/*! 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 16 A mass spectrometer is being use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 16

A mass spectrometer is being used to separate common oxygen- 16 from the much rarer oxygen-18, taken from a sample of old glacial ice. (The relative abundance of these oxygen isotopes is related to climatic temperature at the time the ice was deposited.) The ratio of the masses of these two ions is 16 to 18, the mass of oxygen-16 is \(2.66 \times 10^{-26} \mathrm{~kg}\), and they are singly charged and travel at \(5.00 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a 1.20-T magnetic field. What is the separation between their paths when they hit a target after traversing a semicircle?

Short Answer

Expert verified
Substitute the given values into the formulas to calculate the radii for both isotopes and then find the difference between them to determine the path separation.

Step by step solution

01

Determine the radius of the circular path for each isotope

The radius of the path for a charged particle moving perpendicular to the magnetic field is given by the formula: \( r = \frac{mv}{qB} \), where m is the mass of the particle, v is its velocity, q is the charge of the particle (since the ions are singly charged, q is equal to the elementary charge e), and B is the magnetic field strength. Calculate the radius for both oxygen-16 and oxygen-18 by substituting their respective masses.
02

Calculate the radius for oxygen-16

Insert the given values into the formula for oxygen-16: \( r_{16} = \frac{m_{16}v}{qB} = \frac{2.66 \times 10^{-26} \text{ kg } \times 5.00 \times 10^{6} \text{ m/s}}{1.60 \times 10^{-19} \text{ C } \times 1.20 \text{ T}} = \frac{2.66 \times 10^{-26} \times 5.00 \times 10^{6}}{1.60 \times 10^{-19} \times 1.20} = \text{ radius of oxygen-16} \)
03

Calculate the radius for oxygen-18

Using the same formula for oxygen-18: \( r_{18} = \frac{m_{18}v}{qB} \). Since the mass ratio of oxygen-18 to oxygen-16 is 18 to 16, we have \( m_{18} = \frac{18}{16}m_{16} \). Substitute \( m_{16} \) and the rest of the known values to find \( r_{18} \).
04

Find the difference in the radii to determine the path separation

Since the ions are traveling in a semicircle before hitting the target, the separation between their paths when they hit the target will just be the difference in the radii of their paths: \( \Delta r = r_{18} - r_{16} \). Calculate this difference using the radii from the previous steps.
05

Execute the calculations

Perform the calculations from the previous steps to determine the separation between the paths of the two isotopes.

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.

Isotope Separation in Mass Spectrometry
The process of distinguishing and separating different isotopes of the same element is known as isotope separation. In mass spectrometry, this separation is often essential for various scientific analyses, such as understanding climatic conditions from ice core samples. Isotopes differ in mass but have similar chemical properties, making them difficult to separate by traditional means.

In a mass spectrometer, isotopes are ionized to carry a charge, and then accelerated by an electric field so that they all have the same kinetic energy. Because the isotopes have different masses, they will have different velocities. A magnetic field is then used to deflect them, with the radius of curvature for each isotope鈥檚 path being dependent on their velocity and mass. The larger the mass, the larger the radius of the orbit for a given velocity. By examining the points at which the isotopes strike a detector or target, we can measure their mass-to-charge ratio and thus differentiate between them.

Applications of Isotope Separation

In addition to climatic research, isotope separation is pivotal in nuclear medicine, geology, and archaeology, furnishing valuable information about processes ranging from metabolic pathways to geological timescales.
Magnetic Field in Mass Spectrometry
The role of the magnetic field within mass spectrometry is to separate charged particles based on their mass-to-charge ratio. After ionization, particles are accelerated and then subjected to a perpendicular magnetic field. This magnetic influence causes the charged particles to move in circular or spiral paths due to the Lorentz force, which acts perpendicular to both their velocity vector and the magnetic field.

The radius of the particle's circular path, as revealed by the exercise's step by step solution, is determined by the equation: \( r = \frac{mv}{qB} \), where \( m \) represents the mass of the particle, \( v \) is velocity, \( q \) signifies the charge, and \( B \) denotes the magnetic field strength.

Practical Implications

The ability to manipulate this path with precision is critical for separating and identifying isotopes. These concepts allow researchers to establish precise measurements and analyses of various substances and their isotopic compositions, facilitating advancements in fields such as pharmacology, environmental science, and forensic investigation.
Velocity of Charged Particles
Understanding the velocity of charged particles is crucial in the context of a mass spectrometer. As particles with differing masses but the same charge and energy enter the magnetic field, their varying velocities dictate unique paths within the spectrometer. Based on the exercise provided, we use the formula \( r = \frac{mv}{qB} \), emphasizing how a particle鈥檚 velocity, \( v \), plays a role in determining its circular path radius.

The calculation for oxygen-16 and oxygen-18 demonstrated that even a slight mass difference has an evident effect on the resulting path. Charged particles with higher velocities (for a given magnetic field and charge) will have greater radii of curvature. Consequently, it becomes feasible to achieve separation based on subtle differences within a sample.

Connecting Velocity with Isotope Analysis

By comparing velocity differences, mass spectrometers can effectively distinguish between isotopes, which is key for applications in chemistry, physics, and even medical diagnostics where precise molecular information is necessary.

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

(a) A 0.750-m-long section of cable carrying current to a car starter motor makes an angle of \(60^{\circ}\) with the Earth's \(5.50 \times 10^{-5} \mathrm{~T}\) field. What is the current when the wire experiences a force of \(7.00 \times 10^{-3} \mathrm{~N} ?(\mathrm{~b})\) If you run the wire between the poles of a strong horseshoe magnet, subjecting \(5.00 \mathrm{~cm}\) of it to a \(1.75-T\) field, what force is exerted on this segment of wire?

Why would a magnetohydrodynamic drive work better in ocean water than in fresh water? Also, why would superconducting magnets be desirable?

(a) Triply charged uranium-235 and uranium-238 ions are being separated in a mass spectrometer. (The much rarer uranium-235 is used as reactor fuel.) The masses of the ions are \(3.90 \times 10^{-25} \mathrm{~kg}\) and \(3.95 \times 10^{-25} \mathrm{~kg}\), respectively, and they travel at \(3.00 \times 10^{5} \mathrm{~m} / \mathrm{s}\) in a 0.250-T field. What is the separation between their paths when they hit a target after traversing a semicircle? (b) Discuss whether this distance between their paths seems to be big enough to be practical in the separation of uranium-235 from uranium-238.

(a) A 200-turn circular loop of radius \(50.0 \mathrm{~cm}\) is vertical, with its axis on an east-west line. A current of \(100 \mathrm{~A}\) circulates clockwise in the loop when viewed from the east. The Earth's field here is due north, parallel to the ground, with a strength of \(3.00 \times 10^{-5} \mathrm{~T}\). What are the direction and magnitude of the torque on the loop? (b) Does this device have any practical applications as a motor?

Heroes in movies hide beneath water and breathe through a hollow reed (villains never catch on to this trick). In practice, you cannot inhale in this manner if your lungs are more than \(60.0 \mathrm{cm}\) below the surface. What is the maximum negative gauge pressure you can create in your lungs on dry land, assuming you can achieve \(-3.00 \mathrm{cm}\) water pressure with your lungs \(60.0 \mathrm{cm}\) below the surface?
See all solutions

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Read Explanation

Energy Physics

Read Explanation

Force

Read Explanation

Space Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Translational Dynamics

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.