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.
mv²/2=qU
v=sqrt(2qU/m)
R=mv/qB= (m/qB)• sqrt(2qU/m)=
R3/R4 = sqrt{ 2Um q₄B²/q₃B²2Um)=
=sqrt(q₄/q₃) sqrt(4e/3e)=1.15
To find the ratio r3/r4 of the radii, we can use the equation for the radius of a charged particle moving in a magnetic field:
r = mv / (|q|B)
Where:
- r is the radius of the circular path
- m is the mass of the charged particle
- v is the velocity of the charged particle
- |q| is the absolute value of the charge of the particle
- B is the magnetic field strength
Since both species, X3+ and X4+, are accelerated through the same electric potential difference and experience the same magnetic field, we can consider that the masses, charges, and magnetic fields are constant for both species. Therefore, we can write the equation for each species as:
r3 = m3v3 / (|q3|B) ... (1)
r4 = m4v4 / (|q4|B) ... (2)
Given that the difference in mass between X3+ and X4+ is too small to be detected, we assume that their masses are approximately equal, which means m3 ≈ m4.
Since both species are accelerated through the same electric potential difference, they have the same kinetic energy.
Therefore, m3v3² = m4v4².
Simplifying this equation, we can rewrite v3 and v4 in terms of their respective radii using the equation for the kinetic energy:
v3 = √(2q3V / m3) ... (3)
v4 = √(2q4V / m4) ... (4)
Where:
- V is the electric potential difference
Plugging equations (3) and (4) back into equations (1) and (2), we get:
r3 = √(2m3q3V / (|q3|B)) ... (5)
r4 = √(2m4q4V / (|q4|B)) ... (6)
Now, let's calculate the ratio r3/r4 by dividing equation (5) by equation (6):
r3/r4 = (√(2m3q3V / (|q3|B))) / (√(2m4q4V / (|q4|B)))
= √((2m3q3V / (|q3|B)) / (2m4q4V / (|q4|B)))
= √((m3q3/ m4q4) * (|q4|B / |q3|B))
= √((m3q3/ m4q4) * (|q4| / |q3|))
Since m3 ≈ m4, we can cancel out the mass ratio:
r3/r4 = √(|q4| / |q3|)
Therefore, the ratio of the radii r3/r4 is simply the square root of the ratio of the absolute values of the charges, |q4| / |q3|.