TOF calculations
TOF calculations
🔓 CRACK IT
A TOF calculation is a chain of linked equations. You do not have to guess the whole chain at once. Start with the quantity the question asks for and find the next missing piece.
- Write an equation containing your target, then rearrange it. For time, start with t = d/v.
- Put in the quantities you know. If speed is still unknown, use the kinetic-energy equation to find it.
- Keep going until every quantity needed for the calculation is known.
- Convert to kg, m and s before substituting: mass of one ion in kg, distance in m and time in s. In particular, convert microseconds to seconds.
- Keep the calculator's full values during the working. Round only your final answer.
The equations below are your toolkit. Relative isotopic mass has no units; it is not the mass of one ion in kg. For a singly charged mononuclear ion, its numerical value equals m/z.
d distance travelled by the ion, in m; t flight time of the ion, in s; v speed of the ion, in m s⁻¹.
KE kinetic energy of one ion, in J; m mass of one ion, in kg; v speed of the ion, in m s⁻¹.
L Avogadro constant, 6.022 × 10²³ mol⁻¹.
L Avogadro constant, 6.022 × 10²³ mol⁻¹; m mass of one ion, in kg.
Forward: relative mass → flight time
A singly charged ⁴⁰Ar⁺ ion has relative mass 40.0 and KE 6.00 × 10⁻¹⁶ J. It travels 1.50 m. Use L = 6.022 × 10²³ mol⁻¹.
Calculate the time of flight of the ion, giving your answer in seconds and microseconds.
Worked answer
- Target t: t = d/v. Speed is unknown, so use v = √(2KE/m).
- Mass is still unknown: m = (relative isotopic mass × 10⁻³)/L.
m = (40.0 × 10⁻³)/(6.022 × 10²³) = 6.642 × 10⁻²⁶ kg. - v = √(2KE/m)
v = √[(2 × 6.00 × 10⁻¹⁶)/(6.642 × 10⁻²⁶)] = 1.34 × 10⁵ m s⁻¹. - t = d/v.
t = 1.50/(1.344098… × 10⁵) = 1.12 × 10⁻⁵ s.
Reverse: flight time → m/z
A singly charged ion has KE 5.00 × 10⁻¹⁶ J and travels 1.60 m in 13.0 µs.
Calculate the mass of one ion in kg and its mass-to-charge ratio, m/z. Use L = 6.022 × 10²³ mol⁻¹.
Worked answer
- Target relative mass: relative mass = m × L × 1000.
- Find m from m = 2KE/v². Speed is unknown, so use v = d/t.
- t = 13.0 × 10⁻⁶ s.
v = d/t.
v = 1.60/(13.0 × 10⁻⁶) = 1.230769… × 10⁵ m s⁻¹. - m = 2KE/v².
m = (2 × 5.00 × 10⁻¹⁶)/(1.230769… × 10⁵)² = 6.6015625 × 10⁻²⁶ kg = 6.60 × 10⁻²⁶ kg (3 s.f.). - Relative mass = m × L × 1000.
Relative mass = (6.6015625 × 10⁻²⁶) × (6.022 × 10²³) × 1000 = 39.754609… = 39.8.
Two ions: what cancels?
Two ions of masses m₁ and m₂ have equal kinetic energies and travel equal distances in times t₁ and t₂. Derive an expression for t₂/t₁ in terms of m₁ and m₂.
Worked answer
- ½m₁v₁² = ½m₂v₂². Multiply by 2.
- Substitute v = d/t: m₁(d/t₁)² = m₂(d/t₂)².
- Only the identical d² cancels: m₁/t₁² = m₂/t₂².
- Rearrange: t₂²/t₁² = m₂/m₁, then take the positive square root.
Comparing ²⁴Mg⁺ and ²⁶Mg⁺
²⁴Mg⁺ and ²⁶Mg⁺ ions have the same kinetic energy and travel the same flight distance. State which ion arrives first and calculate the ratio t(²⁶Mg⁺)/t(²⁴Mg⁺).
Worked answer
t₂/t₁ = √(m₂/m₁)
t₂/t₁ = √(26/24) = 1.0408.
🧪 TEST IT
Work on paper. Attempt each question before opening its answer. These are original CrackingChem questions, written in AQA style.
A singly charged ion has relative isotopic mass 72.0 and kinetic energy 8.25 × 10⁻¹⁶ J. It travels through a 1.40 m flight tube. Calculate its time of flight in µs to three significant figures.
Use L = 6.022 × 10²³ mol⁻¹. KE = ½mv²; v = d/t.
Total: 5 marks
Show answer & marking points
- Target t: t = d/v. Find v using v = √(2KE/m).
- m = (72.0 × 10⁻³) / (6.022 × 10²³) = 1.1956 × 10⁻²⁵ kg. [1]
- Rearrange KE: v = √(2KE/m). [1]
- v = √(2KE/m)
v = √[(2 × 8.25 × 10⁻¹⁶)/(1.1956… × 10⁻²⁵)] = 1.1748 × 10⁵ m s⁻¹. [1] - t = d/v.
t = 1.40/(1.1747517… × 10⁵) = 1.1917 × 10⁻⁵ s. [1] - t = 11.9 µs. [1]
An ion travels 1.12 m in 11.4 µs. Its kinetic energy is 6.75 × 10⁻¹⁶ J. Calculate its mass number.
Use L = 6.022 × 10²³ mol⁻¹. KE = ½mv²; v = d/t.
Total: 5 marks
Show answer & marking points
- Target mass: m = 2KE/v². First find the unknown speed.
- v = d/t.
v = 1.12/(11.4 × 10⁻⁶) = 9.8246 × 10⁴ m s⁻¹. [1] - m = 2KE/v². [1]
- m = 2KE/v²
m = (2 × 6.75 × 10⁻¹⁶)/(9.8245614… × 10⁴)² = 1.3986 × 10⁻²⁵ kg. [1] - Molar mass = m × L × 1000.
Molar mass = (1.3986… × 10⁻²⁵) × (6.022 × 10²³) × 1000 = 84.23 g mol⁻¹. [1] - Mass number = 84, a whole number. [1]
A singly charged ion has relative isotopic mass 96.0, kinetic energy 7.40 × 10⁻¹⁶ J and time of flight 18.5 µs. Calculate the flight-tube length to three significant figures.
Use L = 6.022 × 10²³ mol⁻¹. KE = ½mv²; v = d/t.
Total: 5 marks
Show answer & marking points
- Target d: d = vt. Find v using v = √(2KE/m).
- m = (96.0 × 10⁻³) / (6.022 × 10²³) = 1.5942 × 10⁻²⁵ kg. [1]
- v = √(2KE/m). [1]
- v = √(2KE/m)
v = √[(2 × 7.40 × 10⁻¹⁶)/(1.5941547… × 10⁻²⁵)] = 9.6353 × 10⁴ m s⁻¹. [1] - d = vt = (9.6353 × 10⁴)(18.5 × 10⁻⁶). [1]
- d = 1.78 m. [1]
A⁺ and B⁺ have the same kinetic energy and travel the same distance. Relative isotopic mass of A = 64.0; tₐ = 13.6 µs; tᵦ = 15.3 µs.
- Why can you not calculate the mass of B directly from its KE equation alone? [1]
- Set the kinetic energies equal and calculate the relative isotopic mass of B. Show why distance cancels. [5]
- Why can the masses and speeds not be cancelled? [1]
Total: 7 marks
Show answer & marking points
- Mass and speed are both unknown. [1]
- ½mₐvₐ² = ½mᵦvᵦ². [1]
- Multiply by 2: mₐvₐ² = mᵦvᵦ². [1]
- Substitute v = d/t: mₐ(d/tₐ)² = mᵦ(d/tᵦ)². [1]
- The identical d² factors cancel: mₐ/tₐ² = mᵦ/tᵦ². [1]
- mᵦ = mₐ(tᵦ/tₐ)².
mᵦ = 64.0(15.3/13.6)² = 81.0. [1]
- They have different masses and speeds, so these are not identical factors. [1]