Half-life - Formulas For Half-life in Exponential Decay

Formulas For Half-life in Exponential Decay

An exponential decay process can be described by any of the following three equivalent formulas:

where

• N₀ is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
• N(t) is the quantity that still remains and has not yet decayed after a time t,
• t_1/2 is the half-life of the decaying quantity,
• τ is a positive number called the mean lifetime of the decaying quantity,
• λ is a positive number called the decay constant of the decaying quantity.

The three parameters, and λ are all directly related in the following way:

where ln(2) is the natural logarithm of 2 (approximately 0.693).

Click "show" to see a detailed derivation of the relationship between half-life, decay time, and decay constant.

Start with the three equations We want to find a relationship between, and λ, such that these three equations describe exactly the same exponential decay process. Comparing the equations, we find the following condition: Next, we'll take the natural logarithm of each of these quantities. Using the properties of logarithms, this simplifies to the following: Since the natural logarithm of e is 1, we get: Canceling the factor of t and plugging in, the eventual result is:

By plugging in and manipulating these relationships, we get all of the following equivalent descriptions of exponential decay, in terms of the half-life:

Regardless of how it's written, we can plug into the formula to get

• as expected (this is the definition of "initial quantity")
• as expected (this is the definition of half-life)
• , i.e. amount approaches zero as t approaches infinity as expected (the longer we wait, the less remains).