## A commonly used parameter for the rate of decay that is related to the decay constant by:${t}_{\frac{1}{2}}=\frac{ln2}{\lambda}=\tau ln2$.

#### Key Points

• The relationship between time, half-life, and the amount of radionuclide is defined by:$N(t)={N}_{0}{e}^{-\frac{t}{\tau}}={N}_{0}{2}^{-\frac{t}{{t}_{\frac{1}{2}}}}$.

• The relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer.

• Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period.  Half-life has been chosen for convenience and consistency.

#### Terms

• The time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay.

#### Figures

Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms per box (left) or 400 (right). The number at the top is how many half-lives have elapsed. Note the law of large numbers: With more atoms, the overall decay is more regular and more predictable.

2. ##### Nuclear Half Life: Intro and Explanation

Nuclear half life is the time that it takes for one half of a radioactive sample to decay. In this video, we will learn the basics of nuclear half life, and examine graphs and practice problems.

Radioactive decay is a random process at the single-atom level; is impossible to predict exactly when a particular atom will decay. However, the chance that a given atom will decay is constant over time. And for a large number of atoms, the decay rate for the collection as a whole can be computed from the measured decay constants of the nuclides (or equivalently from the half-lives) .

Given a sample of a particular radionuclide, the half-life is the time taken for half of its atoms to decay Figure 2. The following equation is used to predict the number of atoms (N) of a a given radioactive sample that remain after a given time (t):

$N={N}_{0}{e}^{-\lambda t}={N}_{0}{e}^{-\frac {t }{\tau }}$

Where: τ, "tau" is the average lifetime of a radioactive particle before decay. λ, the decay constant "lambda" is the inverse of the mean lifetime, and N0 is the value of N at t=0. The equation indicates that the decay constant λ has units of t-1, and can thus also be represented as 1/τ, where τ is a characteristic time of the process called the time constant.

The half-life is related to the decay constant as follows: set N = N0/2 and t = T1/2 to obtain:

${t}_{1/2}=\frac{ln2}{\lambda}=\tau ln2$

This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives vary widely: that of 209Bi is 1019 years, while unstable nuclides can have half-lives that have been measured as short as 10−23 seconds.

Rewriting the initial equation after converting logarithmic bases, the following equation can be shown to be valid:

$N(t)={N}_{0}{e}^{-\frac{t}{\tau}}={N}_{0}{2}^{-\frac{t}{{t}_{1/2}}}$

Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that gave its "(1/3)-life" (how long until only 1/3 is left) or "(1/10)-life" (a time period until only 10% is left), and so on. Thus, the choice of τ and t1/2 for marker-times, are only for convenience, and from convention.  Mathematically, the nth life for the above situation would be found in the same way as above—by setting N = N0/n, and substituting into the decay solution to obtain:

${t}_{1/n}=\frac{ln (n)}{\lambda}=\tau ln(n)$

#### Key Term Glossary

atom
the smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons
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base
A proton acceptor, or an electron pair donor.
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constant
Consistently recurring over time; persistent
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decay
To change by undergoing fission, by emitting radiation, or by capturing or losing one or more electrons.
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logarithmic
Of, or relating to logarithms.
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nuclide
An atomic nucleus specified by its atomic number and atomic mass.
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period
a horizontal row in the periodic table, which signifies the total number of electron shells in an element's atom
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probability
A number, between 0 and 1, expressing the precise likelihood of an event happening.
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Any of several processes by which unstable nuclei emit subatomic particles and/or ionizing radiation and disintegrate into one or more smaller nuclei.
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solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
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Solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
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substance
Physical matter; material.
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unstable
Fuctuating; not constant.