Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 21: Problem 74

From the definition of curie, calculate Avogadro's number, given that the molar mass of \({ }^{226} \mathrm{Ra}\) is \(226.03 \mathrm{~g} / \mathrm{mol}\) and that it decays with a half-life of \(1.6 \times 10^{3} \mathrm{yr}\)

Short Answer

Expert verified
Avogadro's number is approximately \(6.022 \times 10^{23} \, atoms/mol\).

Step by step solution

01

Calculate Decay Rate from Curie Definition

First, we know that 1 curie (Ci) is defined as \(3.7 \times 10^{10}\) disintegrations per second. Hence, the decay rate is \(3.7 \times 10^{10}\) decays/sec.
02

Calculate Decay Constant Using Half-Life

The decay constant (\(\lambda\)) can be calculated using the formula: \(\lambda = \frac{0.693}{\text{Half-life}}\). Here, the half-life of \({ }^{226} \mathrm{Ra}\) is \(1.6 \times 10^{3} \mathrm{yr}\). Converting this to seconds (\(1 \mathrm{yr} = 3.15 \times 10^{7} \mathrm{s}\)), we get \(1.6 \times 10^{3} \times 3.15 \times 10^{7} = 5.04 \times 10^{10} \mathrm{s}\). Thus, \(\lambda = \frac{0.693}{5.04 \times 10^{10} s}\).
03

Apply Decay Rate formula

The formula for decay rate (Activity) = Number of Atoms x Decay Constant, can be rearranged to find the Number of Atoms = Decay Rate / Decay Constant. Substituting the values we already found, we can calculate the number of atoms.
04

Calculate Avogadro's number

Finally, using the molar mass of \(^{226}\)Ra, \(226.03 \, g/mol\), convert the mass into moles to yield the value of Avogadro's number.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Cobalt- 60 is an isotope used in diagnostic medicine and cancer treatment. It decays with \(\gamma\) ray emission. Calculate the wavelength of the radiation in nanometers if the energy of the \(\gamma\) ray is \(2.4 \times 10^{-13} \mathrm{~J} / \mathrm{photon}\)

The quantity of a radioactive material is often measured by its activity (measured in curies or millicuries) rather than by its mass. In a brain scan procedure, a 70 -kg patient is injected with \(20.0 \mathrm{mCi}\) of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) which decays by emitting \(\gamma\) -ray photons with a halflife of \(6.0 \mathrm{~h}\). Given that the \(\mathrm{RBE}\) of these photons is 0.98 and only two-thirds of the photons are absorbed by the body, calculate the rem dose received by the patient. Assume all of the \({ }^{99 \mathrm{~m}}\) Tc nuclei decay while in the body. The energy of a gamma photon is \(2.29 \times 10^{-14} \mathrm{~J}\).

Each molecule of hemoglobin, the oxygen carrier in blood, contains four Fe atoms. Explain how you would use the radioactive \({ }_{26}^{59} \mathrm{Fe}\left(t_{\frac{1}{2}}=46\right.\) days) to show that the iron in a certain food is converted into hemoglobin.

Explain the functions of a moderator and a control rod in a nuclear reactor.

To detect bombs that may be smuggled onto airplanes, the Federal Aviation Administration (FAA) will soon require all major airports in the United States to install thermal neutron analyzers. The thermal neutron analyzer will bombard baggage with low-energy neutrons, converting some of the nitrogen- 14 nuclei to nitrogen- \(15,\) with simultaneous emission of \(\gamma\) rays. Because nitrogen content is usually high in explosives, detection of a high dosage of \(\gamma\) rays will suggest that a bomb may be present. (a) Write an equation for the nuclear process. (b) Compare this technique with the conventional X-ray detection method.
See all solutions

Recommended explanations on Chemistry Textbooks

Kinetics

Read Explanation

Chemical Analysis

Read Explanation

Physical Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Inorganic Chemistry

Read Explanation

Ionic and Molecular Compounds

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free