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Chapter 20: Problem 60

Sodium- 24 is used in medicine to study the circulatory system. A sample weighing \(5.2 \times 10^{-6} \mathrm{~g}\) has an activity of \(45.3 \mathrm{Ci}\). What is the decay constant of sodium- 24 (in /s)?

Short Answer

Expert verified
The decay constant of sodium-24 is approximately \(1.288 \times 10^{-5} \text{ /s}\).

Step by step solution

01

Understanding the Given

We are given the mass of sodium-24 as \(5.2 \times 10^{-6} \text{ g}\) and its activity as \(45.3 \text{ Ci}\). Our goal is to find its decay constant \(\lambda\).
02

Convert Activity to SI Unit

Activity in Curies (Ci) needs to be converted to disintegrations per second. Since \(1 \text{ Ci = } 3.7 \times 10^{10} \text{ disintegrations/second}\), we convert \(45.3 \text{ Ci}\) as follows: \[A = 45.3 \times 3.7 \times 10^{10} \text{ disintegrations/second} = 1.6761 \times 10^{12} \text{ disintegrations/second}\]
03

Determine the Number of Atoms in the Sample

The number of atoms \(N\) in the sample is found by using the relationship: \[N = \frac{\text{mass}}{\text{molar mass}} \times N_A\] where the molar mass of sodium-24 is \(24 \text{ g/mol}\) and \(N_A\) is Avogadro's number \(6.022 \times 10^{23} \text{ atoms/mol}\).
04

Calculate the Number of Atoms

Using the formula: \[N = \frac{5.2 \times 10^{-6}}{24} \times 6.022 \times 10^{23}\] This gives us \[N = 1.301 \times 10^{17} \text{ atoms}\]
05

Apply the Decay Constant Formula

The decay constant \(\lambda\) is given by the equation: \[ \lambda = \frac{A}{N} \] Substituting the known values: \[ \lambda = \frac{1.6761 \times 10^{12}}{1.301 \times 10^{17}} \approx 1.288 \times 10^{-5} \text{ /s}\]
06

Final Solution

The decay constant of sodium-24 is \(1.288 \times 10^{-5} \text{ /s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decay Constant
The decay constant is a crucial concept in understanding radioactive decay. It is denoted by the Greek letter \(\lambda\). The decay constant represents the probability per unit time that a given nucleus will decay. It quantifies the rate at which a radioactive substance undergoes disintegration.

Mathematically, it is expressed as:
  • \(\lambda = \frac{A}{N}\)
Where:
  • \(A\) is the activity or the disintegration rate of the substance, expressed in disintegrations per second.
  • \(N\) is the number of radioactive nuclei present in the sample.
A higher decay constant means a faster rate of decay for the isotope, whereas a lower decay constant indicates a slower decay. This parameter is essential to understand the longevity and stability of radioactive materials.
Disintegration Rate
The disintegration rate, also known as activity, signifies how many nuclei of a radioactive substance decay per unit time.

It is commonly measured in Curie (Ci) or Becquerel (Bq) where:
  • 1 Ci = \(3.7 \times 10^{10}\) disintegrations per second
  • 1 Bq = 1 disintegration per second
To convert from Curie to disintegrations per second, the conversion factor of \(3.7 \times 10^{10}\) is used, as shown in the exercise. This conversion is essential for computations involving the decay constant, where the activity is a prime contributor.

A higher disintegration rate indicates a more radioactive isotope, meaning it is decaying quickly and thus has a higher short-term activity.
Isotopes
Isotopes are variants of a particular chemical element that have the same number of protons but differ in the number of neutrons. This difference in neutron number leads to different atomic masses.

Sodium-24, as used in the exercise, is an isotope of sodium. Sodium's common isotope is Sodium-23. The number after the element symbol often indicates the total number of protons and neutrons (mass number).

Key characteristics of isotopes include:
  • Similar chemical properties because they have the same electron configuration
  • Diverse physical properties due to differing mass numbers
Radioactive isotopes, such as Sodium-24, are used in various fields, from medical diagnostics to understanding industrial processes, because of their decay properties.
Molar Mass
Molar mass is the mass of one mole of a given substance, typically measured in grams per mole (g/mol). It represents a link between the mass of a substance and the number of particles it contains, thanks to Avogadro's number, \(6.022 \times 10^{23}\) particles/mol.

The molar mass plays a critical role in converting the mass of a substance to the number of moles, as demonstrated in the exercise:
  • \(N = \frac{\text{mass}}{\text{molar mass}} \times N_A\)
Where \(N_A\) is Avogadro's number.

Knowing the molar mass of a radioactive isotope like Sodium-24 (24 g/mol) helps determine the number of atoms present in a sample, which is pivotal in calculating its decay constant.

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Most popular questions from this chapter

What are the two types of nuclear reactions? Give an example of a nuclear equation for each type.

The radioactive isotope phosphorus- 32 is often used in biochemical research. Its half-life is 14.28 days and it decays by beta emission. a Write a balanced equation for the decomposition of \(\mathrm{P}-32\). b If the original sample were \(275 \mathrm{mg}\) of \(\mathrm{K}_{3}{ }^{32} \mathrm{PO}_{4},\) what amount remains after \(37.0 \mathrm{~d}\) ? What percent of the sample has undergone decay?

The bromine isotope \(\mathrm{Br}-75\) is used for imaging in positron emission tomography. Br-75 has a half-life of \(57 \mathrm{~h}\). How much time must elapse for a given dose of \(\mathrm{Br}-75\) to drop to \(25 \%\) of its initial amount?

A sample of sodium- 24 was administered to a patient to test for faulty blood circulation by comparing the radioactivity reaching various parts of the body. What fraction of the sodium- 24 nuclei would remain undecayed after \(12.0 \mathrm{~h} ?\) The half-life is \(15.0 \mathrm{~h}\). If a sample contains \(6.0 \mu \mathrm{g}\) of \({ }^{24} \mathrm{Na}\), how many micrograms remain after \(12.0 \mathrm{~h}\) ?

The gold-198 isotope is used in the treatment of brain, prostate, and ovarian cancer. Au-198 has a half-life of \(2.69 \mathrm{~d}\). If a hospital needs to have \(25 \mathrm{mg}\) of \(\mathrm{Au}-198\) on hand for treatments on a particular day, and shipping takes \(72 \mathrm{~h},\) what mass of \(\mathrm{Au}-198\) needs to be ordered?
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