   Chapter 7.8, Problem 68E

Chapter
Section
Textbook Problem

# As we saw in Section 6.5, a radioactive substance decays exponentially: The mass at time t is m ( t ) = m ( 0 ) e k t , where m(0) is the initial mass and k is a negative constant.The mean life M of an atom in the substance is M = − k ∫ 0 ∞ t e k t d t For the radioactive carbon isotope, C 14 , used in radiocarbon dating, the value of k is − 0.000121 . Find the mean life of a C 14 atom.

To determine

To find:

The mean life of 14C atom.

Explanation

Given:

The mass at time t is m(t)=m(0)ekt, m(0) is the initial mass and k is a negative constant. The mean life M of an atom in the substance is M=k0tektdt, k=0.000121.

Formulae used:

Integration by parts

The mean life M of an atom in the substance is:

M=k0tektdt

This implies:

M=k0tektdt=klimx0xtektdt

Integrate by parts and get:

M=klimx[t×1kekt1kektdt]0x=klimx[t×

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