FIGURE P29.64 shows a mass spectrometer, an analytical instrument used to identify the various molecules in a sample by measuring their charge-to-mass ratio q/m. The sample is ionized, the positive ions are accelerated (starting from rest) through a potential difference ∆V, and they then enter a region of uniform magnetic field. The field bends the ions into circular trajectories, but after just half a circle they either strike the wall or pass through a small opening to a detector. As the accelerating voltage is slowly increased, different ions reach the detector and are measured. Consider a mass spectrometer with a 200.00 mT magnetic field and an 8.0000 cm spacing between the entrance and exit holes. To five significant figures, what accelerating potential differences ∆V are required to detect the ions (a) O₂⁺ (b) N₂⁺ and (c) CO⁺? See Exercise 29 for atomic masses; the mass of the missing electron is less than 0.001 u and is not relevant at this level of precision. Although N₂⁺ and CO⁺ both have a nominal molecular mass of 28, they are easily distinguished by virtue of their slightly different accelerating voltages. Use the following constants: 1 u = 1.6605 x 10⁻²⁷ kg, e = 1.6022 x 10⁻¹⁹ C.

Verified step by step guidance

1. Step 1: Understand the problem setup. The ions are accelerated through a potential difference (ΔV), gaining kinetic energy, and then enter a magnetic field where they follow a circular trajectory. The radius of the trajectory is determined by the balance between the magnetic force and the centripetal force. The goal is to calculate the accelerating potential difference (ΔV) for each ion type (O₂⁺, N₂⁺, CO⁺) based on their charge-to-mass ratio (q/m).
2. Step 2: Write the relationship between the magnetic force and centripetal force. The magnetic force is given by F = qvB, where q is the charge, v is the velocity, and B is the magnetic field. The centripetal force is F = mv²/r, where m is the mass, v is the velocity, and r is the radius of the circular path. Equating these forces gives qvB = mv²/r. Simplify to find v = qBr/m.
3. Step 3: Relate the velocity to the accelerating potential difference. The kinetic energy of the ion is given by KE = 1/2 mv², and this energy comes from the work done by the electric field: KE = qΔV. Substituting v² from the previous step into the kinetic energy equation gives qΔV = 1/2 m(qBr/m)². Simplify to find ΔV = (B²r²q)/(2m).
4. Step 4: Substitute the known values into the formula. The magnetic field B = 200.00 mT = 0.20000 T, the radius r = 8.0000 cm / 2 = 0.040000 m (since the ions travel half a circle), and the charge q = e = 1.6022 x 10⁻¹⁹ C. For each ion, use its respective mass (m) in kilograms, which can be calculated using the atomic mass unit (1 u = 1.6605 x 10⁻²⁷ kg).
5. Step 5: Calculate ΔV for each ion. For O₂⁺, N₂⁺, and CO⁺, substitute their respective masses into the formula ΔV = (B²r²q)/(2m). Ensure that the units are consistent throughout the calculation. This will yield the accelerating potential differences required to detect each ion.

Key Concepts

Charge-to-Mass Ratio (q/m)

The charge-to-mass ratio (q/m) is a fundamental property of ions that describes how much charge an ion carries relative to its mass. In a mass spectrometer, ions are accelerated by an electric field, and their trajectories are then influenced by a magnetic field. The q/m ratio determines the radius of the circular path that the ions take in the magnetic field, allowing for their separation based on mass.

Acceleration through Potential Difference (∆V)

When ions are accelerated through a potential difference (∆V), they gain kinetic energy equal to the work done by the electric field. This relationship is given by the equation KE = q∆V, where KE is the kinetic energy. The resulting speed of the ions is crucial for determining their trajectory in the magnetic field, as it directly affects their radius of curvature and the time taken to reach the detector.

Magnetic Field and Circular Motion

In a uniform magnetic field, charged particles experience a magnetic force that causes them to move in circular paths. The radius of this path is determined by the balance between the magnetic force and the centripetal force required for circular motion. The relationship between the radius, velocity, magnetic field strength, and charge-to-mass ratio is essential for calculating the trajectories of different ions in a mass spectrometer.