(II) Suppose the electric field between the electric plates...

(II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 20-41 is 2.88 $\times$ 10$^4$ V/m and the magnetic fields are $B = B' =$ 0.68 T. The source contains carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate masses of the atoms, multiply by 1.67 $\times$ 10$^{-27} kg.) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged?

(II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 20-41 is 2.88 $\times$ 10$^4$ V/m and the magnetic fields are $B = B' =$ 0.68 T. The source contains
carbon isotopes of mass numbers 12, 13, and 14 from a long-dead piece of a tree. (To estimate masses of the atoms, multiply by 1.67 $\times$ 10$^{-27} kg.) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged?

Key Concepts

Lorentz Force and Ion Trajectories

The Lorentz force, which accounts for the force experienced by a moving charged particle in electromagnetic fields, is central to understanding ion motion in mass spectrometry. It is this force that causes the ions to follow curved paths in a magnetic field, as the magnetic component of the Lorentz force provides the necessary centripetal force for circular motion. This principle is essential for predicting and analyzing the separation of ions based on their dynamic properties.

Charge-to-Mass Ratio (q/m)

The charge-to-mass ratio is a crucial parameter that determines how an ion moves in a magnetic field. In a mass spectrometer, the radius of curvature of an ion's path is given by the relation r = mv/(qB), where m is the mass, q is the charge, and v is the ion’s velocity. Differences in charge (such as between singly and doubly charged ions) and mass result in distinct radii and, consequently, distinct positions on the detector, enabling isotopic separation.

Mass Spectrometry

Mass spectrometry is an analytical technique that ionizes chemical species and sorts the ions based on their mass-to-charge ratio. The method involves ion acceleration, typically achieved through an electric field, followed by the deflection of these ions by a magnetic field. The resulting curved trajectories allow for the separation and detection of ions as they strike a detector, and differences in mass-to-charge ratios produce spatially separated impact positions.

Transcript

00:01 So we know the maximum force, another word, the maximum magnetic force can be equal to qb.

00:07 And this can be also equal to mv square over r, since the particles is having a centrifugal motion.

00:17 So q here is the charge of the charged particle, v is a velocity, b is the magnetic field, and r here is the radius of the curvature.

00:26 And m here is the mass of the particle.

00:28 As you can tell, v here can be canceled out.

00:31 And this will give us qb is equal to m times v over r.

00:35 So we have r is equal to mv over qb.

00:38 And we know in this case, the magnetic force is also equal to the electric force so that the proton can have a straight path.

00:51 So say that the magnetic force for one electrons is fb equal to qv prime.

01:07 So we have qv prime is equal to qe.

01:09 And as you can tell, q here can be canceled on.

01:12 And this will give us vb prime is equal to e, so v is equal to e over b prime.

01:17 And when the r is equal to mv over qb, so we have r is equal to me over qb prime.

01:23 And this will give us 2r is equal to 2me over qbb prime.

01:27 So for carbon 12 isotope, we have the distance for the carbon 12, which is 2r12 here, is equal to 2 times m12 times e over qbb prime.

01:39 So the mass for the carbon 12 isotope is 12 u.

01:43 U is an atomic mass unit, which is approximately equal to 1 .67 times 10 to the power of negative 27 kilograms.

01:53 And if we do some calculation here, eventually you have the mass for the carbon 12 isotope is about 2 .004 times 10 to the power of the negative 26 kilograms.

02:02 And when the strength of the attribute in this case is given as 2 .88 times 10 to the power of 4, volt per meter, and when the b is equal b prime, which equals 0 .68 tesla, and the charge -on -charge particle is 1 .6 times 10 to the power of negative 19 quons.

02:20 If we plug in the values, eventually we have 2r12 is equal to 1 .56 times 10 to the power of a negative meter.

02:30 So now for carbon 13, we have 2r -13 is equal 2m13e over qbb prime...

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