Question

If a substance is radioactive, this means that the nucleus is unstable and will therefore decay by any number of processes (alpha decay, beta decay, etc.). The decay of radioactive elements follows first-order kinetics. Therefore, the rate of decay can be described by the same integrated rate equations and half-life equations that are used to describe the rate of first-order chemical reactions:

lnAtA0=−ktlnAtA0=−kt

and

t1/2=0.693kt1/2=0.693k

where A0A0A_0 is the initial amount or activity, AtAtA_t is the amount or activity at time ttt, and kkk is the rate constant.

By manipulation of these equations (substituting 0.693/t1/20.693/t1/2 for kkk in the integrated rate equation), we can arrive at the following formula:

fraction remaining=AtA0=(0.5)nfraction remaining=AtA0=(0.5)n

where nnn is the number of half-lives. The equation relating the number of half-lives to time ttt is

n=tt1/2n=tt1/2

where t1/2t1/2t_1/2 is the length of one half-life.

In the parts that follow, use the following abbreviations for time

Measure of time	Units of time
minutes	min
hours	h
days	d
years	yr

Part A

You are using a Geiger counter to measure the activity of a radioactive substance over the course of several minutes. If the reading of 400. countscounts has diminished to 100. countscounts after 55.3 minutesminutes , what is the half-life of this substance?

Express your answer with the appropriate units.

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PART B

An unknown radioactive substance has a half-life of 3.20 hourshours . If 40.7 gg of the substance is currently present, what mass A0A0A_0 was present 8.00 hours ago?

Express your answer with the appropriate units.

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PART C

Americium-241 is used in some smoke detectors. It is an alpha emitter with a half-life of 432 yearsyears. How long will it take in years for 45.0 %% of an Am-241 sample to decay?

Express your answer with the appropriate units.

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PART D

A fossil was analyzed and determined to have a carbon-14 level that is 30 %% that of living organisms. The half-life of C-14 is 5730 years. How old is the fossil?

Express your answer with the appropriate units.

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Answer 1