/*! 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 28 The conversion between atomic ma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 19: Problem 28

The conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg}$$ Natural carbon consists of two different isotopes (excluding $^{14} \mathrm{C},$ which is present in only trace amounts). The isotopes have different masses, which is due to different numbers of neutrons in the nucleus; however, the number of protons is the same, and subsequently the chemical properties are the same. The most abundant isotope has an atomic mass of \(12.00 \mathrm{u} .\) When natural carbon is placed in a mass spectrometer, two lines are formed on the photographic plate. The lines show that the more abundant isotope moved in a circle of radius \(15.0 \mathrm{cm},\) while the rarer isotope moved in a circle of radius \(15.6 \mathrm{cm} .\) What is the atomic mass of the rarer isotope? (The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field.)

Short Answer

Expert verified
Answer: The atomic mass of the rarer isotope is 12.43 u.

Step by step solution

01

Write the expression for centripetal force and magnetic force on the ions.

Centripetal force can be written as \(F_c = \frac{mv^2}{r}\), where m is the mass of the ion, v is its speed, and r is the radius of the circle formed in the spectrometer. The force exerted by the magnetic field on the ions can be written as \(F_b = qvB\), where q is the charge of the ion and B is the magnetic field strength. The centripetal force is provided by the magnetic force, so we have \(F_c = F_b\).
02

Set up the ratio between the masses and radii of the isotopes.

Since the forces experienced by both isotopes are the same, we can set up the following ratio: $$\frac{m_1}{m_2}=\frac{r_1}{r_2}= \frac{15.0 \mathrm{cm}}{15.6 \mathrm{cm}}$$
03

Solve for the mass of the rarer isotope.

We know the mass of the most abundant isotope, \(m_1 = 12.00 \, \mathrm{u}\). We can use the above ratio to find the mass of the rarer isotope, \(m_2\): $$m_2 = m_1 \cdot \frac{r_2}{r_1} = 12.00 \, \mathrm{u} \cdot \frac{15.6 \mathrm{cm}}{15.0 \mathrm{cm}}$$ Calculating \(m_2\): $$m_2 = 12.00 \cdot \frac{15.6}{15.0} = 12.43 \, \mathrm{u}$$ So, the atomic mass of the rarer isotope is \(12.43 \, \mathrm{u}\).

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!

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 conversion between atomic mass units and kilograms is $$1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg}$$ A sample containing sulfur (atomic mass 32 u), manganese \((55 \mathrm{u}),\) and an unknown element is placed in a mass spectrometer. The ions have the same charge and are accelerated through the same potential difference before entering the magnetic field. The sulfur and manganese lines are separated by \(3.20 \mathrm{cm},\) and the unknown element makes a line between them that is \(1.07 \mathrm{cm}\) from the sulfur line. (a) What is the mass of the unknown clement? (b) Identify the element.
In an electric motor, a circular coil with 100 turns of radius $2.0 \mathrm{~cm}$ can rotate between the poles of a magnet. When the current through the coil is \(75 \mathrm{~mA},\) the maximum torque that the motor can deliver is \(0.0020 \mathrm{~N} \cdot \mathrm{m} .\) (a) What is the strength of the magnetic field? (b) Is the torque on the coil clockwise or counterclockwise as viewed from the front at the instant shown in the figure?
Two parallel wires in a horizontal plane carry currents \(I_{1}\) and \(I_{2}\) to the right. The wires each have length \(L\) and are separated by a distance \(d\). (a) What are the magnitude and direction of the ficld due to wire 1 at the location of wire \(2 ?\) (b) What are the magnitude and direction of the magnetic force on wire 2 due to this field? (c) What are the magnitude and direction of the field due to wire 2 at the location of wire \(1 ?\) (d) What are the magnitude and direction of the magnetic force on wire 1 due to this field? (e) Do parallel currents in the same direction attract or repel? (f) What about parallel currents in opposite directions?
Point \(P\) is midway between two long, straight, parallel wires that run north- south in a horizontal plane. The distance between the wires is $1.0 \mathrm{cm} .\( Each wire carrics a current of \)1.0 \mathrm{A}$ toward the north. Find the magnitude and direction of the magnetic field at point \(P\)
A straight wire segment of length \(0.60 \mathrm{m}\) carries a current of $18.0 \mathrm{A}$ and is immersed in a uniform external magnetic ficld of magnitude \(0.20 \mathrm{T}\). (a) What is the magnitude of the maximum possible magnetic force on the wire segment? (b) Explain why the given information enables you to calculate only the maximum possible force.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Waves Physics

Read Explanation

Fluids

Read Explanation

Linear Momentum

Read Explanation

Fundamentals of Physics

Read Explanation

Wave Optics

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.