Two of the isotopes of cartoon, carbon-12 and carbon-13,...

Two of the isotopes of cartoon, 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 $(+e),$ each 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.

Key Concepts

Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the mass-to-charge ratio of ions. In this context, it involves accelerating ions (in this case, singly ionized isotopes) into a magnetic field where they follow curved trajectories based on their specific mass and charge, allowing separation and identification of different isotopes.

Lorentz Force

The Lorentz force is the fundamental force acting on charged particles when they move through a magnetic field. It is given by the equation F = q(v × B), meaning that the magnetic force is perpendicular to both the velocity of the ion and the magnetic field. This force causes the ion to follow a curved path, which is critical in the functioning of mass spectrometers.

Circular Motion of Charged Particles

When a charged particle enters a magnetic field perpendicular to its velocity, the Lorentz force acts as a centripetal force, causing the particle to move in a circular path. The radius of this circular motion depends on the particle's mass, charge, velocity, and the strength of the magnetic field, which is used to differentiate isotopes based on their specific properties.

Radius of Curvature Calculation

The radius of curvature of a charged particle's trajectory in a magnetic field is determined by the relationship r = mv/(qB). This formula is essential for understanding how ions with different masses (but the same charge and speed) will have different path radii. The difference in these radii provides a mechanism for spatial separation of isotopes after they traverse a specific portion of their circular paths, such as a half-circle.

Transcript

00:01 In this problem, we have two different particles of differing masses, but with the same charge, they're going to enter the same mass spectrometer at the same speed.

00:10 So ultimately, what's going to happen is that these two particles are going to travel along different circular paths.

00:16 And we're asked to figure out what the difference in the separation between the two particles is after they've already traveled through one half of a circle.

00:25 So let's get a visualization of what this is going to look like.

00:28 We're going to have a region of space with a uniform.

00:30 Magnetic field in it through which our particles are going to travel.

00:36 And they're going to travel with different radii.

00:40 So we're going to have a magnetic field heading out of the page here.

00:44 Each particle is going to enter with the same velocity v around here.

00:50 Remember they have the same charge, but they're going to have different masses.

00:53 So let's say our carbon 12 isotope is going to travel along this circular path.

01:01 So say something like this, and then let's say our carbon 13 particle is going to travel a different circular path.

01:12 We know since the mass is higher and everything else is the same that that radius has to be larger, because if we look at our equation that defines the radii of these different circles in our mass spectrometer, it's given by this, remember? so since the carbon 13 isotope has a larger mass, larger m here, it's going to map out a larger radius.

01:38 And so ultimately, what we're looking for is we have a circular path for our carbon 12 isotope.

01:46 I'm going to use a 12 as a subscript here.

01:48 That's the radius of the circular path for our carbon 12 isotope.

01:52 And then the radius for our circular path for our carbon 13 isotope will be something like this.

01:59 Ultimately, we're looking for this separation here.

02:04 The difference between the two diameters of our different circles...

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