Question 14.4: The half-life of carbon-14, used in radiocarbon dating, is 5......
The half-life of carbon-14, used in radiocarbon dating, is 5730 years. What is the decay constant for carbon-14?
Strategy The half-life and the decay constant are alternative ways of characterizing the rate of decay, so knowing one allows us to find the other. The needed relationship is given in Equation 14.3.
t_{1 / 2}=\frac{\ln 2}{k}=\frac{0.693}{k} (14.3)
\begin{gathered}t_{1 / 2}=\frac{0.693}{k} \\\\k=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{5730\text{ yr} }=1.21 \times 10^{-4}\text{ yr}^{-1}\end{gathered}
Analyze Your Answer Does this answer make sense? First, we need to recognize that the half-life and the decay constant are inversely proportional to each other. So with a half-life on the order of 10^3 yr, we might expect the decay constant to be on the order of 10^{-3} yr. The proportionality constant, 0.693 , is somewhat less than one, so a value of the order of 10^{-4} makes sense.
Check Your Understanding Carbon-15 has a half-life of 2.45 seconds. Calculate the decay constant of carbon-15 and compare it to that of carbon-14 .