/*! 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 73 Bismuth- 214 is an \(\alpha\) -c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 21: Problem 73

Bismuth- 214 is an \(\alpha\) -cmitter with a half-life of 19.7 min. A 5.26 -mg sample of the isotope is placed in a sealed, evacuated flask of volume \(20.0 \mathrm{~mL}\) at \(40^{\circ} \mathrm{C}\). Assuming that all the \(\alpha\) particles generated are converted to helium gas and that the other decay product is nonradioactive, calculate the pressure (in \(\mathrm{mmHg}\) ) inside the flask after 78.8 min. Use 214 amu for the atomic mass of bismuth.

Short Answer

Expert verified
The pressure inside the flask after 78.8 minutes is approximately 2.247 mmHg.

Step by step solution

01

Determine Number of Half-Lives

The half-life of Bismuth-214 is 19.7 minutes. To find how many half-lives have occurred in 78.8 minutes, divide the total time by the half-life:\[\text{Number of half-lives} = \frac{78.8 \text{ minutes}}{19.7 \text{ minutes/half-life}} = 4\]
02

Calculate Remaining Bismuth-214

After each half-life, the quantity of the substance is halved. Starting with 5.26 mg, calculate the amount left after 4 half-lives:\[5.26 \text{ mg} \times \left(\frac{1}{2}\right)^4 = 5.26 \text{ mg} \times 0.0625 = 0.32875 \text{ mg}\]
03

Calculate Decayed Bismuth-214

Subtract the remaining amount from the initial amount to find the decayed mass:\[5.26 \text{ mg} - 0.32875 \text{ mg} = 4.93125 \text{ mg}\]
04

Determine Moles of Decayed Bismuth

Use the atomic mass of bismuth (214 amu) to convert the decayed mass from mg to moles:\[\text{Moles of Bismuth} = \frac{4.93125 \times 10^{-3} \text{ g}}{214 \text{ g/mol}} = 2.304 \times 10^{-5} \text{ moles}\]
05

Calculate Moles of Helium

Since each decay produces one helium atom, the moles of helium produced is equal to the moles of decayed bismuth:\[\text{Moles of Helium} = 2.304 \times 10^{-5}\]
06

Use Ideal Gas Law to Find Pressure

Use the ideal gas law \( PV = nRT \) to find the pressure. Convert volume to liters and use \( R = 0.0821 \text{ L atm/mol K} \) and temperature in Kelvin \( 40^{\circ} C = 313 \, K \):\[P \times 0.020 \text{ L} = 2.304 \times 10^{-5} \times 0.0821 \times 313\]Solving for \( P \):\[P = \frac{2.304 \times 10^{-5} \times 0.0821 \times 313}{0.020} = 2.957 \times 10^{-3} \text{ atm}\]Convert atm to mmHg using \( 1 \text{ atm} = 760 \text{ mmHg} \):\[P = 2.957 \times 10^{-3} \times 760 = 2.247 \text{ mmHg}\]
07

Verify Calculations

Review each step to confirm calculations and ensure that all assumptions align with provided data, especially regarding the decay reaction and gas law application.

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.

Alpha Decay
Alpha decay is a type of radioactive decay where an atomic nucleus emits an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, essentially forming a helium nucleus. This emission reduces the atomic mass by 4 units and the atomic number by 2, changing the element into a new one.
  • In alpha decay, heavy elements often become different elements since their nucleus loses both mass and protons.
  • This process contributes to the stability of an atom by helping it achieve a more balanced energy state.
  • While alpha particles are comparatively large and do not penetrate materials deeply, they nonetheless pose significant health risks if inhaled or ingested.
Understanding alpha decay is essential in calculations involving isotopes like Bismuth-214, which is an alpha emitter, meaning it releases alpha particles during its decay process.
Half-life Calculation
The half-life of a radioactive substance is the time it takes for half of any given sample of the substance to decay. In exercises involving half-life, we often calculate how many half-lives have occurred to determine the remaining undecayed portion of the substance.
  • Half-life is a constant for a particular substance and does not change. For instance, the half-life of Bismuth-214 is always 19.7 minutes.
  • In decay calculations, one can use the formula: \[ \text{Remaining Amount} = \text{Initial Amount} \times \left(\frac{1}{2}\right)^{\text{Number of Half-Lives}} \]
  • This exponential decay implies that with each passing half-life, the rate of decrease slows down proportionately.
In the exercise, understanding half-life allowed us to determine the remaining mass of Bismuth-214 after a specified period, which aids further calculations.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry that relates pressure, volume, temperature, and the amount of gas. The equation can be written as: \[ PV = nRT \] where
  • \( P \) represents pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature in Kelvin.
  • It assumes that gas particles are in constant, random motion and do not attract or repel each other significantly.
  • Using this law, we can find unknown variables, such as pressure, if the rest are known.
In the Bismuth-214 decay scenario, the Ideal Gas Law was instrumental in calculating the pressure of the helium produced from the decayed bismuth in the flask.
Bismuth-214 Decay
Bismuth-214 is a radioactive isotope of bismuth, decaying primarily through alpha decay. It is part of the uranium series and eventually decays into stable lead-206.
  • Bismuth-214's decay releases energy in the form of alpha particles, along with non-radioactive products such as helium gas.
  • The decay process is rapid with a short half-life of just 19.7 minutes, indicating that it transforms significantly within a brief period.
  • In radiological terms, such a short half-life implies high activity, meaning it undergoes many decay events in a given time frame.
Understanding the decay of Bismuth-214 aids in solving problems involving radioactive samples and their transformation into non-radioactive forms.

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

The radioactive potassium- 40 isotope decays to argon- 40 with a half-life of \(1.2 \times 10^{9}\) yr. (a) Write a balanced equation for the reaction. (b) A sample of moon rock is found to contain 18 percent potassium -40 and 82 percent argon by mass. Calculate the age of the rock in years.

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

These cquations are for nuclear reactions that are known to occur in the explosion of an atomic bomb. Identify \(X\). (a) \({ }^{235} \mathrm{U}+{ }^{1} \mathrm{n} \longrightarrow{ }_{56}^{140} \mathrm{Ba}+3_{0}^{1} \mathrm{n}+\mathrm{X}\) (b) \({ }^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{53}^{144} \mathrm{Cs}+{ }_{37}^{90} \mathrm{Rb}+2 \mathrm{X}\) (c) \({ }^{235} \mathrm{U}+{ }^{1} \mathrm{n} \longrightarrow{ }^{\mathrm{s} 7} \mathrm{Br}+3{ }^{1} \mathrm{n}+\mathrm{X}\) (d) \({ }^{235} \mathrm{U}+{ }^{1} \mathrm{n} \longrightarrow{ }^{160} \mathrm{Sm}+\frac{72}{30} \mathrm{Zn}+4 \mathrm{X}\)

Define nuclear binding energy, mass defect, and nucleon.

Tritium, \({ }^{3} \mathrm{H}\), is radioactive and decays by electron emission. Its half-life is \(12.5 \mathrm{yr}\). In ordinary water the ratio of \({ }^{1} \mathrm{H}\) to \({ }^{3} \mathrm{H}\) atoms is \(1.0 \times 10^{17}\) to \(1 .(\mathrm{a})\) Write \(\mathrm{a}\) balanced nuclear equation for tritium decay. (b) How many disintegrations will be observed per minute in a \(1.00-\mathrm{kg}\) sample of water?
See all solutions

Recommended explanations on Chemistry Textbooks

Making Measurements

Read Explanation

Chemical Reactions

Read Explanation

Organic Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation

Inorganic Chemistry

Read Explanation

Physical 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.