\( ln \dfrac=-kt \) \( \dfrac=t \) A From Equation 14.5.5, we know that : \( \dfrac= \dfrac=\dfrac=\dfrac \) Now we need only calculate the rate constant for the reaction from its half-life (5730 yr) using Equation 14.5.3: \( t_=\dfrac \) This equation can be rearranged as follows: \( k=\dfrac=\dfrac=1.22 \times 10^ \; yr^ \) B Substituting into the equation for , \( t=\dfrac=\dfrac=5.2 \times 10^ \; yr \) From our calculations, the man died 5200 yr ago.

Exercise It is believed that humans first arrived in the Western Hemisphere during the last Ice Age, presumably by traveling over an exposed land bridge between Siberia and Alaska.

The half-life of a first-order reaction is (to indicate a half-life) into Equation 14.5.1 gives \( ln \dfrac = ln 2=kt_ \tag \) The natural logarithm of 2 (to three decimal places) is 0.693.

Exercise In Example 4 you found that ethyl chloride decomposes to ethylene and HCl in a first-order reaction that has a rate constant of 1.6 × 10 at 650°C.Comparing the disintegrations per minute per gram of carbon from an archaeological sample with those from a recently living sample enables scientists to estimate the age of the artifact, as illustrated in Example 11.ratio in the atmosphere is constant, which is not strictly correct.The half-lives of several isotopes are listed in Table 14.5.1, along with some of their applications.In our earlier discussion, we used the half-life of a first-order reaction to calculate how long the reaction had been occurring.

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