Spectrometer Secret Lab Notes Article The Magnetism Review
Volume 12, Issue 4 • Advanced Physics Research Digest
Article: Mass Spectrometry
Weighing the Invisible: The Mechanics of Mass Spectrometry
An in-depth analysis of ion trajectory control and atomic mass determination through electrodynamic fields.
Technical Schematic: Three-Stage Mass Spectrometer
Stage I Stage II Stage III Filament Source ΔV B₁ (IN) + Plate - Plate F_m F_e B₂ (OUT) Mass m₁ Mass m₂ DETECTOR ARRAY
Figure 1.1: Standard 180° Mass Spectrometer. Red plate is positive; blue plate is negative. Velocity is rightward with B₁ field directed into the page.
I. What is a Mass Spectrometer?
A mass spectrometer is an analytical instrument used to identify the chemical composition of a sample by measuring the mass-to-charge ratio (\(m/q\)) of its constituent ions. It is the "gold standard" for isotopic analysis, forensic drug testing, and identifying unknown organic compounds. The process transforms a sample into a beam of high-speed ions and then uses magnetic and electric fields to sort them.
II. Stage I: Ionization and Acceleration
The process begins with ionization . In an electron impact source, a high-energy electron beam strikes the sample vapor, knocking electrons off the atoms to create positive ions. These ions are then accelerated by an electric potential difference \(\Delta V\).
Energy Conservation Derivation:
\[W = \Delta K \implies q\Delta V = \frac{1}{2}mv^2\]
\[v = \sqrt{\frac{2q\Delta V}{m}}\]
III. Stage II: The Velocity Selector
To ensure every ion enters the analysis chamber at the same speed, they pass through a Velocity Selector . This region uses crossed electric (\(\vec{E}\)) and magnetic (\(\vec{B}_1\)) fields. In our refined schematic (Figure 1.1 ), the red plate is maintained at a positive potential, while the blue plate is negative.
For a positive ion, the electric force (\(F_e = qE\)) pulls it toward the negative blue plate. Simultaneously, the magnetic force (\(F_m\)) acts in the opposite direction. For a particle to satisfy the "straight-path" condition, the forces must be in perfect translational equilibrium:
\(qvB_1 = qE \implies v = \frac{E}{B_1}\)
IV. Stage III: Deflection and Detection
Finally, the ions enter a second uniform magnetic field (\(\vec{B}_2\)). Since the magnetic force is always perpendicular to the velocity, it provides centripetal acceleration , forcing the ions into a circular arc.
\(qvB_2 = \frac{mv^2}{r} \implies r = \frac{mv}{qB_2}\)
Substituting the selected velocity (\(v = E/B_1\)) gives: \(r = \frac{m(E/B_1)}{qB_2}\). This relationship shows that the radius of the path is strictly proportional to the mass-to-charge ratio . By measuring where the ions hit the detector array, the specific mass of the isotope can be identified.
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AP Physics C: Electrodynamics Section 4.5
Magnetic Force Slides Mass Spectrometry
AP Physics C: Analysis of Atomic Motion
The Analytical Pipeline
B₁ (IN) POSITIVE PLATE B₂ (OUT) DETECTOR ARRAY
The Speed Filter
Newton's 2nd Law
\[\Sigma F = F_e - F_m = 0\]
\[v = \frac{E}{B_1}\]
The Velocity Selector uses crossed fields. In our setup, the Red Plate (+) is on top and the Blue Plate (-) is on the bottom.
F_m F_e
Stage III Analysis: Mass Filtering
Centripetal Control
\[r = \frac{mv}{qB_2}\]
"Path radius depends directly on mass."
Resolution Rules:
Larger Mass → Larger Radius
Constant Speed → Mass Sorting
Identifying Peaks
The output Mass Spectrum shows the abundance of each isotope detected at different radii.
m₁ (¹²C) m₂ (¹⁴C) Mass / Charge (m/z)
Isotopic Ratios:
The ratio of the peak heights (areas) directly correlates to the natural abundance of isotopes in the original sample.
Mass Spectrometry Worksheet Mass Spectrometry Worksheet
AP Physics C: Electrodynamics Analysis
Name:
Date:
STAGE I STAGE II STAGE III ΔV ? RED PLATE (+) BLUE PLATE (-) B₂ (OUT) DETECTOR ARRAY
Diagram 1: Three-Stage Quantitative Mass Spectrometer Schematic. Paths omitted in Stage III.
01
Stage II: Velocity Selector Inquiry
A beam of positive ions enters Stage II traveling horizontally to the right. The region between the plates contains a uniform electric field \(\vec{E}\) created by the red and blue plates shown. A magnetic field \(\vec{B}_1\) of \(0.60 \, \text{T}\) is also present but hidden.
A. Based on the plate polarity shown in Diagram 1, determine the direction of the magnetic field \(\vec{B}_1\) required to ensure the ions travel in a straight line toward the exit slit. Provide a clear justification using a free-body diagram and the appropriate Right-Hand Rule.
B. If the electric field strength is \(4.5 \times 10^4 \, \text{V/m}\), calculate the magnitude of the velocity required for an ion to pass through the selector.
02
Stage III: Atomic Weight Prediction
The "velocity-filtered" ions enter Stage III where they encounter a uniform magnetic field \(B_2 = 0.55 \, \text{T}\) directed out of the page .
A. On Diagram 1, draw the trajectory for two isotopes (\(m_1\) and \(m_2\), where \(m_2 > m_1\)) as they enter Stage III. Explicitly show where they hit the detector array. Justify your choice of deflection direction.
B. Derive the expression for the ion's mass \(m\) in terms of charge \(q\), velocity \(v\), field \(B_2\), and the impact distance \(x\) (the horizontal distance from the entrance slit to where the ion strikes the detector).
03
Challenge: Negative Ion Substitution
A beam of negative chloride ions (\(\text{Cl}^-\)) is substituted into the instrument.
A. If the potentials in the velocity selector (Stage II) remain the same (Red+, Blue-), in which direction must the magnetic field \(\vec{B}_1\) point for the negative ions to pass through undeflected? Justify your answer by analyzing how both \(\vec{F}_e\) and \(\vec{F}_m\) change with the sign of the charge.
B. Upon entering Stage III (\(B_2\) out of page), will the negative ions deflect toward the TOP or BOTTOM of the detector array? Draw this trajectory on a separate sketch below and explain your reasoning.
Advanced Electrodynamics Laboratory • Unit 4
Mass Spectrometry Answer Key Answer Key: Mass Spectrometry
Teacher Reference • AP Physics C • Unit 4
01. Stage II Inquiry
Part A: B-Field Direction
Direction: Into the page ($\times$).Justification: Positive ions feel a downward electric force $\vec{F}_e$ (Red+ to Blue-). To maintain a straight trajectory, the magnetic force $\vec{F}_m$ must be UP. By RHR ($v$ right, thumb up), the fingers/palm must point INTO the page.
Part B: Velocity
$v = E/B_1 = 4.5 \times 10^4 / 0.60 = \mathbf{7.5 \times 10^4 \, \text{m/s}}$.
02. Stage III Prediction
Part A: Trajectory
Students should draw two semicircles curving DOWNWARD from the slit. The larger radius circle corresponds to $m_2$ hitting lower on the detector array.
Part B: Derivation
$qvB_2 = \frac{mv^2}{r} \implies m = \frac{qrB_2}{v}$
03. Chlorine Ratio (Conceptual Synthesis)
$m \propto x \implies m_2/m_1 = x_2/x_1 = 16.91 / 16.0 = \mathbf{1.0569}$.
04. Negative Ion Challenge
Part A: Selector Field
Direction: Still INTO THE PAGE .Justification: For a negative ion, $\vec{F}_e$ flips UP (toward red plate). However, $\vec{F}_m = q(\vec{v} \times \vec{B})$ also flips DOWN (opposite to a positive ion). Since both forces flip, the equilibrium condition is identical.
Part B: Stage III Deflection
Direction: Toward the TOP of the chamber.Justification: $\vec{F}_m = q(\vec{v} \times \vec{B})$. For a negative charge, the force is in the opposite direction of the cross product result for a positive charge. Since positive ions deflected down, negative ions deflect UP.
Mass Spectrometry Mastery Problem Mass Spectrometry Mastery Problem
AP Physics C: Electrodynamics Assessment
Student:
Date:
STAGE I STAGE II STAGE III ΔV B₁ = ? RED (+) BLUE (-)
Diagram 1: System Context
A beam of singly ionized Uranium isotopes ($^{235}\text{U}^+$ and $^{238}\text{U}^+$) is accelerated in Stage I. In Stage II, they encounter an electric field $E = 4.0 \times 10^4 \, \text{V/m}$ and a magnetic field $B_1$. In Stage III, they are sorted by a uniform magnetic field $B_2 = 0.50 \, \text{T}$ (out of page).
1. Vector Analysis
(A) Determine the direction of the magnetic field $B_1$ required to pass the positive ions through Stage II undeflected. Justify with a sketch of the forces.
2. Trajectory Prediction
(A) Sketch the trajectories of both isotopes in Stage III on Diagram 1. Identify which isotope hits the detector further from the inlet.
03
Quantitative Resolution
Given the acceleration potential in Stage I is $\Delta V = 2.5 \times 10^3 \, \text{V}$, and the mass of $^{238}\text{U}$ is $238u$, calculate the following:
(A) Calculate the magnitude of $B_1$ required to filter the $^{238}\text{U}^+$ ions correctly.
(B) Calculate the physical separation $\Delta x$ (in mm) between the impact points of $^{235}\text{U}$ and $^{238}\text{U}$ on the detector array.
B₂ DETECTOR
Diagram 2: Stage III Work Area
Case Study: Nuclear Isotope Enrichment • Unit 4
Mass Spectrometry Mastery Answer Key Answer Key: Mastery Problem
Teacher Reference • AP Physics C • Unit 4
01. Stage II Selection Logic
Part A: B₁ Field Direction
Direction: INTO the page ($\times$).Justification: The electric field $\vec{E}$ points from Red (+) to Blue (-), which is DOWN. For a positive ion, $\vec{F}_e$ is downward. To maintain a straight path, the magnetic force $\vec{F}_m$ must be UP. By RHR ($v$ right, thumb up), the palm/fingers must point INTO the page.
Part B: Equilibrium Relationship
$\Sigma F = F_m - F_e = 0 \implies qvB_1 = qE \implies \mathbf{v = E/B_1}$.
Mastery Solution Diagram (Composite)
B₁ (IN) m₂₃₅ m₂₃₈ DETECTOR
02. Stage III Analysis
Part A: (See Diagram Above) Ions curve DOWN due to $\vec{v} \times \vec{B}$ direction for a positive charge.
Part B: Mass Formula: $qvB_2 = \frac{mv^2}{r} \implies m = \frac{qrB_2}{v}$. Since $x = 2r$, $\mathbf{m = \frac{qxB_2}{2v}}$.
03. Numerical Challenge
Part A: B1 Calculation
1. Find $v$ from Stage I: $v = \sqrt{2q\Delta V / m}$.
Part B: Physical Separation Δx
$x = 2mv / qB_2$
