what is the equation for exponetial decay(radioactive decay)

and how to use the half life in the equation

The equation for exponential decay, including radioactive decay, is:

N(t) = N₀ * e^(-kt)

In this equation:
- N(t) represents the final amount of a substance at time t
- N₀ represents the initial amount of the substance
- e is the mathematical constant approximately equal to 2.71828
- k is the decay constant, which is a characteristic constant specific to each decay process
- t represents the time elapsed

To incorporate the concept of half-life into the equation, you need to understand that the half-life (T½) of a substance is the time it takes for half of the substance to decay. Knowing the half-life, you can determine the value of k.

The relationship between the decay constant (k) and the half-life is given by:

k = ln(2) / T½

In this equation:
- ln(2) represents the natural logarithm of 2, approximately equal to 0.693

Once you have determined the decay constant, you can substitute it in the equation for exponential decay (N(t) = N₀ * e^(-kt)) to calculate the quantity of the substance at a given time.