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Chapter 19: Problem 75

Iodine-131 has a half-life of \(8.0\) days. How many days will it take for \(174 \mathrm{~g}\) of \({ }^{131}\) I to decay to \(83 \mathrm{~g}\) of \({ }^{131} \mathrm{I}\) ?

Short Answer

Expert verified
It will take approximately \(5.65\) days for \(174 \mathrm{~g}\) of Iodine-131 to decay to \(83 \mathrm{~g}\).

Step by step solution

01

Write down the half-life formula

First, let's recall the formula for half-life: \[N_t = N_0 \times \left(\frac{1}{2}\right)^\frac{t}{t_{1/2}}\] where: - \(N_t\) is the final mass after decay (in this case, 83 g) - \(N_0\) is the initial mass before decay (in this case, 174 g) - \(t\) is the time it takes to reach the final mass \(N_t\) - \(t_{1/2}\) is the half-life (in this case, 8.0 days) We need to find \(t\), so let's rearrange the formula: \[\frac{N_t}{N_0} = \left(\frac{1}{2}\right)^\frac{t}{t_{1/2}}\]
02

Substitute the given values into the formula

Now, substitute the known values into the formula: \[\frac{83}{174} = \left(\frac{1}{2}\right)^\frac{t}{8.0}\]
03

Solve for t

Next, we need to solve for \(t\). To simplify the expression, let's use the logarithm function: \[\log_2 \left(\frac{83}{174}\right) = \frac{t}{8.0}\] Now, multiply both sides by 8.0 to isolate \(t\): \[t = 8.0 \times \log_2 \left(\frac{83}{174}\right)\]
04

Calculate the value of t

Finally, use a calculator or logarithm table to find the value of \(t\): \[t = 8.0 \times \log_2 \left(\frac{83}{174}\right) \approx 5.65\]
05

Interpret the result

The result, \(t \approx 5.65\) days, tells us that it will take approximately 5.65 days for 174 g of Iodine-131 to decay to 83 g.

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Most popular questions from this chapter

Many transuranium elements, such as plutonium-232, have very short half-lives. (For \({ }^{232} \mathrm{Pu}\), the half-life is 36 minutes.) However, some, like protactinium-231 (half-life \(=3.34 \times\) \(10^{4}\) years), have relatively long half-lives. Use the masses given in the following table to calculate the change in energy when 1 mole of \({ }^{232}\) Pu nuclei and 1 mole of \({ }^{231}\) Pa nuclei are each formed from their respective number of protons and neutrons.

Phosphorus-32 is a commonly used radioactive nuclide in biochemical research, particularly in studies of nucleic acids. The half-life of phosphorus-32 is \(14.3\) days. What mass of phosphorus-32 is left from an original sample of \(175 \mathrm{mg}\) \(\mathrm{Na}_{3}{ }^{32} \mathrm{PO}_{4}\) after \(35.0\) days? Assume the atomic mass of \({ }^{32} \mathrm{P}\) is \(32.0 \mathrm{u}\).

The radioactive isotope \({ }^{247} \mathrm{Bk}\) decays by a series of \(\alpha\) -particle and \(\beta\) -particle productions, taking \({ }^{247} \mathrm{Bk}\) through many transformations to end up as \({ }^{207} \mathrm{~Pb}\). In the complete decay series, how many \(\alpha\) particles and \(\beta\) particles are produced?

Assume a constant \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) ratio of \(13.6\) counts per minute per gram of living matter. A sample of a petrified tree was found to give \(1.2\) counts per minute per gram. How old is the tree? \(\left(\right.\) For \({ }^{14} \mathrm{C}, t_{1 / 2}=5730\) years. \()\)

When using a Geiger-Müller counter to measure radioactivity, it is necessary to maintain the same geometrical orientation between the sample and the Geiger-Müller tube to compare different measurements. Why?
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