By Mark Kennan – Updated Mar 24, 2022
Decay describes the rapid decline of a quantity over time, commonly seen in bacterial populations, radioactive isotopes, and even financial depreciation. When the rate of decline is directly proportional to the remaining amount, the process follows an exponential decay model, expressed mathematically as N(t) = N₀ e^(k t), where k is the decay constant (negative for decay). Knowing the initial (N₀) and final (N(t)) populations allows you to determine k and predict future values.
Divide the final count by the initial count. For instance, if you start with 100 bacteria and find 80 after 2 hours, the ratio is 80 ÷ 100 = 0.8.
Take the natural logarithm (ln) of the ratio. Using the example, ln(0.8) ≈ -0.223143551.
Divide the logarithm result by the elapsed time to obtain the decay rate (k). Here, -0.223143551 ÷ 2 hours = -0.111571776 per hour.
With the decay constant known, you can forecast the population at any time t using the formula:
N(t) = N₀ e^(k t)
Example: To estimate the bacteria count after 5 hours, compute 5 × -0.111571776 = -0.55785888. Then e^(-0.55785888) ≈ 0.57243340. Finally, 0.57243340 × 100 = 57.24 bacteria.
The negative sign indicates decay. Multiply your desired time by the decay rate, exponentiate e, and then multiply by the initial population to find the future value.
