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Chapter 21: Problem 34

It takes 5.2 min for a 1.000 -g sample of \({ }^{210} \mathrm{Fr}\) to decay to \(0.250 \mathrm{~g}\). What is the half-life of \({ }^{210} \mathrm{Fr}\) ?

Short Answer

Expert verified
The half-life of \({ }^{210} \mathrm{Fr}\) is approximately 3.201 minutes.

Step by step solution

01

Identifying the Equations

In order to find the half-life of the radioactive substance, we will use the radioactive decay equation: \[N = N_0e^{-\lambda t}\] Where: - \(N\) represents the final mass of the substance - \(N_0\) represents the initial mass of the substance - \(e\) is the base of the natural logarithm (approximately 2.71828) - \(\lambda\) is the decay constant - \(t\) is time We also know that the half-life can be calculated using the following equation: \[T_{1/2} = \frac{\ln2}{\lambda}\] Where: - \(T_{1/2}\) = half-life - \(\lambda\) is the decay constant
02

Finding the Decay Constant

We will first find the decay constant (\(\lambda\)). To do so, we need to rearrange the radioactive decay equation: \(\lambda = -\frac{\ln(\frac{N}{N_0})}{t}\) Using the given information, we have: \(N\) = 0.250 g \(N_0\) = 1.000 g \(t\) = 5.2 minutes Now, we can plug in these values into the rearranged equation to find \(\lambda\): \(\lambda = -\frac{\ln(\frac{0.250}{1.000})}{5.2}\) \[\lambda \approx 0.2164\,\text{min}^{-1}\]
03

Calculating the Half-life

Now that we have found the decay constant, we can use the equation for half-life to find \(T_{1/2}\): \[T_{1/2} = \frac{\ln2}{\lambda}\] Replacing \(\lambda\) with the calculated value: \[T_{1/2} = \frac{\ln2}{0.2164}\] \[T_{1/2} \approx 3.201\,\text{min}\] So the half-life of \({ }^{210} \mathrm{Fr}\) is approximately 3.201 minutes.

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Most popular questions from this chapter

Complete and balance the following nuclear equations by supplying the missing particle: (a) \({ }_{98}^{252} \mathrm{Cf}+{ }_{5}^{10} \mathrm{~B} \longrightarrow 3{ }_{0}^{1} \mathrm{n}+?\) (b) \({ }_{1}^{2} \mathrm{H}+{ }_{2}^{3} \mathrm{He} \longrightarrow{ }_{2}^{4} \mathrm{He}+\) ? (c) \({ }_{1}^{1} \mathrm{H}+{ }_{5}^{11} \mathrm{~B} \longrightarrow 3\) ? (d) \({ }_{53}^{122} \mathrm{I} \longrightarrow{ }_{54}^{122} \mathrm{Xe}+?\) (e) \(\frac{59}{26} \mathrm{Fe} \longrightarrow{ }_{-1}^{0} \mathrm{e}+?\)

Based on the following atomic mass values \(-1 \mathrm{H}\), 1.00782 amu; \({ }^{2} \mathrm{H}, 2.01410 \mathrm{amu} ;{ }^{3} \mathrm{H}, 3.01605 \mathrm{amu} ;{ }^{3} \mathrm{He}\) 3.01603 amu; \({ }^{4}\) He, 4.00260 amu- and the mass of the neutron given in the text, calculate the energy released per mole in each of the following nuclear reactions, all of which are possibilities for a controlled fusion process: (a) \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \longrightarrow{ }_{2}^{4} \mathrm{He}+{ }_{0}^{1} \mathrm{n}\) (b) \({ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{2}^{3} \mathrm{He}+{ }_{0}^{1} \mathrm{n}\) (c) \({ }_{1}^{2} \mathrm{H}+{ }_{2}^{3} \mathrm{He} \longrightarrow{ }_{2}^{4} \mathrm{He}+{ }_{1}^{1} \mathrm{H}\)

Decay of which nucleus will lead to the following products: (a) bismuth-211 by beta decay; (b) chromium-50 by positron emission; (c) tantalum-179 by electron capture; (d) radium226 by alpha decay?

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: (a) tritium, \({ }_{1}^{3} \mathrm{H},(\mathbf{b}){ }_{38}^{89} \mathrm{Sr}\), (c) iodine-120, (d) silver-102.

Cobalt- 60 , which undergoes beta decay, has a half-life of 5.26 yr. (a) How many beta particles are emitted in \(600 \mathrm{~s}\) by a \(3.75-\mathrm{mg}\) sample of \({ }^{60} \mathrm{Co} ?\) (b) What is the activity of the sample in \(\mathrm{Bq}\) ?
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