example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. Solution of the differential equation The equation that describes exponential decay is or, by rearranging (applying the technique called separation of variables), Integrating, we have where C is the constant of integration, and hence where the final substitution, N0 = eC, is obtained by evaluating the equation at t = 0, as N0 is defined as being the quantity at t = 0. This is the form of the equation that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime, or half-life is sufficient to characterise the decay. The notation λ for the decay constant is a remnant of the usual notation for an eigenvalue. In this case, λ is the eigenvalue of the negative of the differential operator with N(t) as the corresponding eigenfunction. The units of the decay constant are s−1. Derivation of the mean lifetime Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, , (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. Specifically, if the individual lifetime of an element of the assembly is the time elapsed between some reference time and the removal of that element from the assembly, the mean lifetime is the arithmetic mean of the individual lifetimes. Starting from the population formula first let c be the normalizing factor to convert to a probability density function: or, on rearranging, Exponential decay is a scalar multiple of the exponential distribution (i.e. the individual lifetime of each object is exponentially distributed), which has a well-known expected value. We can compute it here using integration by parts. Decay by two or more processes 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 N is given by the sum of the decay routes; thus, in the case of two processes: The solution to this equation is given in the previous section, where the sum of is treated as a new total decay constant . Partial mean life associated with individual processes is by definition the multiplicative