/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 It takes \(5.2\) minutesfor a \(... [FREE SOLUTION] | 91影视

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Chapter 21: Problem 34

It takes \(5.2\) minutesfor 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 \(^{210}\text{Fr}\) is approximately \(2.492\) minutes.

Step by step solution

01

Identify given values

The given values are: - Initial mass, \(N_0\) = 1.000 g - Decayed mass, \(N(t)\) = 0.250 g - Decay time, \(t\) = 5.2 minutes
02

Calculate the decay constant

We'll use the decay formula to find the decay constant: \[ N(t) = N_0 e^{-\lambda t} \] \[ \frac{N(t)}{N_0} = e^{-\lambda t} \] \[ \lambda = \frac{- \ln \left( \frac{N(t)}{N_0}\right)}{t} \] Now substitute the given values: \[ \lambda = \frac{- \ln \left( \frac{0.250}{1.000}\right)}{5.2} \] \[ \lambda \approx 0.2785~\text{min}^{-1} \]
03

Calculate the half-life

Now we'll use the half-life formula and the calculated decay constant: \[ t_{1/2} = \frac{\ln 2}{\lambda} \] Substitute the decay constant: \[ t_{1/2} = \frac{\ln 2}{0.2785} \] \[ t_{1/2} \approx 2.492 ~\text{minutes} \] So the half-life of \(^{210}\text{Fr}\) is approximately \(2.492\) minutes.

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Most popular questions from this chapter

The cloth shroud from around a mummy is found to have a \({ }^{14} \mathrm{C}\) activity of \(9.7\) disintegrations per minute per gram of carbon as compared with living organisms that undergo \(16.3\) disintegrations per minute per gram of carbon. From the half-life for \({ }^{14} \mathrm{C}\) decay, \(5715 \mathrm{yr}, \mathrm{cal}-\) culate the age of the shroud.

The accompanying graph illustrates the decay of \({ }_{42}^{88}\) Mo, which decays via positron emission. (a) What is the halflife of the decay? (b) What is the rate constant for the decay? (c) What fraction of the original sample of \(\frac{88}{42}\) Mo remains after 12 minutes? (d) What is the product of the decay process? [Section 21.4]

A wooden artifact from a Chinese temple has a \({ }^{14} \mathrm{C}\) activity of \(38.0\) counts per minute as compared with an activity of \(58.2\) counts per minute for a standard of zero age. From the half-life for \({ }^{14} \mathrm{C}\) decay, 5715 yr, determine the age of the artifact.

Complete and balance the following nuclear equations by supplying the missing particle: (a) \(_{16}^{32} \mathrm{~S}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{1}^{1} \mathrm{p}+?\) (b) \({ }_{4}^{7} \mathrm{Be}+{ }_{-\mathrm{j}}^{0}\) (orbital electron) \(\longrightarrow\) ? (c) \(? \longrightarrow{ }_{76}^{187} \mathrm{Os}+{ }_{-1}^{0} \mathrm{e}\) (d) \({ }_{2}^{98} \mathrm{Mo}+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{0}^{1} \mathrm{n}+?\) (e) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{54}^{135} \mathrm{Xe}+2{ }_{0}^{1} \mathrm{n}+?\)

One of the nuclides in each of the following pairs is radioactive. Predict which is radioactive and which is stable: (a) \({ }_{19}^{39} \mathrm{~K}\) and \({ }_{19}^{40} \mathrm{~K},(\mathrm{~b})^{209} \mathrm{Bi}\) and \({ }^{208} \mathrm{Bi}\), (c) nickel-58 and nickel-65. Explain.
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