Predict whether each of the following nuclides is stable or unstable (radioactive). If the nuclide is unstable, predict the type of radioactivity you would expect it to exhibit. a. \(_{19}^{45} \mathrm{K}\) b. \(_{56}^{26} \mathrm{Fe}\) c. \(_{20}^{11} \mathrm{Na}\) d. \(_{194}^{81} \mathrm{TI}\)

Short Answer

Expert verified
a. $_{19}^{45} \mathrm{K}$: Unstable, likely to undergo beta-minus decay. b. $_{56}^{26} \mathrm{Fe}$: Highly unstable, expected to undergo beta-plus decay or electron capture. c. $_{20}^{11} \mathrm{Na}$: Highly unstable, expected to undergo beta-plus decay or electron capture. d. $_{194}^{81} \mathrm{Tl}$: Unstable, likely to undergo beta-minus decay or, less likely, alpha decay.

Step by step solution

01

a. \(_{19}^{45} \mathrm{K}\)#

To analyze potassium-45 (K-45), we first determine the number of neutrons and the neutron-to-proton ratio. The neutron number (N) is found by subtracting the atomic number (Z) from the mass number (A): N = A - Z = 45 - 19 = 26. Thus, the N/Z ratio is 26/19 ≈ 1.37. Since Z ≤ 20 and the N/Z ratio is larger than 1, K-45 is expected to be unstable and undergo beta-minus decay, as it has more neutrons than required for stability.
02

b. \(_{56}^{26} \mathrm{Fe}\)#

For iron-26 (Fe-26), we follow the same process. We have N = A - Z = 26 - 56 = -30. In this case, the N/Z ratio is negative (-30/56), which is an abnormality and indicates this nuclide is highly unstable. Due to the low number of neutrons, Fe-26 is expected to undergo beta-plus decay or electron capture to increase its neutron-to-proton ratio and reach stability.
03

c. \(_{20}^{11} \mathrm{Na}\)#

When analyzing sodium-11 (Na-11), we determine N = A - Z = 11 - 20 = -9. The N/Z ratio is therefore -9/20. Similar to Fe-26, this nuclide is highly unstable and has too few neutrons. Na-11 is also expected to undergo beta-plus decay or electron capture in order to achieve stability.
04

d. \(_{194}^{81} \mathrm{Tl}\)#

Lastly, for thallium-194 (Tl-194), we obtain N = A - Z = 194 - 81 = 113. The N/Z ratio is 113/81 ≈ 1.4. Since Z > 20, the stable N/Z ratio should be around 1.5. Due to the imbalance in the N/Z ratio and the large value of Z, Tl-194 is expected to undergo beta-minus decay or, less likely, alpha decay, in order to reach stability.

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Most popular questions from this chapter

Consider the following graph of binding energy per nucleon as a function of mass number a. What does this graph tell us about the relative half-lives of the nuclides? Explain your answer. b. Which nuclide shown is the most thermodynamically stable? Which is the least thermodynamically stable? c. What does this graph tell us about which nuclides undergo fusion and which undergo fission to become more stable? Support your answer.

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