Half-life decay (t1⁄2) is the time required for a quantity to reduce to half of its initial value. This is actually described as a probability, as illustrated below. In nuclear physics this demonstrates how quickly unstable isotopes undergo radioactive decay. It also shows the nature of exponential or non-exponential decay. This model is extensively used in medical, physics and chemistry to predict lifetimes of substances, radioactive elements and concentrations of observed liquids.
What is Half-life (radioactive decay)?
The half-life is the time taken for an entity to reduce to half of its original value and is used to illustrate how unstable atoms undergo radioactive decay. It also describes exponential or non-exponential decay, such as the biological half-life of pharmaceutical drugs in the body.
Rutherford’s discovery of the concept in 1907, namely half-life period, was subsequently shortened to half-life in the 1950s. He successfully used the principle of a radioactive element’s e.g. Radium’s half-life to determine the age and decay to liberate Lead 206. Half-life is constant over the lifetime of an exponentially decaying quantity.
Half-life of example materials and elements
uranium 238 – 4,460,000,000 yrs
uranium 235 – 713,000,000 yrs
carbon 14 – 5,730 yrs
cobalt 60 – 5.27 yrs
silver 94 – 0.42 seconds
Half-life formula and related equations
The half-life is actually defined as a probability. Specifically, it is the time for exactly half of the entities to decay, on average. The probability of a radioactive atom decaying within its half-life is 50%.
In a random simulation of many identical atoms undergoing radioactive decay, after one half-life there are not exactly half of the atoms left, linked to the random variation in the process. However, with a large sample, the law of large numbers suggests that it is a very good approximation to say that half of the atoms will remain.
In a chemical reaction, the half-life is related to the time it takes for the concentration to fall to half of its initial value. In a first-order reaction the half-life of the reactant is ln(2)/λ, where λ is the reaction rate constant. An exponential decay can be described below:
where N0 is the initial quantity of the substance that will decay, N(t) is the quantity and has not yet decayed after a time t, t1⁄2 is the half-life of the decaying quantity, τ is the mean lifetime of the decaying quantity, λ is the decay constant of the decaying quantity.
The three parameters t1⁄2, τ, and λ are all directly related in the following way:
where ln(2) is the natural logarithm of 2 (approximately 0.693)
Radioactive decay by +2 processes
Some quantities decay by two exponential-decay processes simultaneously and the actual half-life T1⁄2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:
For +3 the analogous formula is:
What happens to the mass in half-life decay?
As an example, if the half-life is 10 yrs, its highly probable that half of the radioactive atoms will decay, however, there will not be a significant change in mass.
Half-life usage within Radiotherapy and Biology
An example in Radiotherapy, the isotope cobalt-60 has a half-life of 5.26 years. After this time period, if the original quantity contained 16 g, it would now contain only 8 g of cobalt-60. After 10.52 years, the sample would be 4 g etc. The newly formed nickel-60 nuclei remain with the undecayed cobalt-60.
A biological half-life is the time for a substance (drug, radioactive nuclide) to lose one-half of its pharmacologic activity. This may also describe the time that it takes for the concentration in blood plasma to reach one-half of its steady-state. The understanding of biological and plasma half-lives can be complicated, linked to factors including accumulation in tissues and receptor interactions.
A radioactive isotope decays according to first order kinetics, where the rate constant is a fixed number. However, the elimination of a substance from an organism follows more complex chemical kinetics. In epidemiology, the concept of half-life can refer to the length of time for the number of incident cases in a disease outbreak to drop by half.
An example of a drug administrated to the body:
A 1000 mg dose of an intravenous drug with a half-life of 30 minutes as an example.
- 30 minutes after the drug is injected, 500 mg of the drug is abundant in the body.
- 60 mins, 250 mg evident
- 120 mins, 125 mg evident
- 240 mins, 62.5 mg evident
Non-exponential half-life decay
In a decay that isn’t exponential, the half-life will change during the decay. It is unusual to describe half-life in this scenario, however, it can be used to describe the decay in terms of its first half-life, second half-life and so on. The first half-life is defined as decay to 50%, the second half-life is from 50% to 25% etc.
Please see a link to a half-life calculator. Note this is a third party website and we are not responsible for the content on it.
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