Editing Exponential decay (section)

=== Decay by two or more processes ===<!-- This section is linked from [[Half-life]] -->
{{see also|Branching fraction}}
A quantity may decay via two or more different processes simultaneously. In general, these processes (often called "decay modes", "decay channels", "decay routes" etc.) have different probabilities of occurring, and thus occur at different rates with different half-lives, in parallel. The total decay rate of the quantity&nbsp;''N'' is given by the ''sum'' of the decay routes; thus, in the case of two processes:

:<math>-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N.</math>

The solution to this equation is given in the previous section, where the sum of <math>\lambda _1 + \lambda _2\,</math> is treated as a new total decay constant <math>\lambda _c</math>.

:<math>N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}.</math>

'''Partial mean life''' associated with individual processes is by definition the [[multiplicative inverse]] of corresponding partial decay constant: <math>\tau = 1/\lambda</math>. A combined <math>\tau_c</math> can be given in terms of <math>\lambda</math>s:

:<math>\frac{1}{\tau_c} = \lambda_c = \lambda_1 + \lambda_2 = \frac{1}{\tau_1} + \frac{1}{\tau_2}</math>

:<math>\tau_c = \frac{\tau_1 \tau_2}{\tau_1 + \tau_2}. </math>

Since half-lives differ from mean life <math>\tau</math> by a constant factor, the same equation holds in terms of the two corresponding half-lives:

:<math>T_{1/2} = \frac{t_1 t_2}{t_1 + t_2} </math>

where <math>T _{1/2}</math> is the combined or total half-life for the process, <math>t_1</math> and <math>t_2</math> are so-named '''partial half-lives''' of corresponding processes. Terms "partial half-life" and "partial mean life" denote quantities derived from a decay constant as if the given decay mode were the only decay mode for the quantity. The term "partial half-life" is misleading, because it cannot be measured as a time interval for which a certain quantity is [[one half|halved]].

In terms of separate decay constants, the total half-life <math>T _{1/2}</math> can be shown to be

:<math>T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2}.</math>

For a decay by three simultaneous exponential processes the total half-life can be computed as above:

:<math>T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda_1 + \lambda_2 + \lambda_3} = \frac{t_1 t_2 t_3}{(t_1 t_2) + (t_1 t_3) + (t_2 t_3)}.</math>