Number of halflives elapsed 
Fraction remaining 
Percentage remaining 


0  ^{1}/_{1}  100  
1  ^{1}/_{2}  50  
2  ^{1}/_{4}  25  
3  ^{1}/_{8}  12  .5 
4  ^{1}/_{16}  6  .25 
5  ^{1}/_{32}  3  .125 
6  ^{1}/_{64}  1  .563 
7  ^{1}/_{128}  0  .781 
...  ...  ...  
n  ^{1}/_{2n}  100/(2^{n}) 
Halflife (symbol t_{1/2}) is the time required for a quantity to reduce to half its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay. The term is also used more generally to characterize any type of exponential or nonexponential decay. For example, the medical sciences refer to the biological halflife of drugs and other chemicals in the human body. The converse of halflife is doubling time.
The original term, halflife period, dating to Ernest Rutherford's discovery of the principle in 1907, was shortened to halflife in the early 1950s.^{[1]} Rutherford applied the principle of a radioactive element's halflife to studies of age determination of rocks by measuring the decay period of radium to lead206.
Halflife is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic unit for the exponential decay equation. The accompanying table shows the reduction of a quantity as a function of the number of halflives elapsed.
A halflife usually describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition that states "halflife is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom, and its halflife is one second, there will not be "half of an atom" left after one second.
Instead, the halflife is defined in terms of probability: "Halflife is the time required for exactly half of the entities to decay on average". In other words, the probability of a radioactive atom decaying within its halflife is 50%.
For example, the image on the right is a simulation of many identical atoms undergoing radioactive decay. Note that after one halflife there are not exactly onehalf of the atoms remaining, only approximately, because of the random variation in the process. Nevertheless, when there are many identical atoms decaying (right boxes), the law of large numbers suggests that it is a very good approximation to say that half of the atoms remain after one halflife.
There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.^{[2]}^{[3]}^{[4]}
An exponential decay can be described by any of the following three equivalent formulas:
where
The three parameters t_{1/2}, ?, and ? are all directly related in the following way:
where ln(2) is the natural logarithm of 2 (approximately 0.693).
By plugging in and manipulating these relationships, we get all of the following equivalent descriptions of exponential decay, in terms of the halflife:
Regardless of how it's written, we can plug into the formula to get
Some quantities decay by two exponentialdecay processes simultaneously. In this case, the actual halflife T_{1/2} can be related to the halflives t_{1} and t_{2} that the quantity would have if each of the decay processes acted in isolation:
For three or more processes, the analogous formula is:
For a proof of these formulas, see Exponential decay § Decay by two or more processes.
There is a halflife describing any exponentialdecay process. For example:
The half life of a species is the time it takes for the concentration of the substance to fall to half of its initial value.
The decay of many physical quantities is not exponentialfor example, the evaporation of water from a puddle, or (often) the chemical reaction of a molecule. In such cases, the halflife is defined the same way as before: as the time elapsed before half of the original quantity has decayed. However, unlike in an exponential decay, the halflife depends on the initial quantity, and the prospective halflife will change over time as the quantity decays.
As an example, the radioactive decay of carbon14 is exponential with a halflife of 5,730 years. A quantity of carbon14 will decay to half of its original amount (on average) after 5,730 years, regardless of how big or small the original quantity was. After another 5,730 years, onequarter of the original will remain. On the other hand, the time it will take a puddle to halfevaporate depends on how deep the puddle is. Perhaps a puddle of a certain size will evaporate down to half its original volume in one day. But on the second day, there is no reason to expect that onequarter of the puddle will remain; in fact, it will probably be much less than that. This is an example where the halflife reduces as time goes on. (In other nonexponential decays, it can increase instead.)
The decay of a mixture of two or more materials which each decay exponentially, but with different halflives, is not exponential. Mathematically, the sum of two exponential functions is not a single exponential function. A common example of such a situation is the waste of nuclear power stations, which is a mix of substances with vastly different halflives. Consider a mixture of a rapidly decaying element A, with a halflife of 1 second, and a slowly decaying element B, with a halflife of 1 year. In a couple of minutes, almost all atoms of element A will have decayed after repeated halving of the initial number of atoms, but very few of the atoms of element B will have done so as only a tiny fraction of its halflife has elapsed. Thus, the mixture taken as a whole will not decay by halves.
A biological halflife or elimination halflife is the time it takes for a substance (drug, radioactive nuclide, or other) to lose onehalf of its pharmacologic, physiologic, or radiological activity. In a medical context, the halflife may also describe the time that it takes for the concentration of a substance in blood plasma to reach onehalf of its steadystate value (the "plasma halflife").
The relationship between the biological and plasma halflives of a substance can be complex, due to factors including accumulation in tissues, active metabolites, and receptor interactions.^{[5]}
While a radioactive isotope decays almost perfectly according to socalled "first order kinetics" where the rate constant is a fixed number, the elimination of a substance from a living organism usually follows more complex chemical kinetics.
For example, the biological halflife of water in a human being is about 9 to 10 days,^{[6]} though this can be altered by behavior and various other conditions. The biological halflife of cesium in human beings is between one and four months.