Question

the radioisotope $^{14}c$ (half - life = 5730 years) is used for carbon dating. what is the first - order rate constant for $^{14}c$?

Explanation:

Step1: Recall the formula for half - life of first - order reaction

For a first - order reaction, the half - life ($t_{1/2}$) and the rate constant ($k$) are related by the formula $t_{1/2}=\frac{\ln 2}{k}$.

Step2: Rearrange the formula to solve for $k$

We can rewrite the formula as $k = \frac{\ln 2}{t_{1/2}}$.

Step3: Substitute the given half - life value

Given $t_{1/2}=5730$ years, and $\ln 2\approx0.693$. Then $k=\frac{0.693}{5730\ years}$.
$k = 1.21\times 10^{- 4}\text{ year}^{-1}$

Answer:

$1.21\times 10^{-4}\text{ year}^{-1}$