t, ${N}_{o}$and ${N}_{t}$;

Half-life (${t}_{1/2}$) =$\frac{2\mathrm{log}2}{\mathrm{log}\left(\frac{{N}_{o}}{{N}_{t}}\right)}$                                                                                       2

Example 1

Given the 800 mg of a radioactive sample decayed to 50 mg in 20 minutes, what is the half-life of this nuclide?

Solution

Clearly, ${N}_{o}$= 800 mg, ${N}_{t}$= 50 mg, and t = 20 minutes

${t}_{1/2}$$\frac{2\mathrm{log}2}{\mathrm{log}\left(\frac{800}{50}\right)}$

= 5 minutes

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## How to calculate half-life using the calculator

There are two scenarios the calculator can calculate the half-life, the first scenario is the when you have ${N}_{t}$, ${N}_{o}$ and t as in the example above. You will be required to key in the initial mass/number of atoms ${N}_{o}$, the remaining mass/ number of atoms/nuclei after decay ${N}_{t}$ and total time that elapsed (t). The calculator will automatically compute half-life (${t}_{1/2}$) using equation 2 above.

The second scenario is when you have the percentage of the radioactive material remaining and the total time t (see example 2), we can get ${t}_{1/2}$ using equation 2 which can as well be written as;

${t}_{1/2}$$\frac{-2\mathrm{log}2}{\mathrm{log}\left(\frac{{N}_{t}}{{N}_{o}}\right)}$

Example 2

Calculate the half-life of a radioactive nuclide that takes 689.28 years to decay to 92.0% of its original mass.

${t}_{1/2}$$\frac{-689.28\mathrm{log}2}{\mathrm{log}\frac{92}{100}}$

= 5730 years

In order for the calculator to give you the half-life from such a scenario, you just need to input the total time that elapsed, t=689.28 years, and the percentage remaining, in this case, 92. (Hint: The calculator in the steps expresses the percentage remaining as a fraction, 100 becomes the initial quantity before decay, 92 becomes amount remaining after time t)

The calculator automatically gives decay constant $\lambda$ and mean-lifetime τ once you already have half-life.

4

## What is decay constant?

Decay constant (λ) is the probability per unit time that a given nucleus of a radioactive nuclide will decay. Decay constants are independent of temperature, pressure and the strengths of bonds in the radioactive nuclide. The units of λs are ${s}^{-1}$. Decay constant and half-life are inversely proportional, the shorter the ${t}_{1/2}$ the larger the decay constant (λ) (equation 3).

Decay constant (λ) and half-life (${t}_{1/2}$) are connected by the equation:

λ=                                                                                         3

= $\frac{0.693}{{t}_{1/2}}$

Decay constant can also be expressed in terms of ${N}_{o}$ and ${N}_{t}$ using the exponential decay equation;

${N}_{t}$${N}_{o}$ ${e}^{-\lambda t}$                                                                                        4

Where ${N}_{t}$ is the amount remaining after time t, is the initial amount/mass ${N}_{o}$, λ is the decay constant. Rearranging equation 4 to make λ the subject of the formula, we obtain the equation;

Decay constant (λ) = $\frac{1}{t}$ln ($\frac{{N}_{o}}{{N}_{t}}$)                                                                                   5

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## How to calculate decay constant in the calculator

There are three ways the calculator can compute the decay constant.

i) If you have half-life, decay constant is computed using equation 3, you just need to key in the half-life and automatically get the decay constant.

ii)  If you have mean lifetime τ, the calculator will give you Decay constant (λ) using the equation

Decay constant (λ)  = $\frac{1}{\tau }$                                                                        6

iii) If you have total time t, ${N}_{o}$and ${N}_{t}$, the calculator will give you λ using equation 5. You can always view in the calculator the steps as well as the formula used to compute the missing quantity.

6

## Mean lifetime (τ) and mean life time formula

The decay of a radioactive nuclide is ordinarily expressed in terms of half-life, decay constant and mean lifetime. Mean lifetime is the average lifetime of all the nuclei of a particular unstable atomic species. It can also be described as the average amount of time before an unstable nucleus undergoes radioactive decay. It is a statistically derived time that a radioactive nuclide can exist before it decays to form stable nuclides. Mean lifetime can be related to decay constant and half-life using the following formulae

τ = $\frac{1}{\lambda }$                                                                                            7

and

τ = $\frac{1}{{t}_{1/2}}$                                                                                           8

And from equation 3 and 7, we can deduce that;

τ =                                                                                             9

Since decay constant and mean life time are inversely proportional to (reciprocal of) each other, we can write equation 4 in terms of τ;

${N}_{t}$ = ${N}_{o}$ ${e}^{-\frac{t}{\tau }}$                                                                                    10

We can rearrange eqn 10 to get τ in terms of ${N}_{o}$,${N}_{t}$ and t.

7

## How to calculate mean-life in the calculator

The calculator can give mean lifetime using equations 7, 9 and 11. The calculator will help you to calculate τ if you have the following options

i) If you have λ, then simply key in its value in the calculator and click calculate. The correct value of τ will be displayed; Calculation steps using of equation 6 will be displayed.

ii) If you have half-life, key in the value in the calculator and click calculate, the calculator will give the answer by utilizing equation 9. The equation will be displayed

iii) If you have ${N}_{o}$, ${N}_{t}$ and total time t, the calculator will give you τ and will display steps using equation 11.

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