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Chapter 22: Problem 71

A mixture of two isotopes is injected into a mass spectrometer. One isotope follows a curved path of radius \(R_{1}=48.9 \mathrm{cm}\) the other follows a curved path of radius \(R_{2}=51.7 \mathrm{cm} .\) Find the mass ratio, \(m_{1} / m_{2},\) assuming that the two isotopes have the same charge and speed.

Short Answer

Expert verified
The mass ratio \( \frac{m_1}{m_2} \) is approximately 0.946.

Step by step solution

01

Understanding the context

In a mass spectrometer, isotopes with the same charge and speed undergo circular motion in a magnetic field. The radius of curvature of their path is related to their mass.
02

Deriving the formula

The radius of curvature () for a charged particle moving in a magnetic field is given by the formula \( R = \frac{mv}{qB} \), where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field.
03

Setting the equations for both isotopes

Since the isotopes have the same charge \( q \) and speed \( v \), and are subject to the same magnetic field \( B \), we have: \( R_1 = \frac{m_1v}{qB} \) and \( R_2 = \frac{m_2v}{qB} \).
04

Solving for the mass ratio

Dividing the equation for \( R_1 \) by the equation for \( R_2 \), we eliminate \( v \), \( q \), and \( B \), leading to \( \frac{R_1}{R_2} = \frac{m_1}{m_2} \).
05

Calculating the mass ratio

Insert the radii into the derived equation: \( \frac{48.9}{51.7} = \frac{m_1}{m_2} \). Calculate this ratio to find \( \frac{m_1}{m_2} \).
06

Perform the final calculation

Converting the ratio of radii \( \frac{48.9}{51.7} \), we get approximately \( 0.946 \). Therefore, \( \frac{m_1}{m_2} = 0.946 \).

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Key Concepts

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

Isotopes
Isotopes are different forms of the same element. They have the same number of protons but different numbers of neutrons. This difference in neutrons results in isotopes having distinct mass numbers. For example, carbon has isotopes like carbon-12 and carbon-14.
Each isotope behaves similarly in chemical reactions because they have the same electron configuration. However, the difference in mass may affect physical behaviors. This is particularly useful in applications like mass spectrometry, where different isotopes of an element can be separated and identified based on their mass.
In a mass spectrometer, isotopes are distinguished by their masses as they move through a magnetic field. The varying paths they take due to their mass allows the spectrometer to differentiate between them, illustrating their differing mass-to-charge ratios.
Circular Motion
Circular motion occurs when an object follows a curved path. In physics, when a charged particle enters a magnetic field perpendicular to its velocity, it experiences a force pointing towards the center of its curved path. This force, known as the centripetal force, keeps the particle in circular motion.
For any charged particle moving in a magnetic field, the formula describing its path is given as \( R = \frac{mv}{qB} \). Here, \( R \) is the radius of the path, \( m \) is the particle's mass, \( v \) is its velocity, \( q \) is its charge, and \( B \) is the magnetic field strength. The radius of the path is directly proportional to the particle's mass and velocity, and inversely proportional to the charge and magnetic field.
  • In the case of isotopes in a mass spectrometer, the same charge and speed ensure that the radius is solely influenced by the mass, facilitating the separation between different isotopes.
Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. In the context of a mass spectrometer, it plays a crucial role in inducing circular motion in charged particles, such as isotopes.
The magnetic field exerts a perpendicular force on charged particles causing them to move in circular paths. The strength of this field, denoted as \( B \), is a significant factor in determining the radius of the particle's path. It can be manipulated to adjust the separation between isotopes, as precise as the specific isotopes being studied.
  • The magnetic field works in tandem with the charge and speed of the isotopes to influence the curvature of their paths. This separation is essential for identifying isotopes based on their masses in scientific analysis.

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Most popular questions from this chapter

The number of turns in a solenoid is doubled, and at the same time its length is doubled. Does the magnetic field within the solenoid increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. Doubling the number of turns in a solenoid doubles its magnetic field, and hence the field increases. II. Making a solenoid longer decreases its magnetic field, and therefore the field decreases. III. The magnetic field remains the same because the number of turns per length is unchanged.

A charged particle moves in a horizontal plane with a speed of \(8.70 \times 10^{6} \mathrm{m} / \mathrm{s} .\) When this particle encounters a uniform magnetic field in the vertical direction it begins to move on a circular path of radius \(15.9 \mathrm{cm}\). (a) If the magnitude of the magnetic field is \(1.21 \mathrm{T}\), what is the charge-to-mass ratio \((q / m)\) of this particle? (b) If the radius of the circular path were greater than \(15.9 \mathrm{cm},\) would the corresponding charge-to-mass ratio be greater than, less than, or the same as that found in part (a)? Explain. (Assume that the magnetic field remains the same.)

An electron moves at right angles to a magnetic field of 0.18 T. What is its speed if the force exerted on it is \(8.9 \times 10^{-15} \mathrm{N}\) ?

A recently developed method to study brain function is to produce a rapidly changing magnetic field within the brain. When this technique, known as transcranial magnetic stimulation (TMS), is applied to the prefrontal cortex, for example, it can reduce a person's ability to conjugate verbs, though other thought processes are unaffected. The rapidly varying magnetic field is produced with a circular coil of 21 turns and a radius of \(6.0 \mathrm{cm}\) placed directly on the head. The current in this loop increases at the rate of \(1.2 \times 10^{7} \mathrm{A} / \mathrm{s}\) (by discharging a capacitor). \((\mathrm{a}) \mathrm{At}\) what rate does the magnetic field at the center of the coil increase? (b) Suppose a second coil with half the area of the first coil is used instead. Would your answer to part (a) increase, decrease, or stay the same? By what factor?

A wire with a length of \(3.6 \mathrm{m}\) and a mass of \(0.75 \mathrm{kg}\) is in a region of space with a magnetic field of 0.84 T. What is the minimum current needed to levitate the wire?
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