In one type of mass spectrometer, ions having the same...

In one type of mass spectrometer, ions having the same velocity move through a uniform magnetic field. The spectrometer is being used to distinguish ${ }^{12} \mathrm{C}$ and ${ }^{14} \mathrm{C}$ ions that have the same charge. The ${ }^{12} \mathrm{C}$ ions move in a circle of diameter $25 \mathrm{~cm}$. (a) What is the diameter of the orbit of ${ }^{14} \mathrm{C}$ ions? (b) What is the ratio of the frequencies of revolution for the two types of ion? (c) Repeat parts (a) and (b) if the ions are given the same kinetic energy rather than the same velocity.

In one type of mass spectrometer, ions having the same velocity move through a uniform magnetic field. The spectrometer is being used to distinguish ${ }^{12} \mathrm{C}$ and ${ }^{14} \mathrm{C}$ ions that have the same charge. The ${ }^{12} \mathrm{C}$ ions move in a circle of diameter $25 \mathrm{~cm}$. (a) What is the diameter of the orbit of ${ }^{14} \mathrm{C}$ ions? (b) What is the ratio of the frequencies of revolution for the two types of ion?
(c) Repeat parts (a) and (b) if the ions are given the same kinetic energy rather than the same velocity.

Key Concepts

Cyclotron Motion and Frequency

In a uniform magnetic field, a charged particle undergoing circular motion rotates with a specific frequency, known as the cyclotron frequency. This frequency is determined by the ratio of the charge to the mass of the particle and is independent of the velocity when the particles are given the same velocity; however, it can vary under conditions of equal kinetic energy for different masses.

Mass Spectrometry Principles

Mass spectrometry involves the separation and measurement of ions based on their mass-to-charge ratios. By analyzing the curvature of an ion's path in a magnetic field, one can distinguish between ions of different masses. The diameter of the circular orbit provides direct information about this ratio, allowing isotopic differentiation.

Magnetic Deflection of Charged Particles

Charged particles moving through a magnetic field experience a force perpendicular to both their velocity and the magnetic field, as described by the Lorentz force. This force causes the particles to follow circular trajectories, and the radius of these paths depends on the particle's momentum, charge, and the strength of the magnetic field.

Kinetic Energy and Velocity Relationship

The relationship between kinetic energy and velocity is crucial in the analysis of ion motion in magnetic fields. When ions are accelerated such that they have the same kinetic energy rather than the same velocity, differences in mass result in different velocities. This variation affects the ion trajectories and the corresponding frequencies of circular motion under the influence of a magnetic field.

Transcript

00:01 In this question we have a mass spectrometer that is trying to distinguish between the two isotopes of carbon, carbon 12 and carbon 14.

00:12 We already have the mass of carbon 12 and carbon 14 in atomic units, and in order to convert them to kilograms, we're going to multiply them by 1 .6.

00:23 So 1 .6 times 10 to the power, negative 27 kilograms.

00:33 By atomic units.

00:35 This is going to be equals to 1 .992 times 10 to the power negative 26 kilograms.

00:48 While for carbon 14, we're going to do the same thing.

00:54 This is a transferring factor, kilogram per atomic unit, and this is going to be equal to 2 .324 times to 2 .324 times to the power negative 26 kilograms.

01:11 And of course the radius of the mass spectrometer for carbon 12 is 0 .25 meters.

01:20 And in part a, we wish to find the radius for carbon 14.

01:27 Therefore, so this is going to be part a.

01:32 I'm going to say the net force on the beam of these parts, particles, it's going to be equal to mass times acceleration, and since it's going to in a circular path, it's centripetal acceleration.

01:52 Therefore we can write here fb equals mass times centripetal acceleration, and fb is qvb, which equals to mass, and the centripetal acceleration is v squared divided by r.

02:10 And here we have vs will cancel from both sides and r14.

02:17 It's going to be equal to m times v divided by qb.

02:30 Now here only we have m14, so let me make it looks more clear...

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