ssm When beryllium-7 ions (m=11.65 ×10^-27 kg) pass through a mass spectrometer, a uniform magne

Ssm When beryllium-7 ions (m=11.65 ×10^-27 kg) pass through...

Key Concepts

Mass Spectrometry

Mass spectrometry is an analytical technique that measures the mass-to-charge ratio of ions. It is commonly used to identify chemical substances by accelerating ions in an electric field and then deflecting them using magnetic or electric fields to separate based on their masses. This concept is fundamental for understanding how particles are segregated and analyzed in a mass spectrometer.

Motion of Charged Particles in a Magnetic Field

When charged particles move through a uniform magnetic field, they experience a magnetic force perpendicular to both their velocity and the magnetic field direction. This results in circular or helical trajectories depending on the initial angle of motion. The analysis of this motion is essential for predicting particle paths in devices like mass spectrometers and cyclotrons.

Centripetal Force and Radius of Curvature

The circular motion of a charged particle in a magnetic field is characterized by the balance between the magnetic (Lorentz) force and the required centripetal force. The radius of curvature of the particle's path can be determined by equating these forces. This relationship is crucial as it allows for the calculation of particle properties such as velocity and mass based on the path curvature in the magnetic field.

Energy Conservation in Particle Acceleration

Energy conservation principles play a key role in particle acceleration, where particles gain kinetic energy when accelerated by a potential difference. The relation between the kinetic energy acquired and the accelerating potential helps in determining the velocity of the ions before entering the magnetic field. This concept connects the electric acceleration phase with the subsequent magnetic deflection, forming an integral part of analysis in mass spectrometry.

Transcript

0:00 Hello.

00:01 In this problem, we want two different particles to travel along the same circular path in a mass spectrometer.

00:08 So we know that the radius of the circular path that a particle will travel in a mass spectrometer is given by mass of the particle times velocity, divided by the charge of that particle and the magnitude of the magnetic field through which the particle is traveling.

00:26 Now, we're considering two different particles, a beryllium -7 ion and a beryllium -10 ion.

00:33 They both have the same charge.

00:35 They're singly ionized, but they have different masses.

00:38 So, ultimately, we can write out the expression for the radius that these will travel through in terms of the respective masses and velocities, and then since these need to hit the same detector same distance away, those radii need to be the same traveled through each of those particles.

00:54 So what i'm going to do is use a subscript of 7 for beryllium 7 things and a subscript of 10 for beryllium 10 things but we want the radius to those two to travel to be the same.

01:06 So what is this going to look like? well for the radius of the beryllium 7 atom we have the mass of beryllium 7 multiply by how fast it's going in the mass spectrometer divided by its charge which will do as a plus a plus e charge and then we have the magnetic field through which it is traveling for that particle.

01:27 Now we can do the same thing and set them equal to each other for the beryllium 10 ion.

01:33 And i'm going to use a subscript of 10 for that.

01:36 So they both have that same plus e charge.

01:39 And then put some parentheses around this.

01:42 And then it's going to be traveling through a different magnetic field.

01:46 And it must, because we have to alter the field such that it still curves around the same radius in our mass spectrometer.

01:54 So ultimately, what are we are looking for, we're looking for this new magnetic field that we need for a mass spectrometer to make sure that that burium 10 hits the same target that the brillium 7 did.

02:06 So let's rearrange that equation and see what we ultimately need to solve for.

02:10 So let's solve the magnetic field that the brilium 10 ion is going through.

02:14 We'll notice that when we set these equal to each other, our charges actually cancel out.

02:19 So those are going to go away.

02:20 So that's nice.

02:21 So let's keep writing this expression for brilium 10.

02:23 When we do that, we see that we have this is going to be equal to the magnetic field that the beryllium 7 ion travels through multiplied by the mass of the beryllium 10 ion, the velocity of the billion 10 ion, divided by the mass of the beryllium 10 ion, divided by the velocity of the beryllium 7 ion.

02:49 Yes, that is our expression.

02:51 So basically, we have to figure out the ratio of the momentum of these two ions.

02:56 In order to figure out how we're going to scale upward down that magnetic field so that these two particles hit the exact same spot on the detector.

03:03 Now, then that tells us what we really need to figure out, since we don't know the velocities of these particles, is a relationship between the two velocities.

03:11 But we know that they go through the exact same potential difference to be accelerated up to a particular velocity.

03:19 So what does that mean? that means that the potential difference that both of these charges go through in order to be accelerated is given by the charge multiplied by the potential, sorry, the potential energy that these two lose when being accelerated is given by the charge of the particles multiplied by the potential difference and that is going to be equal to an increase in the kinetic energy of our particles.

03:43 Because these two have the same charge and go through the same potential difference, that means this value is the same for each particle and therefore this gain of kinetic energy is the same for each particle.

03:52 So we can write out one half mv squared the change in kinetic energy, the gain in kinetic energy, for each particle respectively.

04:01 And these two have to be equal to each other...

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