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Chapter 21: Problem 33

Each statement that follows refers to a comparison between two radioisotopes, \(A\) and \(X .\) Indicate whether each of the following statements is true or false. (a) If the half-life for \(\mathrm{A}\) is shorter than the half-life for \(\mathrm{X}, \mathrm{A}\) has a larger decay rate constant. (b) If \(X\) is "not radioactive," its half-life is essentially zero. (c) If A has a half-life of 10 yr, and \(X\) has a half-life of $10,000 \mathrm{yr}$, A would be a more suitable radioisotope to measure processes occurring on the 40 -yr time scale.

Short Answer

Expert verified
(a) True - A shorter half-life implies a larger decay rate constant. (b) False - A non-radioactive isotope has an infinite half-life, not zero. (c) True - Isotope A is more suitable for measuring processes on a 40-year time scale.

Step by step solution

01

(a: Half-life vs. Decay rate constant relationship)

Since the relationship between half-life and the decay constant is \(T=\frac{ln(2)}{\lambda}\), we can deduce that the half-life and the decay rate constant are inversely related. Therefore, if the half-life of isotope A is shorter than the half-life of isotope X, the decay rate constant for A would be larger. So, the statement is true.
02

(b: Half-life of non-radioactive isotope)

If isotope X is "not radioactive," that means it does not decay or disintegrate by emitting radiation, and it has a stable nucleus; hence, its half-life is not essentially zero. Instead, it would be infinite since it would not undergo decay. The statement is false.
03

(c: Suitability for measuring 40-year time scale processes)

We need to determine whether A or X would be more suitable for measuring processes occurring on a 40-year time scale. Given that isotope A has a half-life of 10 years, and isotope X has a half-life of 10,000 years, isotope A with a half-life closer to the given time scale (40 years) would be better suited for measuring the processes of interest, as that would mean more significant changes in its decay in a 40-year period compared to isotope X. The statement is true.

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Most popular questions from this chapter

One nuclide in each of these pairs is radioactive. Predict which is radioactive and which is stable: \((\mathbf{a}){ }_{20}^{40} \mathrm{Ca}\) and \({ }_{20}^{45} \mathrm{Ca},\) (b) \({ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C}\) (c) lead-206 and thorium-230. Explain your choice in each case.

Which type or types of nuclear reactors have these characteristics? (a) Does not use a secondary coolant (b) Creates more fissionable material than it consumes (c) Uses a gas, such as \(\mathrm{He}\) or \(\mathrm{CO}_{2}\), as the primary coolant

The Sun radiates energy into space at the rate of $3.9 \times 10^{26} \mathrm{~J} / \mathrm{s} .$ (a) Calculate the rate of mass loss from the Sun in kg/s. (b) How does this mass loss arise? (c) It is estimated that the Sun contains \(9 \times 10^{56}\) free protons. How many protons per second are consumed in nuclear reactions in the Sun?

A \(26.00-g\) sample of water containing tritium, \({ }_{1}^{3} \mathrm{H},\) emits \(1.50 \times 10^{3}\) beta particles per second. Tritium is a weak beta emitter with a half-life of \(12.3 \mathrm{yr}\). What fraction of all the hydrogen in the water sample is tritium?

The isotope \({ }_{28}^{62} \mathrm{Ni}\) has the largest binding energy per nucleon of any isotope. Calculate this value from the atomic mass of nickel-62 \((61.928345 \mathrm{u})\) and compare it with the value given for iron- 56 in Table 21.7 .
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