Foundation · Lesson 4
Follow the ion, one force at a time
Prerequisites: m/z and charge and ion formation. Allow about 20 minutes with the exercises. This is a physical model, not an instrument operating procedure.
- Predict how charge sign changes the force from an electric field
- Separate accelerating an ion, steering it, and measuring its m/z
- Use energy and flight-time ratios without confusing mass with m/z
- Explain why gas is removed in some regions and deliberately added in others
Predict: which ion arrives first?
Two ideal ions start from rest, gain energy through the same accelerating potential difference, and then traverse the same field-free distance. Ion A has ion mass 500 Da and charge +1. Ion B has ion mass 1000 Da and charge +2. Ignore collisions, initial energy spread, and acceleration time.
Does A arrive first, B arrive first, or do they arrive together? Before calculating, write down the m/z value for each ion. These are deliberately chosen ion masses, not neutral masses that need protons added.
Reveal the prediction and its reasoning
They arrive together in this ideal model. Both have m/z 500. Ion B is twice as massive, but its charge magnitude is also twice as large, so it receives twice the kinetic energy through the same potential difference. The ratio of kinetic energy to mass is unchanged; their final speeds are equal.
“The lighter ion is faster” is safe only when the relevant conditions, including charge, are held fixed. A spectrum's m/z coordinate already encodes the crucial ratio.
Electric fields change ion motion
An electric field exerts a force on an ion. In a simple classical description, F = qE, where F is force in newtons, q is signed charge in coulombs, and E is electric field in volts per metre (equivalently newtons per coulomb). Combining this with F = ma gives a = qE/m, with ion mass m in kilograms.
For a positive ion, the electric force points along the field. For a negative ion, it points opposite the field. Doubling the charge magnitude doubles the force in the same field; doubling the mass halves the acceleration at the same force. Both mass and charge matter.
The field is related to how voltage changes with position. A large voltage value by itself does not specify a force: the geometry and the voltage differences between electrodes determine the field. A uniform region at one potential can be field-free even when that potential is far from laboratory ground.
Worked example: force is not speed
Two positive ions, 500 Da at +1 and 1000 Da at +1, enter the same field with the same initial velocity. Their forces have equal magnitude because their charges are equal. The 1000 Da ion has half the acceleration because its mass is twice as large.
If the ions instead pass through the same potential drop from rest, both gain the same kinetic energy, not the same speed. The heavier ion then travels at 1/√2 of the lighter ion's speed. The word “same” needs an attached quantity.
A neutral particle has q = 0, so this charge-based force is zero. That does not mean neutrals never enter an instrument or can never interact with fields. It means the ordinary charged-particle steering model cannot guide a neutral in the same way as an ion. Gas flow, collisions and geometry can still transport neutral material.
Voltage becomes kinetic energy
In a static electrostatic field, an ion moving from potential Vstart to Vend changes its kinetic energy by ΔK = q(Vstart − Vend), if no other work or collisions intervene. The sign matters: a positive ion gains energy moving toward lower potential; a negative ion gains energy moving toward higher potential.
For convenient timing calculations, let Vacc > 0 be the magnitude of an accelerating potential difference, with the electrode polarity chosen appropriately. Starting from rest,
K = |z|eVacc = ½mv²
v = √(2|z|eVacc/m)
Here |z| is charge magnitude, e is the elementary charge, and v is speed. One elementary charge accelerated through 1 V gains 1 electronvolt (eV) of energy. An electronvolt is an energy unit; it is not a voltage and it is not restricted to electrons.
| Ion | Charge magnitude | Kinetic-energy gain | What this says |
|---|---|---|---|
| A | 1 | 1000 eV | Energy per ion is charge × potential difference |
| B | 2 | 2000 eV | Same voltage does not imply same energy for different charge states |
Check: a negative ion moves from −100 V to 0 V. Does it gain energy?
Yes. With q = −e, ΔK = (−e)[−100 V − 0 V] = +100 eV. Reversing the charge sign without changing electrode polarity would reverse the acceleration. Negative-ion m/z is still usually displayed as a positive magnitude; electrode polarity is a separate issue.
From speed to flight time
In an ideal field-free drift region of length L, an ion with constant speed v takes t = L/v. Substituting the energy relation gives:
t = L√[m/(2|z|eVacc)]
With path and accelerating voltage fixed, drift time scales with √(m/|z|), the square root of the ion's m/z coordinate. Multiplying m/z by four doubles the drift time. Doubling the flight path doubles it. Quadrupling the accelerating voltage halves it. These are model predictions you can test below. The Shimadzu TOF derivation explains this energy-to-time relationship.
Worked example: put the units through the equation
- Choose an ion mass of 500 Da, |z| = 1, Vacc = 1000 V, and L = 1.00 m
- Convert the ion mass: m = 500 × 1.66053906892 × 10−27 kg = 8.3026953446 × 10−25 kg
- Use e = 1.602176634 × 10−19 C. Energy gain is 1.602176634 × 10−16 J
- Speed is approximately 19,645.4 m/s, so drift time is approximately 50.90 µs
This is a field-free drift-time calculation. Real measured arrival times include extraction timing, flight optics, finite acceleration regions, electronic offsets and other instrument-specific effects. It is not a calibration equation for a real instrument.
A longer path also gives a larger absolute time separation for two different m/z values in this model. That alone does not guarantee better real resolving power: the packet's arrival-time spread and the measurement electronics also matter. A reflectron uses electrostatic fields for time focusing; our straight drift model does not simulate it.
Why the instrument has pressure stages
Residual gas can collide with an ion and change its energy, direction or chemical identity. The chance of an unwanted encounter depends on the gas density, collision cross section and distance travelled. Lower pressure generally means a longer distance between encounters at fixed gas composition and temperature. In a mass-analysis region, fewer unintended collisions help preserve the motion being measured.
Vacuum does not identify an ion, and “more vacuum” is not a universal optimization rule. Atmospheric-pressure sources admit gas through an interface into progressively lower-pressure regions. Separate pumping stages manage that gas load. Some ion guides use collisions for cooling; collision or reaction cells deliberately introduce gas for a particular measurement task. Different regions have different jobs. Agilent's vacuum overview gives the background-gas rationale; the TOF technical overview illustrates gas-filled collision and low-pressure analysis regions in one instrument.
| Region | Intended role | Useful question |
|---|---|---|
| Atmospheric-pressure interface | Transfer a fraction of the ions while managing gas and droplets | Are source/interface conditions and pressure readings consistent with the method? |
| Collision cell | Use controlled collisions, for example to produce fragments | Is the specified gas and collision-energy condition appropriate for this experiment? |
| Mass-analysis region | Preserve predictable motion long enough to measure or select ions | Does its pressure satisfy the instrument's own ready-state criteria? |
These are conceptual checks. Vacuum faults, high voltage, hot components and pressurized gases require the instrument procedure and trained personnel. Do not open a vacuum system or alter protective interlocks to test a lesson prediction.
Transport is not the same as mass analysis
An ion lens can focus or steer a beam through an aperture. A guide can confine ions over a broad m/z range. A mass analyzer supplies a relationship between motion and m/z that can be used to separate, select or measure ions. Improving transmission may make a peak taller without improving the evidence for its chemical identity.
A quadrupole applies time-varying fields to the ion's transverse motion. Its useful behavior cannot be reduced to a static “opposite charges attract” drawing: RF phase, trajectory and DC/RF settings matter. Start with the instrument anatomy lesson, then move to the Mathieu-equation derivation when you are ready for the mathematics.
Likewise, increasing an interface potential may change desolvation, transmission or in-source fragmentation, depending on the design. A higher signal does not by itself tell you which of these changed. Manufacturer controls with similar names need not refer to identical physical quantities.
Apply: explain before adjusting
A laboratory sees lower intact-ion signal after an interface voltage change. Fragment-like peaks rise, while a reference standard still gives the expected mass positions. Give two hypotheses and one useful comparison. This is a reasoning exercise, not a diagnosis of a specific instrument.
Compare your investigation plan
One hypothesis is increased in-source fragmentation. Another is altered transmission that changes the detected ion population; a coincident source or sample change is also possible. Compare the saved original condition with the changed condition using the same suitable standard and controlled sample introduction, under the approved method. Inspect intact and candidate-fragment signals together with relevant status information.
Correct mass positions do not prove that transmission or ion chemistry is unchanged. Conversely, a signal change does not prove that mass-axis calibration has failed. Preserve the original settings and record what was changed so the comparison is interpretable.
Transfer checks
- At fixed charge and accelerating voltage, the ion mass increases from 500 to 2000 Da. What happens to drift time?
- Ion C has mass 750 Da and |z| = 3. Ion D has mass 500 Da and |z| = 1. Which reaches the end of the same ideal drift first?
- Why is a collision cell not evidence that “mass spectrometers do not need vacuum”?
- Can equal arrival times in this model identify two ions as the same molecule?
Worked answers
- The mass-to-charge ratio is four times larger, so drift time is twice as long
- C has m/z 250 and D has m/z 500. C arrives first, with tC/tD = √(250/500) ≈ 0.707 under the stated conditions
- Gas is intentionally controlled in that region; other regions need to preserve motion with fewer unwanted collisions. Pressure is part of a staged design
- No. Equal m/z can produce equal ideal times for different ion compositions and charge states. Timing supplies a measurement coordinate, not a unique chemical identity
Your four-line explanation
- Fields act on charge: qE sets force; qE/m sets acceleration
- Voltage changes energy: through an accelerating potential, K = |z|eVacc
- Motion encodes a ratio: ideal drift time scales as √(m/|z|) at fixed path and voltage
- Pressure has a purpose: suppress unwanted collisions and enable intended ones in their own regions
Primary references and model scope
- NIST CODATA 2022 constants: e and the atomic mass constant used in the calculation
- Shimadzu, Introduction to LC-MS, part 6: ideal time-of-flight derivation; instrument comparisons on that page are not used as universal claims here
- Agilent, Vacuum Technology for Mass Spectrometry Instruments: residual gas and mean free path
- Agilent, Time-of-Flight Mass Spectrometry: instrument-specific example of collision, transfer and TOF regions
The numerical examples and explorer are synthetic and nonrelativistic. They omit initial energy, space charge, finite acceleration time, packet width, collisions, reflectrons and detector/electronic delays. Constants are declared calculation inputs; displayed decimals are not an uncertainty statement.
Choose the next question
You have completed the four foundation lessons. Use Resolution is not mass accuracy to ask how good the measurement is; use Mass spectrometer anatomy to follow the physical instrument; or explore quadrupole stability for a deeper mathematical model.