Decay and Half-life Help

By — McGraw-Hill Professional
Updated on Sep 12, 2011

Decay and Half-Life

Radioactive substances gradually lose “potency” as time passes. Unstable nuclei degenerate one by one. Sometimes an unstable nucleus decays into a stable one in a single event. In other cases, an unstable nucleus changes into another unstable nucleus, which later degenerates into a stable one. Suppose that you have an extremely large number of radioactive nuclei, and you measure the length of time required for each one to degenerate and then average all the results. The average decay time is called the mean life and is symbolized by the lowercase Greek letter tau (τ).

Some radioactive materials give off more than one form of emission. For any given ionizing radiation form (alpha particles, beta particles, gamma rays, or other), there is a separate decay curve , or function of intensity versus time. A radioactive decay curve always has a characteristic shape: It starts out at a certain value and tapers down toward zero. Some decay curves decrease rapidly, and others decrease slowly, but the characteristic shape is always the same and can be defined in terms of a time span known as the half-life , symbolized t 1/2 .

Suppose that the intensity of radiation of a particular sort is measured at time t 0 . After a period of time t 1/2 has passed, the intensity of that form of radiation decreases to half the level it was at t 0 . After the half-life passes again (total elapsed time 2 t 1/2 ), the intensity goes down to one-quarter of its original value. After yet another half-life passes (total elapsed time 3 t 1/2 ), the intensity goes down to one-eighth its original value. In general, after n half-lives pass from the initial time t 0 (total elapsed time nt 1/2 ), the intensity goes down to 1/(2 n ), or 0.5 n , times its original value. If the original intensity is x 0 units and the final intensity is x f units, then

x f = 0.5 n x 0

The general form of a radioactive decay curve is shown in Fig. 18-9. The half-life t 1/2 can vary tremendously depending on the particular radioactive substance involved. Sometimes t 1/2 is a tiny fraction of 1 second; in other cases it is millions of years. For each type of radiation emitted by a material, there is a separate value of t 1/2 and therefore a separate decay curve.

Forms of Radiation Radioactivity Decay And Half-life

Fig. 18-9 . General form of a radioactive decay curve.

Another way to define radioactive decay is in terms of a number called the decay constant , symbolized by the lowercase Greek letter lambda (λ). The decay constant is equal to the natural logarithm of 2 (approximately 0.69315) divided by the half-life in seconds. This is expressed as follows:

λ = 0.69315/ t 1/2

The symbol for the radioactive decay constant happens to be the same as the symbol for EM wavelength. Don’t confuse them; they are entirely different and independent quantities. Also, when determining the decay constant, be sure that t 1/2 is expressed in seconds. This will ensure that the decay constant is expressed in the proper units ( s −1 ). If you start with t 1/2 expressed in units other than seconds, you’ll get a decay constant that is the wrong number because it is expressed inappropriately.

The decay constant is the reciprocal of the mean life in seconds. Therefore, we can state these equations:

λ = 1/τ and τ = 1/λ

From these equations we can see that the mean life τ is related to the half-life t 1/2 as follows:

τ = t 1/2 /0.69315

= 1.4427 t 1/2


t 1/2 = 0.69315τ

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