/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 112 The gold-198 isotope is used in ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 112

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 \(15 \mathrm{mg}\) of Au- 198 on hand for treatments on a particular day, and shipping takes \(72 \mathrm{~h}\), what mass of Au- 198 needs to be ordered?

Short Answer

Expert verified
32.69 mg of Au-198 should be ordered.

Step by step solution

01

Understand the Half-life Concept

The half-life of a radioactive isotope is the time required for one-half of the radioactive nuclei in a sample to decay. For Gold-198 (Au-198), the half-life is given as 2.69 days.
02

Convert Time Units

We know the shipping time is 72 hours. First, convert this time to days since the half-life is given in days. There are 24 hours in a day, so 72 hours is equivalent to \( \frac{72}{24} = 3 \) days.
03

Calculate Number of Half-lives Passed

Determine the number of half-lives that have passed during the shipping period. This is calculated by dividing the shipping time in days by the half-life of Au-198. Number of half-lives = \( \frac{3}{2.69} \approx 1.115 \) half-lives.
04

Apply the Radioactive Decay Formula

Use the formula for radioactive decay: \[ N = N_0 \left( \frac{1}{2} \right)^{n} \] where \( N \) is the final amount of substance, \( N_0 \) is the initial amount, and \( n \) is the number of half-lives. Since we need 15 mg after decay, and \( n \approx 1.115 \), solve for \( N_0 \): \[ 15 = N_0 \left( \frac{1}{2} \right)^{1.115} \].
05

Calculate the Initial Amount

Rearrange the formula to find \( N_0 \): \[ N_0 = \frac{15}{\left( \frac{1}{2} \right)^{1.115}} \]. Calculating further, we find \( N_0 \approx 15 \times 2.179 \approx 32.685 \) mg.
06

Conclusion

Considering the decay during shipping, approximately 32.69 mg of Au-198 needs to be ordered to ensure 15 mg is available for treatment after the shipping time.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Half-life
Half-life is a fundamental concept in nuclear chemistry, particularly relevant for understanding how radioactive substances decay over time. It refers to the period it takes for half the nuclei in a radioactive sample to undergo decay. In simpler terms, if you start with 100 units of a radioactive substance, after one half-life, you will have 50 units remaining.

This concept is essential for calculating how much of a substance remains after a certain amount of time, as it provides a predictable pattern of decay. When dealing with medical isotopes like Gold-198, knowing the half-life ensures precise dosages can be calculated for treatments.
  • Predictable: Half-life provides a dependable way to anticipate the decay process.
  • Consistent: The time span for each half-life remains constant.
  • Measurement: Often reported in seconds, minutes, hours, days, or even years, depending on the substance.
Gold-198
Gold-198, symbolized as Au-198, is a radioactive isotope of gold. It is commonly used in medical therapies, particularly for cancer treatment, due to its decay properties and half-life.

Au-198 emits beta particles and gamma rays, which are effective in damaging cancer cells. It has a half-life of 2.69 days, making it suitable for medical situations requiring radiation over a short period.
  • Application: Used in targeted radiation treatments, minimizing damage to surrounding tissues.
  • Handling: Requires careful transport and usage, given its radioactive nature.
Understanding the properties of Au-198 is critical for healthcare providers to plan and administer precise treatments.
Radioactive Isotopes
Radioactive isotopes, or radioisotopes, are variants of a chemical element that exhibit radioactivity. They decay over time, emitting radiation in the form of alpha, beta, or gamma rays.

These isotopes have diverse applications, from medical treatments to industrial uses like radiographic inspection.
  • Stability: Radioisotopes are inherently unstable, leading to the emission of radiation as they seek a stable state.
  • Usage: In medicine, they track and diagnose conditions, as well as in treatment protocols.
Understanding radioactive isotopes is fundamental in fields like medical imaging and cancer therapy, where controlled radiation is utilized for diagnostics and treatment.
Nuclear Chemistry
Nuclear chemistry is a branch of chemistry that deals with the reactions and properties of atomic nuclei. It plays a critical role in understanding radioactive decay processes and the use of isotopes in various fields.

Nuclear chemistry involves the study of:
  • Nuclear reactions: Such as fission and fusion, where atomic nuclei are split or combined.
  • Radioactivity: The spontaneous emission of particles from an unstable nucleus.
  • Nuclear decay: The process of a radioactive substance losing energy over time.
In real-world applications, nuclear chemistry is essential for the development of nuclear energy, medical isotopes for treatment and diagnostics, and in environmental monitoring. It helps scientists to harness the power of the nucleus for beneficial uses, while also offering insights into the risks associated with radiation exposure.

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

You have two piles of different unknown radioactive substances: pile \(A\) with a mass of \(200 \mathrm{~g}\), and pile \(\mathrm{B}\) with a mass of \(100 \mathrm{~g} .\) Would it be possible for these two piles to have the same rate of radioactive decay? Explain.

A solution of sodium iodide containing iodine-131 was given to a patient to test for malfunctioning of the thyroid gland. What fraction of the iodine-131 nuclei would remain undecayed after \(7.0 \mathrm{~d}\) ? If a sample contains \(2.5 \mu \mathrm{g}\) of \({ }^{131} \mathrm{I}\), how many micrograms remain after \(7.0 \mathrm{~d}\) ? The half-life of I-131 is \(8.07 \mathrm{~d}\).

A bismuth-209 nucleus reacts with a deuteron to produce a polonium nucleus and a neutron. Write the complete nuclear equation for this reaction.

The half-life of cesium-137 is \(30.2 \mathrm{y}\). How long will it take for a sample of cesium-137 to decay to \(1 / 8\) of its original mass?

Cobalt-60 has a half-life of \(5.26\) y. Gamma radiation from this isotope is used to treat malignant tumors. What mass of cobalt- 60 is required to generate a 500 -Ci gamma-ray source?
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Analysis

Read Explanation

Nuclear Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Chemistry Branches

Read Explanation

Organic Chemistry

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.