What is half-life of radioactive decaying material? | How it is mathematically related to decay constant and initial concentration of substance?

 What is half-life of radioactive decaying material?

The half-life of a radioactive substance is the amount of time it takes for half of the original amount of the substance to decay. It is a characteristic property of a particular radioactive material and is independent of the amount of the substance present.

For example, if you have 100 grams of a radioactive material with a half-life of 10 years, after 10 years you would have 50 grams remaining. After another 10 years (a total of 20 years), you would have 25 grams remaining. After 30 years, you would have 12.5 grams remaining, and so on.

The half-life of a radioactive substance can be used to calculate how much of the substance will remain after a certain amount of time has passed. This is important in fields such as nuclear physics, radiology, and medicine, where radioactive substances are used for various purposes.

How it is mathematically related to decay constant and initial concentration of substance?

The relationship between the half-life of a radioactive substance, the decay constant, and the initial concentration of the substance can be expressed mathematically using the radioactive decay law, which states that the rate of decay of a radioactive substance is proportional to the amount of the substance present.

The radioactive decay law can be written as:

dN/dt = -λN

where N is the number of radioactive nuclei present, t is the time, λ is the decay constant, and dN/dt is the rate of decay.

Integrating this equation gives:

ln(N/N0) = -λt

where N0 is the initial number of radioactive nuclei present.

Solving for t when N/N0 = 1/2 gives:

t1/2 = ln(2)/λ

where t1/2 is the half-life of the substance.

The initial concentration of the substance can be related to N0 by:

N0 = NA * n

where NA is Avogadro’s number and n is the number of moles of the substance present.

Therefore, the relationship between the half-life, decay constant, and initial concentration of a radioactive substance can be expressed mathematically as:

t1/2 = ln(2)/(λ * NA * n)

Half-life of radioactive material