The half-life is the time required for a specific quantity of a substance to reduce to half of its initial value. This concept applies across various fields, including the decay of radioactive isotopes, the breakdown of medicines, or chemical reactions. While complex calculations can determine this value, a graph provides a straightforward, visual method for quick determination. This graphical approach allows the half-life (\(t_{1/2}\)) to be found by directly observing the decay process over time.
Understanding the Decay Curve
A graph used to determine half-life plots two variables: the quantity of the substance remaining (Y-axis) and the time elapsed (X-axis). The Y-axis represents the amount, measured in units like concentration or mass. The X-axis represents time, which varies from seconds to millions of years depending on the substance.
The resulting plot, known as a decay curve, begins at the initial, full amount of the substance at Time = 0. As time progresses, the quantity decreases, forming a characteristic curve that drops steeply and then gradually flattens out. This shape illustrates exponential decay, where the substance loses half of its present amount over a consistent time interval.
Determining the First Half-Life
To find the half-life from the curve, first identify the initial quantity of the substance at time zero, where the curve intersects the Y-axis. Next, calculate the halfway point, which is half of the initial quantity.
Locate this halfway value on the Y-axis and draw a horizontal line until it meets the decay curve. This intersection point marks when half of the original substance remains. From this point, draw a vertical line straight down to the X-axis (the time axis).
The value where this vertical line meets the X-axis is the half-life (\(t_{1/2}\)) of the substance. This time interval is the duration required for the initial quantity to be reduced by fifty percent. For instance, if the initial amount was 100 grams, the half-life is the time it took to drop to 50 grams.
Consistency Check Using Subsequent Intervals
The half-life is a constant value in exponential decay, meaning the time interval for the quantity to halve remains the same throughout the process. To confirm the initial measurement, a consistency check should be performed by calculating subsequent half-lives.
To perform this check, take the quantity remaining after the first half-life (50%) and calculate the next halfway point, which is one-quarter (25%) of the original amount. Locate this new quantity on the Y-axis, draw a horizontal line to the curve, and then a vertical line down to the X-axis.
The time interval between the first half-life mark and this second half-life mark must equal the first half-life measurement. Repeating this process, such as measuring the time from 25% remaining down to 12.5%, should yield the same time value. This consistent interval confirms the accuracy of the graphical determination.