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Chapter 21: Problem 17

Two isotopes of carbon, carbon- 12 and carbon- \(13,\) have masses of \(19.93 \times 10^{-27} \mathrm{~kg}\) and \(21.59 \times 10^{-27} \mathrm{~kg},\) respectively. These two isotopes are singly ionized \((+e)\) and each is given a speed of \(6.667 \times 10^{5} \mathrm{~m} / \mathrm{s}\). The ions then enter the bending region of a mass spectrometer where the magnetic field is \(0.8500 \mathrm{~T}\). Determine the spatial separation between the two isotopes after they have traveled through a half-circle.

Short Answer

Expert verified
The spatial separation is approximately \(1.63 \times 10^{-2} \mathrm{~m}\).

Step by step solution

01

Identify the relevant formula

To find the spatial separation between the two isotopes, we need to use the formula for the radius of the path of a charged particle in a magnetic field. This is given by the equation \( r = \frac{mv}{qB} \), where \( r \) is the radius, \( m \) is the mass of the ion, \( v \) is its velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.
02

Calculate radius for carbon-12

For carbon-12, \( m = 19.93 \times 10^{-27} \mathrm{~kg} \), \( v = 6.667 \times 10^{5} \mathrm{~m/s} \), \( q = 1.6 \times 10^{-19} \mathrm{~C} \) (charge of an electron), and \( B = 0.8500 \mathrm{~T} \). Using the formula, the radius \( r_{12} \) is:\[r_{12} = \frac{19.93 \times 10^{-27} \mathrm{~kg} \times 6.667 \times 10^{5} \mathrm{~m/s}}{1.6 \times 10^{-19} \mathrm{~C} \times 0.8500 \mathrm{~T}} \approx 9.785 \times 10^{-2} \mathrm{~m}\]
03

Calculate radius for carbon-13

For carbon-13, \( m = 21.59 \times 10^{-27} \mathrm{~kg} \), and using the same values for the other constants, calculate radius \( r_{13} \):\[r_{13} = \frac{21.59 \times 10^{-27} \mathrm{~kg} \times 6.667 \times 10^{5} \mathrm{~m/s}}{1.6 \times 10^{-19} \mathrm{~C} \times 0.8500 \mathrm{~T}} \approx 1.06 \times 10^{-1} \mathrm{~m}\]
04

Determine path spatial separation

The spatial separation is the difference in diameters of their semicircular paths. The diameter is twice the radius, so:\[\text{Separation} = 2(r_{13} - r_{12}) = 2(1.06 \times 10^{-1} \mathrm{~m} - 9.785 \times 10^{-2} \mathrm{~m}) \approx 1.63 \times 10^{-2} \mathrm{~m}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isotopes
Isotopes are variations of the same chemical element that have different numbers of neutrons but the same number of protons. This means that isotopes of an element will have the same atomic number but different mass numbers. These mass variations do not affect the chemical behavior of the element significantly but can affect its physical properties, such as mass.
  • For example, carbon has isotopes such as carbon-12 and carbon-13.
  • Both have 6 protons, but carbon-12 has 6 neutrons, while carbon-13 has 7 neutrons.
These small differences in mass are crucial when it comes to techniques like mass spectrometry, where the mass difference affects the path of the ions during analysis.
Magnetic Field
A magnetic field is a vector field surrounding magnetic material and moving electric charges. It exerts a force on charges and affects their motion if they are moving.
  • This field is measured in Tesla (T).
  • In mass spectrometry, a magnetic field is used to bend the paths of ions, causing them to travel in circular trajectories.
The force exerted by a magnetic field on a moving charged particle is perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for the circular motion of ions in devices like mass spectrometers.
Charged Particles
Charged particles are atoms or molecules that have gained or lost electrons and thus have a net electric charge. This occurs through ionization.
  • Ions are the charged particles that enter a mass spectrometer for analysis.
  • The carbon isotopes, for example, are ionized by losing an electron, becoming positively charged.
  • The sign and magnitude of the charge affect how the particle moves in a magnetic field.
The charge of the particles determines their interaction with the magnetic field. This interaction is part of what allows mass spectrometers to differentiate isotopes based on their mass-to-charge ratio.
Radius of Path
The radius of the path describes the size of the circular trajectory a charged particle takes when moving through a magnetic field. The relationship is governed by the formula \( r = \frac{mv}{qB} \), where \( r \) is the radius, \( m \) is the mass of the ion, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.
  • Heavier ions (greater mass \( m \)) will have a larger radius.
  • Faster ions (greater velocity \( v \)) result in a larger radius as well.
  • A stronger magnetic field \( B \) or a greater charge \( q \) reduce the radius size.
In mass spectrometry, calculating the radius helps to determine the spatial separation of isotopes, as heavier isotopes will follow larger radii than lighter ones, leading to different paths through the magnetic field.

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Most popular questions from this chapter

In a television set, electrons are accelerated from rest through a potential difference of 19 \(\mathrm{kV}\). The electrons then pass through a 0.28 - T magnetic field that deflects them to the appropriate spot on the screen. Find the magnitude of the maximum magnetic force that an electron can experience.

A wire carries a current of 0.66 A. This wire makes an angle of \(58^{\circ}\) with respect to a magnetic field of magnitude \(4.7 \times 10^{-5} \mathrm{~T}\). The wire experiences a magnetic force of magnitude \(7.1 \times 10^{-5} \mathrm{~N}\). What is the length of the wire?

In the operating room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients undergoing surgery. One gas that is often monitored is the anesthetic isoflurane (molecular mass \(\left.=3.06 \times 10^{-25} \mathrm{~kg}\right)\). In a spectrometer, a singly ionized molecule of isoflurane (charge \(=+e\) ) moves at a speed of \(7.2 \times 10^{3} \mathrm{~m} / \mathrm{s}\) on a circular path that has a radius of \(0.10 \mathrm{~m}\). What is the magnitude of the magnetic field that the spectrometer uses?

The 1200 -turn coil in a dc motor has an area per turn of \(1.1 \times 10^{-2} \mathrm{~m}^{2}\). The design for the motor specifies that the magnitude of the maximum torque is \(5.8 \mathrm{~N} \cdot \mathrm{m}\) when the coil is placed in a 0.20 -T magnetic field. What is the current in the coil?

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio \(\tau\) square \(/ \tau_{\text {rectangle}}\) of the maximum torques that these coils experience in the same magnetic field when they contain the same current.
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