Problem 18 - Physics

Question

The ion source in a mass spectrometer produces both singly and doubly ionized species, \(\mathrm{X}^{+}\) and \(\mathrm{X}^{2+}\). 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 \(\mathrm{X}^{+}\) is \(r_{1}\), while the radius for species \(\mathrm{X}^{2+}\) is \(r_{2} .\) Find the ratio \(r_{1} / r_{2}\) of the radii.

Step-by-Step Solution

Answer

The ratio of the path radii is \( \sqrt{2} \).

Step 1: Understand the forces acting on the ions

In a mass spectrometer, ions are accelerated by an electric potential and then move in a circular path due to the magnetic field. The magnetic force provides the centripetal force for the ions, where the Lorentz force equation can be used: \( qvB = \frac{mv^2}{r} \). This can be rearranged to \( r = \frac{mv}{qB} \), where \( r \) is the radius of the path, \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field.

Step 2: Determine the velocity of ions

The velocity \( v \) of both ions can be found using the kinetic energy equation. Since both ions are subjected to the same potential difference \( V \), their kinetic energy is given by \( \frac{1}{2}mv^2 = qV \), thus \( v = \sqrt{\frac{2qV}{m}} \). Substitute this into the radius formula to get \( r = \frac{m\sqrt{2qV/m}}{qB} \).

Step 3: Apply to both singly and doubly ionized ions

Both \( \mathrm{X}^{+} \) and \( \mathrm{X}^{2+} \) have the same mass \( m \), but different charges. For \( \mathrm{X}^{+} \), the charge \( q_1 = e \), and for \( \mathrm{X}^{2+} \), the charge \( q_2 = 2e \). Substituting these into the formula for the radius, we get \( r_1 = \frac{m\sqrt{2eV/m}}{eB} = \frac{\sqrt{2mV}}{eB} \) for \( \mathrm{X}^{+} \) and \( r_2 = \frac{m\sqrt{4eV/m}}{2eB} = \frac{\sqrt{mV}}{eB} \) for \( \mathrm{X}^{2+} \).

Step 4: Calculate the ratio of the radii

Find the ratio \( \frac{r_1}{r_2} \). Substitute the expressions for \( r_1 \) and \( r_2 \) from Step 3: \[ \frac{r_1}{r_2} = \frac{\frac{\sqrt{2mV}}{eB}}{\frac{\sqrt{mV}}{eB}} = \frac{\sqrt{2mV}}{\sqrt{mV}} = \sqrt{2}. \]

Step 5: Provide the final answer

The ratio \( r_1 / r_2 \) of the radii for the singly and doubly ionized species is \( \sqrt{2} \).

Key Concepts

Ionization

Ionization is a fundamental process in mass spectrometry, where atoms or molecules are converted into ions. An ion is an atom that has either lost or gained electrons, resulting in a net positive or negative charge.
In a mass spectrometer, these ions are created in an ion source. Here, species like \( \mathrm{X}^+ \) and \( \mathrm{X}^{2+} \) are formed. \( \mathrm{X}^+ \) represents a singly ionized species with a charge of \( e \), while \( \mathrm{X}^{2+} \) is doubly ionized, possessing a charge of \( 2e \).
Ionization is crucial as it allows the particles to be manipulated by electric and magnetic fields. This manipulation is key to analyzing the ions based on their mass-to-charge ratio. The mass spectrometer uses this principle to distinguish between ions of different masses.

Electric Potential Difference

The electric potential difference, or simply voltage, is a measure of the work needed to move a charge from one point to another. In a mass spectrometer, ions are accelerated through this potential difference, which provides them with kinetic energy.
The energy provided to the ion is related to the voltage by the formula \( qV = \frac{1}{2}mv^2 \), where \( q \) is the charge of the ion, \( V \) is the potential difference, \( m \) is the mass, and \( v \) is the resulting velocity.

• For singly ionized species \( \mathrm{X}^+ \), the charge \( q \) is \( e \).
• For doubly ionized species \( \mathrm{X}^{2+} \), the charge \( q \) is \( 2e \).

The same potential difference allows both ions to gain kinetic energy and subsequently achieve velocity depending on their charge.

Magnetic Field

A magnetic field is a region where a magnetic force can be exerted on moving charges, such as ions in a mass spectrometer. When ions enter a magnetic field, they experience a force perpendicular to their direction of motion as well as to the magnetic field itself.
This is described by the Lorentz force law, \( F_B = qvB \, \sin \theta \), where \( F_B \) is the magnetic force, \( q \) is the ion's charge, \( v \) is the velocity, and \( B \) is the magnetic field strength. Since the ions move perpendicular to the field (\( \sin 90^\circ = 1 \)), the force becomes \( qvB \), which acts as the centripetal force guiding the ions on a circular path.
The radius of this path depends on the ion's velocity, mass, and charge, allowing the spectrometer to distinguish between ions of different charges while keeping the mass constant.

Circular Motion

Circular motion describes the path that ions take when they move through a magnetic field in a mass spectrometer. As ions are subjected to magnetic force, they are guided into a circular trajectory.
This is because the magnetic force acts as a centripetal force, directing the ions towards the center of their path. The radius \( r \) of this path is given by \( r = \frac{mv}{qB} \,\) where \( m \) is the ion's mass, \( v \) its velocity, \( q \) the charge, and \( B \) the magnetic field strength.
From this relationship, it's evident that ions with greater charge or mass will have different radii, influencing how they separate out in a mass spectrometer. This principle allows the measurement of mass-to-charge ratios critical for discrimination between closely related ions.

Centripetal Force

Centripetal force is a necessary component for circular motion, acting towards the center of the circle to keep an object moving in a curved path. In a mass spectrometer, the magnetic force plays the role of the centripetal force.

• Expressed as \( F_c = \frac{mv^2}{r}, \) where \( m \) is mass, \( v \) is velocity, and \( r \) is the radius.
• This force ensures that ions continue on their circular paths within the magnetic field.

Balancing the centripetal force with the magnetic force gives the equation \( qvB = \frac{mv^2}{r} \.\)
This balance is crucial to determining the radius of ion paths in the spectrometer, distinguishing between ions with different charges while recognizing that their mass may be too similar to differentiate otherwise.

Lorentz Force

The Lorentz force is the combination of electric and magnetic forces on a charged particle, essential for the functioning of a mass spectrometer. Specifically, when a charged particle like an ion moves perpendicular through a magnetic field, it experiences the magnetic component of Lorentz force.
This force is given by the formula \( F = qvB \,\) where \( q \) is the ion's charge, \( v \) is its velocity, and \( B \) is the magnetic field's strength. It causes the ion to deflect into a circular path.

• Helps in achieving the necessary centripetal force for circular motion.
• The direction of force is perpendicular to both the velocity and the magnetic field.

Understanding the Lorentz force is fundamental to grasping how mass spectrometers separate ions, demonstrating the interplay between electric fields, magnetic fields, and induced ion trajectories.