Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 40: Problem 69

\(^{214} \mathrm{Pb}\) has a half-life of \(26.8 \mathrm{~min}\). How many minutes must elapse for \(90.0 \%\) of a given sample of \({ }^{214} \mathrm{~Pb}\) atoms to decay?

Short Answer

Expert verified
Answer: It takes approximately 89.16 minutes for 90% of a given sample of 214Pb atoms to decay.

Step by step solution

01

Understand the given information and the half-life formula

We are given the half-life of \(^{214}\mathrm{Pb}\) atoms as 26.8 minutes, and we need to find the time it takes for 90% of the sample to decay. We can use the half-life formula: \(N_t = N_0 \times \left( \frac{1}{2} \right)^\frac{t}{t_{1/2}}\) where: - \(N_t\) is the remaining amount of substance at time \(t\) - \(N_0\) is the initial amount of substance - \(t\) is the time elapsed - \(t_{1/2}\) is the half-life of the substance Since 90% of the substance has decayed, it means that 10% is remaining. So, we can write \(N_t\) as \(0.1N_0\). Now, we need to find the time \(t\).
02

Setup the equation and isolate time

We can rewrite the half-life formula with the given values for the initial and remaining amounts of \(^{214}\mathrm{Pb}\) atoms: \(0.1N_0 = N_0 \times \left( \frac{1}{2} \right)^\frac{t}{26.8}\) Since we are looking for the time, we can eliminate the initial amount \(N_0\) from both sides: \(0.1 = \left( \frac{1}{2} \right)^\frac{t}{26.8}\) Now, we need to solve for \(t\).
03

Solve for time

To solve for \(t\), we can take the natural logarithm of both sides of the equation: \(\ln(0.1) = \ln \left(\frac{1}{2}\right)^{\frac{t}{26.8}}\) Using the properties of logarithms, we can bring the exponent to the front: \(\ln(0.1) = \frac{t}{26.8} \times \ln \left( \frac{1}{2} \right)\) Now, we can solve for \(t\) by isolating it: \(t = \frac{26.8 \times \ln(0.1)}{\ln\left(\frac{1}{2}\right)}\) Calculating the value for \(t\): \(t \approx 89.16 \mathrm{~minutes}\)
04

Conclusion

It takes approximately 89.16 minutes for 90% of a given sample of \(^{214}\mathrm{Pb}\) atoms to decay.

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

In a simple case of chain radioactive decay, a parent radioactive species of nuclei, A, decays with a decay constant \(\lambda_{1}\) into a daughter radioactive species of nuclei, B, which then decays with a decay constant \(\lambda_{2}\) to a stable element C. a) Write the equations describing the number of nuclei in each of the three species as a function of time, and derive an expression for the number of daughter nuclei, \(N_{2}\), as a function of time, and for the activity of the daughter nuclei, \(A_{2},\) as a function of time. b) Discuss the results in the case when \(\lambda_{2}>\lambda_{1}\left(\lambda_{2} \approx 10 \lambda_{1}\right)\) and when \(\lambda_{2}>>\lambda_{1}\left(\lambda_{2} \approx 100 \lambda_{1}\right)\).

Apart from fatigue, what is another reason the Federal Aviation Administration limits the number of hours intercontinental pilots can travel annually?

Before you look it up, make a prediction of the spin (intrinsic spin, i.e., actual angular momentum) of the deuteron, \({ }^{2} \mathrm{H}\). Explain your reasoning.

Consider the following fusion reaction, which allows stars to produce progressively heavier elements: \({ }_{2}^{3} \mathrm{He}+{ }_{2}^{4} \mathrm{He} \rightarrow{ }_{4}^{7} \mathrm{Be}+\gamma\). The mass of \({ }_{2}^{3} \mathrm{He}\) is \(3.016029 \mathrm{u}\), the mass of \({ }_{2}^{4}\) He is \(4.002603 \mathrm{u}\), and the mass of \({ }_{4}^{7} \mathrm{Be}\) is \(7.0169298 \mathrm{u}\). The atomic mass unit is \(u=1.66 \cdot 10^{-27} \mathrm{~kg} .\) Assuming the Be atom is at rest after the reaction and neglecting any potential energy between the atoms and kinetic energy of the He nuclei, calculate the minimum possible energy and maximum possible wavelength of the photon \(\gamma\) that is released in this reaction.

According to the standard notation, find the number of protons, nucleons, neutrons, and electrons of \({ }_{54}^{134}\) Xe.
See all solutions

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Read Explanation

Energy Physics

Read Explanation

Radiation

Read Explanation

Linear Momentum

Read Explanation

Electrostatics

Read Explanation

Waves Physics

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