Chemistry: An Atoms-Focused Approach
Chemistry: An Atoms-Focused Approach
14th Edition
ISBN: 9780393912340
Author: Thomas R. Gilbert, Rein V. Kirss, Natalie Foster
Publisher: W. W. Norton & Company
Question
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Chapter 21, Problem 21.80QA
Interpretation Introduction

To explain:

If the radioactive dating distinguishes between debris from the eruptions of Mt. Vesuvius that occurred in years 472 and 512.

Expert Solution & Answer
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Answer to Problem 21.80QA

Solution:

Yes; we can distinguish between the radioactive dating debris from the eruptions of Mt. Vesuvius that occurred in years 472 and 512.

Explanation of Solution

1) Concept:

The half-life of C 14 is taken from the Table 21.2. Using the half-life, we get the value of rate constant. Plugging the values of rate constant and given time in the first order radioactive decay equation, we can calculate the ratio C 14C 12  for both years. Then the difference between both ratios and given uncertainty in determining the ratio is compared.

2) Formula:

i) t12=0.693k

ii)  lnXtX0=-kt

where, Xt is the final concentration, X0 is initial concentration, k is rate constant,  t is time, and t12 is half-life.

3) Given:

i) Uncertainty in determining the ratio =0.1 %

ii) t1=472 years

iii)  t2=512 years

4) Calculations:

The half-life of C 14 is taken from the Table 21.2 as 5730 years.

So, the rate constant is calculated as follows:

t12=0.693k

5730 yr=0.693k

k=0.6935730 yr=1.209×10-4 yr-1

Calculating C/  14C 12 for t1=472 years:

lnXtX0=-kt

lnC 14C 12=-1.209×10-4yr-1×472 yr

C 14C 12= e-1.209×10-4 yr-1×472 yr=0.9445

C 14C 12=94.5 %

Calculating C/  14C 12 for t1=512 years:

lnXtX0=-kt

ln[C 14C 12]=-1.209×10-4 yr-1×512 ys

C 14C 12= e-1.209×10-4 yr-1×512 yr=0.9400

C 14C 12=94.0 %

Therefore, the difference in C 14C 12 for both time is =94.5%-94.0%=0.5%

While the uncertainty in determining the ratio is 0.1 %, this is less than the difference in the values of ratio C 14C 12. So, we can distinguish between the radioactive dating debris from the eruptions of Mt. Vesuvius that occurred in years 472 and 512.

Conclusion:

The difference between carbon-14: carbon 12 for both years is larger than the uncertainty in determining the ratio.

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Chapter 21 Solutions

Chemistry: An Atoms-Focused Approach

Ch. 21 - Prob. 21.12VPCh. 21 - Prob. 21.13QACh. 21 - Prob. 21.14QACh. 21 - Prob. 21.15QACh. 21 - Prob. 21.16QACh. 21 - Prob. 21.17QACh. 21 - Prob. 21.18QACh. 21 - Prob. 21.19QACh. 21 - Prob. 21.20QACh. 21 - Prob. 21.22QACh. 21 - Prob. 21.23QACh. 21 - Prob. 21.24QACh. 21 - Prob. 21.25QACh. 21 - Prob. 21.26QACh. 21 - Prob. 21.27QACh. 21 - Prob. 21.28QACh. 21 - Prob. 21.29QACh. 21 - Prob. 21.30QACh. 21 - Prob. 21.31QACh. 21 - Prob. 21.32QACh. 21 - Prob. 21.33QACh. 21 - Prob. 21.34QACh. 21 - Prob. 21.35QACh. 21 - Prob. 21.36QACh. 21 - Prob. 21.37QACh. 21 - Prob. 21.38QACh. 21 - Prob. 21.39QACh. 21 - Prob. 21.40QACh. 21 - Prob. 21.41QACh. 21 - Prob. 21.42QACh. 21 - Prob. 21.43QACh. 21 - Prob. 21.44QACh. 21 - Prob. 21.45QACh. 21 - Prob. 21.46QACh. 21 - Prob. 21.47QACh. 21 - Prob. 21.48QACh. 21 - Prob. 21.49QACh. 21 - Prob. 21.50QACh. 21 - Prob. 21.51QACh. 21 - Prob. 21.52QACh. 21 - Prob. 21.53QACh. 21 - Prob. 21.54QACh. 21 - Prob. 21.55QACh. 21 - Prob. 21.56QACh. 21 - Prob. 21.57QACh. 21 - Prob. 21.58QACh. 21 - Prob. 21.59QACh. 21 - Prob. 21.60QACh. 21 - Prob. 21.61QACh. 21 - Prob. 21.62QACh. 21 - Prob. 21.63QACh. 21 - Prob. 21.64QACh. 21 - Prob. 21.65QACh. 21 - Prob. 21.66QACh. 21 - Prob. 21.67QACh. 21 - Prob. 21.68QACh. 21 - Prob. 21.69QACh. 21 - Prob. 21.70QACh. 21 - Prob. 21.71QACh. 21 - Prob. 21.72QACh. 21 - Prob. 21.73QACh. 21 - Prob. 21.74QACh. 21 - Prob. 21.75QACh. 21 - Prob. 21.76QACh. 21 - Prob. 21.77QACh. 21 - Prob. 21.78QACh. 21 - Prob. 21.79QACh. 21 - Prob. 21.80QACh. 21 - Prob. 21.81QACh. 21 - Prob. 21.82QACh. 21 - Prob. 21.83QACh. 21 - Prob. 21.84QACh. 21 - Prob. 21.85QACh. 21 - Prob. 21.86QACh. 21 - Prob. 21.87QACh. 21 - Prob. 21.88QACh. 21 - Prob. 21.89QACh. 21 - Prob. 21.90QACh. 21 - Prob. 21.91QACh. 21 - Prob. 21.92QACh. 21 - Prob. 21.93QACh. 21 - Prob. 21.94QACh. 21 - Prob. 21.95QACh. 21 - Prob. 21.96QACh. 21 - Prob. 21.97QACh. 21 - Prob. 21.98QACh. 21 - Prob. 21.99QA
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