Exponential Decay - Solution of The Differential Equation

Solution of The Differential Equation

The equation that describes exponential decay is

or, by rearranging,

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 opposite of the differentiation operator with N(t) as the corresponding eigenfunction. The units of the decay constant are s−1.

Read more about this topic:  Exponential Decay

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