The ion source in a mass spectrometer produces both triply and quadruply ionized species, X3+ and X4+. The difference in mass between these species is too small to be detected. Both species are accelerated through the same electric potential difference, and both experience the same magnetic field, which causes them to move on circular paths. The radius of the path for the species X3+ is r3, while the radius for species X4+ is r4. Find the ratio r3 / r4 of the radii.
r3 / r4 =
To find the ratio r3 / r4 of the radii, we can use the principles of mass spectrometry along with the behavior of charged particles in electric and magnetic fields.
In a mass spectrometer, charged particles are accelerated through an electric potential difference and then subjected to a magnetic field, which causes them to move on circular paths. The radius of the circular path is related to the particle's mass and charge.
Let's denote the mass of each species as m and q3, q4 as the charges of X3+ and X4+, respectively.
The centripetal force acting on a charged particle moving in a magnetic field is given by:
F = qvB,
where F is the centripetal force, q is the charge, v is the velocity of the particle, and B is the magnetic field strength.
Since the accelerated particles experience the same electric potential difference, their velocities are equal, so we can compare the centripetal forces for the species X3+ and X4+:
F3 = q3vB,
F4 = q4vB.
The centripetal force can also be expressed in terms of mass and radius:
F = (mv^2) / r.
Comparing the centripetal forces for X3+ and X4+, we have:
(mv3^2) / r3 = (mv4^2) / r4.
Since the masses and velocities are the same for both species, we can cancel them out:
v3^2 / r3 = v4^2 / r4.
Now, let's rearrange the equation to find the ratio r3 / r4:
v3^2 / v4^2 = r3 / r4.
Since the particles are accelerated through the same electric potential difference, the kinetic energy of each particle is equal. Therefore, the velocities v3 and v4 can be related using the kinetic energy equation:
(1/2)mv3^2 = (1/2)mv4^2.
This equation shows that the velocities v3 and v4 are equal.
Substituting v3 = v4 in the earlier equation:
v3^2 / v4^2 = r3 / r4,
we find that r3 / r4 = 1.
Therefore, the ratio of the radii is r3 / r4 = 1.