This all-in-one online Half-Life Calculator evaluates the parameters of the continuous exponential decay process. The parameters describing this process are: initial and final amount of a substance subject to decay, half-life of the substance, and time elapsed since the beginning of the observation. You can enter the values of any three parameters in the input fields of this calculator and find the missing parameter.
Note, that the initial and final amount as well as the half-life and elapsed time should be in the same units of measurement for amount of substance (grams, atoms, etc.) and time (seconds, years, etc.) respectively.
Half-Life Definition
Half-life is the time required for some quantity (of a substance, for example) to reduce to half of its initial value.
Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic parameter for the exponential decay process. Exponential decay is found in phenomena (mostly natural) when the amount of something decreases at a rate proportional to its current value.
The term is also used to describe non-exponential decay. For example, this concept is used in chemistry and biology. In medicine, pharmacology and toxicology they often say about the biological half-life of drugs and other chemicals in the human body.
This term is commonly used in nuclear physics to describe the process of radioactive decay of unstable nuclei. When describing the decay of discrete objects, such as radioactive atoms, a probabilistic definition of the half-life seems to be more appropriate. According to this definition, the probability of decay of a radioactive atom during its half-life is 50%.
Half-Life Formula
All calculations that performs this half-life calculator refer to exponential decay, described by the following formula:
$$A(t)=A(0) { e }^{ -rt }$$
where \(A(t)\) and \(A(0)\) are amounts of some quantity at time \(t\) and \(0\) respectively, \(r\) is the decay rate and \(t\) is the time elapsed. (Note, that in case \(r\) is known or is to be found, it may be more convenient to use our Exponential Decay Calculator.)
On the other hand, using the concept of half-life, the process of exponential decay can be described by the following half-life formula:
$$A(t)=A(0) \left ( \frac{1}{2} \right )^{t/t_{1/2}}$$
where \(t_{1/2}\) is the half-life. This very formula is used in our Half-Life Calculator.
It is easy to see that both of the above formulas describe the same process, and the parameters present in them are related by the simple equation for half-life:
$$t_{1/2} = \frac{ln(2)}{r}.$$
Half-Life in Radioactive Decay
Probably the best-known example of the practical use of half-life is carbon-14 dating.
The 14C radioactive isotope is constantly formed in the Earth’s stratosphere as a result of the bombardment of nitrogen atoms by neutrons that are part of cosmic rays. Within a few years, the “newborn” 14C, along with stable isotopes 12C and 13C, enters the Earth’s carbon cycle in the atmosphere, hydrosphere and biosphere.
The equilibrium concentration of 14C is reached when the rate of entry of this isotope into the atmosphere becomes equal to the rate of its decay. It is assumed that the equilibrium isotopic composition of the atmosphere remains unchanged over time.
As long as a living organism is in a state of metabolism with its environment, the content of 14C in it remains constant and is in equilibrium with the concentration of this isotope in the atmosphere. When the organism dies, the exchange of carbon with the environment stops; the content of the radioactive isotope begins to decrease, since there is no longer an influx of “fresh” 14C from outside.
The half-life of the 14C isotope is about 5730 years. Therefore, knowing the initial amount of 14C in an organism in relation to the stable isotopes 12C and 13C in a state of equilibrium (when the organism is alive) and the content of 14C in fossil remains, it is possible to establish how much time has elapsed since the death of a carbon-containing organism.
In other words, based on the remains of plants and animals, and other objects containing carbon, it is possible to establish the age of these objects using the radiocarbon method. Radiocarbon dating can date carbon-containing substances up to about 50,000 years ago.
Recently, recalibration of 14C-dates are being carried out. The need for this is due to the fact that the amount of the 14C isotope in the atmosphere, hydrosphere and biosphere did not remain constant, but changed under the influence of a number of external factors.
Related calculators
Check out our other chemistry calculators such as Moles to Atoms Converter or Mass to Mole Calculator.