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Chapter 19: Problem 31

The rate constant for a certain radioactive nuclide is $1.0 \times 10^{-3} \mathrm{h}^{-1} .$ What is the half-life of this nuclide?

Short Answer

Expert verified
The half-life (T) of the radioactive nuclide can be found using the formula \(T = \frac{\ln(2)}{k}\), where k is the rate constant. Given the rate constant \(k = 1.0 \times 10^{-3} h^{-1}\), the half-life of the nuclide is approximately \(T = \frac{\ln(2)}{1.0 \times 10^{-3} h^{-1}} \approx 693\) hours.

Step by step solution

01

1. Write down the given information and formula.

We are given the rate constant (k) for a certain radioactive nuclide: \(k = 1.0 \times 10^{-3} h^{-1}\). We want to find the half-life (T) of this nuclide. The formula connecting these values is: \(T = \frac{\ln(2)}{k}\)
02

2. Substitute the rate constant into the formula.

Now we can plug in the value of the rate constant k into the formula: \(T = \frac{\ln(2)}{1.0 \times 10^{-3} h^{-1}}\)
03

3. Calculate the half-life (T).

Divide the natural logarithm of 2 by the rate constant to find the half-life of the nuclide: \(T = \frac{\ln(2)}{1.0 \times 10^{-3} h^{-1}} \approx \frac{0.693}{1.0 \times 10^{-3} h^{-1}} = 693 \, hours\)
04

4. State the final answer.

The half-life of this radioactive nuclide is approximately 693 hours.

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Most popular questions from this chapter

In the bismuth-214 natural decay series, Bi-214 initially undergoes \(\beta\) decay, the resulting daughter emits an \(\alpha\) particle, and the succeeding daughters emit a \(\beta\) and a \(\beta\) particle in that order. Determine the product of each step in the Bi-214 decay series.

A \(0.10-\mathrm{cm}^{3}\) sample of a solution containing a radioactive nuclide $\left(5.0 \times 10^{3} \text { counts per minute per milliter) is injected }\right.\( into a rat. Several minutes later 1.0 \)\mathrm{cm}^{3}$ of blood is removed. The blood shows 48 counts per minute of radioactivity. Calculate the volume of blood in the rat. What assumptions must be made in performing this calculation?

The number of radioactive nuclides in a sample decays from $1.00 \times 10^{20}\( to \)2.50 \times 10^{19}$ in 10.0 minutes. What is the half-life of this radioactive species?

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.

Breeder reactors are used to convert the nonfissionable nuclide 238 \(\mathrm{U}\) to a fissionable product. Neutron capture of the 238 \(\mathrm{U}\) is followed by two successive beta decays. What is the final fissionable product?
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