Exponential Decay 

Need help preparing for the General Chemistry section of the MCAT? MedSchoolCoach expert, Ken Tao, will teach everything you need to know about Exponential Decay for Atomic Structure. Watch this video to get all the MCAT study tips you need to do well on this section of the exam!
When we examine radioactive decay, it is important to able to identify the difference between different rates of decay, as this can be an indicator of the stability of the isotope. The simplest form of decay is a linear decay. Linear decay has a constant slope, meaning that the rate of decay does not change over time. If the first unit of time sees a decay quantity of five, then each successive unit of equivalent time would see an additional decay quantity of five.
However, most decay you may see on the MCAT is not linear, but exponential. Exponential decay refers to any decay process where the rate of decay is proportional to the amount of decaying material, or, phrased in a different way, the rate of decay is variable, meaning that if the first decay quantity per unit of time is five, subsequent periods of time will experience different decay quantities. One of the most important of decay measures on the MCAT is known as half-life. Half-life is the period of time required for half of a given substance to decay. Consider an example with a 100g sample of a radioactive isotope phosphate-32, with a half-life of 14 days. After the first interval of 14 days, 50 grams will remain. After the second interval of 14 days, or 28 days total, 25 grams will remain. After the third interval of 14 days, or 42 days total, 12.5 grams of undecayed phosphate-32 will remain, and so on. Note that the amount of undecayed phosphate-32 will never technically reach zero, only becoming progressively smaller and smaller and approaching zero, a graphical term known as an asymptote.
One can plot the radioactive decay of phosphate-32 in a linear format. What this means is that the x and y-axes themselves were scaled linearly, with each x or y-interval of equivalent distance to the intervals above and below it. Note that the decay itself was not linear, but exponential: only the graph is linear. On the MCAT, you may also see exponential decay plotted on what is called a semi-log plot. On a semi-log plot, the x-axis of the graph will remain on a linear scale, while the y-axis is changed to a logarithmic scale, or a scale ranging by increasing magnitudes of ten. Interestingly, exponential decay appears linearly on a logarithmic scale, so these plots are often used scientifically to evaluate changes in the curve of change for an exponentially increasing or decreasing value. For example, during the 2020 COVID-19 pandemic, semi-log scales were used early in the spread of the disease to evaluate the effectiveness of mitigation efforts: even though case numbers were still increasing, semi-log plots of the spread of the disease could be used to look for a leveling off of spread, which is easier to see on a logarithmic scale than a linear one.
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