Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 18: Problem 2546

The activity of a radioactive sample is measured as \(\mathrm{N}_{0}\) counts per minute at \(\mathrm{t}=0\) and \(\left(\mathrm{N}_{0} / \mathrm{e}\right)\) counts Per minute at \(\mathrm{t}=5 \mathrm{~min} .\) The time (in min) at which activity reduces to half its value is (A) log e \((2 / 5)\) (B) \(5 \log _{10} 2\) (C) \(5 \log _{\mathrm{e}} 2\) (D) \(\log _{10}^{(2 / 5)}\)

Short Answer

Expert verified
The time at which the activity reduces to half of its initial value is \(5\ln2\) minutes. The correct answer is (C) \(5 \log_{e} 2\).

Step by step solution

01

Understand the decay law formula

The decay law formula is given by \[N(t) = N_{0} \cdot e^{-\lambda t}\], where - \(N(t)\) is the remaining activity at time \(t\), - \(N_{0}\) is the initial activity, - \(\lambda\) is the decay constant, and - \(t\) is the elapsed time. We are given \(N_0\) and \(N(t)\) at different points in time. We will use this information to determine the decay constant \(\lambda\).
02

Calculate the decay constant \(\lambda\)

We have the following information: - At \(t = 0\), the sample activity is \(N_0\). - At \(t = 5\) minutes, the sample activity is \(N_0 / e\). Using the decay law formula for \(t = 5\) minutes, we get: \[\frac{N_0}{e} = N_0 \cdot e^{-\lambda (5)}\] Divide both sides by \(N_0\): \[\frac{1}{e} = e^{-5\lambda}\] Now take the natural logarithm of both sides: \[-1 = -5\lambda\] Solving for \(\lambda\), we get: \[\lambda = \frac{1}{5}\]
03

Calculate the half-life time

The half-life time, denoted as \(t_{1/2}\), is the time required for the activity to reduce to half its initial value. Mathematically, this means that \(N(t_{1/2}) = \frac{N_0}{2}\). Using the decay law formula, we can write: \[\frac{N_0}{2} = N_0 \cdot e^{-\lambda t_{1/2}}\] Divide both sides by \(N_0\): \[\frac{1}{2} = e^{-\frac{1}{5}t_{1/2}}\] Take the natural logarithm of both sides: \[-\ln2 = -\frac{1}{5}t_{1/2}\] Solving for \(t_{1/2}\), we get: \[t_{1/2} = 5\ln2\] So, the time at which the activity reduces to half of its initial value is \(5\ln2\) minutes. The correct answer is (C) \(5 \log_{e} 2\).

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

which of the following series in the spectrum of hydrogen atom lies in the visible legion of the electro magnetic spectrum? (A) Paschen (B) Lyman (C) Brakett (D) Balmer

The innermost orbit of the hydrogen atom has a radius \(0.53 \mathrm{~A}\). what is radius of \(2^{\text {nd }}\) orbit is ? (A) \(2.12 \AA\) (B) \(1.06 \AA\) (C) \(21.2 \AA\) (D) \(10.6 \AA\)

It the radius of a nucleus of mass number 3 is \(\mathrm{R}\). then the radius of a nucleus of mass number 81 is (A) \(27 \mathrm{R}\) (B) \(9 \mathrm{R}\) (C) \(3 \mathrm{R}\) \((\mathrm{D})(27)^{(1 / 2)} \mathrm{R}\)

A nucleus of \({ }^{210}{ }_{84}\) Po originally at rest emits \(\alpha\) -particle with speed \(\mathrm{v}\) what will be the recoil speed of the daughter nucleus (A) \([\mathrm{v} /(214)]\) (B) \([(4 \mathrm{v}) /(214)]\) (C) \([(4 \mathrm{v}) /(206)]\) (D) \([\mathrm{v} /(206)]\)

Match column I and II column I \(\frac{\text { column I }}{\text { fnucleus }}\) (a) size of (b) number of Proton in a nucleus column II (p) Z (q) \(10^{-15} \mathrm{~m}\) (r) \((\mathrm{A}-\mathrm{Z})\) (s) \(10^{-10} \mathrm{~m}\) 0 (c) size of Atom (d) Number of neutrons in a nucleus (A) $\mathrm{a} \rightarrow \mathrm{r}, \mathrm{b} \rightarrow \mathrm{s}, \mathrm{c} \rightarrow \mathrm{q}, \mathrm{d} \rightarrow \mathrm{p}$ (B) $\mathrm{a} \rightarrow \mathrm{q}, \mathrm{b} \rightarrow \mathrm{p}, \mathrm{c} \rightarrow \mathrm{s}, \mathrm{d} \rightarrow \mathrm{r}$ (C) $\mathrm{a} \rightarrow \mathrm{s}, \mathrm{b} \rightarrow \mathrm{r}, \mathrm{c} \rightarrow \mathrm{q}, \mathrm{d} \rightarrow \mathrm{p}$ (D) $\mathrm{b} \rightarrow \mathrm{s}, \mathrm{c} \rightarrow \mathrm{p}, \mathrm{c} \rightarrow \mathrm{q}, \mathrm{d} \rightarrow \mathrm{r}$
See all solutions

Recommended explanations on English Textbooks

Text Comparison

Read Explanation

Linguistic Terms

Read Explanation

Sociolinguistics

Read Explanation

Graphology

Read Explanation

Language Acquisition

Read Explanation

Discourse

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