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Chapter 18: Problem 78

Which of the following statement(s) is(are) true? a. A radioactive nuclide that decays from \(1.00 \times 10^{10}\) atoms to \(2.5 \times 10^{9}\) atoms in 10 minutes has a half-life of 5.0 minutes.b. Nuclides with large \(Z\) values are observed to be \(\alpha\) -particle producers. c. As \(Z\) increases, nuclides need a greater proton-to-neutron ratio for stability. d. Those "light" nuclides that have twice as many neutrons as protons are expected to be stable.

Short Answer

Expert verified
Statements B and C are true. Statement A is false as the calculated half-life is not equal to 5.0 minutes, and statement D is false because having twice as many neutrons as protons does not guarantee stability even for light nuclides.

Step by step solution

01

Statement A

To verify this statement, we need to calculate the half-life of the nuclide. The relationship between initial number of atoms \(N_0\), number of atoms after time 't' \(N_t\), decay constant \(\lambda\) and time 't' is given by: \( N_t = N_0 \cdot e^{-\lambda t} \) Here, \(N_0 = 1.00 \times 10^{10}\), \(N_t = 2.5 \times 10^{9}\), and \(t = 10\) minutes. Substituting these values and solving for \(\lambda\), we get: \(\lambda = 1/t \cdot ln(N_0/N_t) \) The half-life \(T_{1/2}\) is given by: \( T_{1/2} = ln(2) / \lambda \) If the calculated \(T_{1/2}\) is equal to 5.0 minutes, then the statement is true; otherwise, it is false.
02

Statement B

This statement is generally true. Nuclides with large atomic numbers are frequently unstable and undergo alpha decay. Alpha decay is a common form of radioactive decay in which a nucleus emits an alpha particle, reducing its atomic number by two and atomic mass by four. So, we can consider this statement as true.
03

Statement C

This statement is also true. In general, for stable nuclides, the number of neutrons equals (or is slightly more than) the number of protons for light elements. However, as the atomic number increases (which implies large 'Z' values), nuclides need more neutrons than protons for maintaining stability. This is due to the fact that the repulsive forces between protons (which are all positively charged) need to be balanced by the strong nuclear force provided by the neutrons.
04

Statement D

This statement is false. Although as the atomic number increases, nuclides require more neutrons than protons for maintaining stability, having twice as many neutrons as protons does not guarantee stability, even for light nuclides. It is generally seen that stable nuclides have a neutron-proton ratio close to 1 for light nuclides and somewhat greater than 1 for heavy nuclides.

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Most popular questions from this chapter

Define "third-life" in a similar way to "half-life," and determine the "third- life" for a nuclide that has a half-life of 31.4 years.

In addition to the process described in the text, a second process called the carbon-nitrogen cycle occurs in the sun:a. What is the catalyst in this process? b. What nucleons are intermediates? c. How much energy is released per mole of hydrogen nuclei in the overall reaction? (The atomic masses of \(i \mathrm{H}\) and 4 He are 1.00782 u and 4.00260 u, respectively.)

In 1994 it was proposed (and eventually accepted) that element 106 be named seaborgium, \(\mathrm{Sg}\), in honor of Glenn T. Seaborg, discoverer of the transuranium elements.a. \(^{263} \mathrm{Sg}\) was produced by the bombardment of \(^{249} \mathrm{Cf}\) with a beam of \(^{18} \mathrm{O}\) nuclei. Complete and balance an equation for this reaction. b. \(^{263}\) Sg decays by \(\alpha\) emission. What is the other product resulting from the \(\alpha\) decay of \(^{263} \mathrm{Sg} ?\)

A certain radioactive nuclide has a half-life of 3.00 hours. a. Calculate the rate constant in \(s^{-1}\) for this nuclide. b. Calculate the decay rate in decays/s for 1.000 mole of this nuclide.

Write an equation describing the radioactive decay of each of the following nuclides. (The particle produced is shown in parentheses, except for electron capture, where an electron is a reactant.) a. \(\frac{3}{1} \mathrm{H}(\beta)\) b. \(\frac{8}{3} L i(\beta \text { followed by } \alpha)\) c. \(^{7}_{4}\) Be (electron capture)d. \(^{8}_{5} B\) (positron)
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