How do you calculate exponential decay?

To calculate exponential decay, use the formula \( N(t) = N_0 e^{-kt} \).

Exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. The formula \( N(t) = N_0 e^{-kt} \) helps us calculate the remaining quantity \( N(t) \) after time \( t \). Here, \( N_0 \) is the initial amount, \( e \) is the base of the natural logarithm (approximately 2.71828), \( k \) is the decay constant, and \( t \) is the time elapsed.

To use this formula, you first need to identify the initial amount \( N_0 \) and the decay constant \( k \). The decay constant \( k \) can often be found from the problem statement or calculated if you know the half-life of the substance. The half-life is the time it takes for the quantity to reduce to half its initial value. The relationship between the half-life \( T_{1/2} \) and the decay constant \( k \) is given by \( k = \frac{\ln(2)}{T_{1/2}} \), where \( \ln \) is the natural logarithm.

Once you have \( N_0 \) and \( k \), you can substitute these values into the formula along with the time \( t \) to find the remaining quantity \( N(t) \). For example, if you start with 100 grams of a substance with a decay constant of 0.1 per year, and you want to know how much remains after 5 years, you would calculate \( N(5) = 100 e^{-0.1 \times 5} \).

Understanding exponential decay is useful in various real-world contexts, such as radioactive decay, population decline, and cooling of objects.