CHEM 5181 – Fall 2006
Homework Assignment #3
Assigned 21-Sept / Due 3-Oct
Problem 1. Calculate the mass analyzer resolving power (using FWHM for m, as we
defined it in class) needed to separate CO+ and N2+, and also to separate C3H7+ and
C2H3O+. Assume that (a) the peaks have Gaussian shapes; (b) both peaks have the same
total intensity; (c) the separation is acceptable for our analysis purposes when the valley
between two peaks in the spectrum is less than 10% of the peak value.
Problem 2. The figure below represents the simplified stability diagram of a quadrupole
for three ions of masses m1, m2, and m3 (singly charged) vs. U and V. Explain which
ions will / will not be observed as the values of U and V are changed from point A to
point G in the figure.
A E
D
F
B
D
B
G
A
E
G
C
Problem 3. In a 1997 paper (J. Am. Soc. Mass Spec. 8(3), 209, 1997) Martin Vestal and
co-workers used a linear TOFMS with a delayed extraction source (modified PerSeptive
Biosystems Voyager RP) to measure the initial velocity distribution of ions generated by
MALDI.
In this problem, you will predict the distribution of recorded flight times for the singly
charged insulin ion (MW = 5733.50) based on initial velocity distributions and TOFMS
parameters reported in that paper. The basic schematic is identical to the sketch of the
linear TOFMS that you drew in lab #1.
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The distance (d1) between the back plate (V0) and the first grid (V1) is 0.39 cm
The distance (d2) between the first grid and the grid at the beginning of the drift
region (V2) is 1.48 cm
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The total length of the drift region (D) between V2 and the detector is 128.9 cm
The voltage applied V0 is 20 kV
The voltage applied to V1 for extraction is 90% of the voltage applied to V0, or
18 kV.
V2 and the detector are held at ground.
The insulin ions have a mean initial velocity along the TOF axis of 300 m / sec,
and range from 0 to 600 m/sec.
An Excel spreadsheet has been provided (on the course web page) as a guide for solving
this problem. You are not required to use it, but the printed results you submit should
include tables similar to those in the spreadsheet and a plot of the flight times of 7 ions
(of different initial velocity) as a function of delay time. Cells of the Excel spreadsheet
are colored in order to indicate what is provided as an input, and what requires
calculation by you.
These steps will guide you through the spreadsheet:
a. Within the spreadsheet, fill in the cells corresponding to the field strengths in the
regions between V0 and V1 (E1) and V1 and V2 (E2), the charge of the ion in
Coulombs, and the mass of a single ion in kg. These will be critical values for
the calculations that follow.
b. s’ denotes the distance between an ion and V1 at the moment of acceleration.
Ions originate at V0; prior to the extraction event (i.e., during the delay), ions
drift away from the V0 at an initial velocity originating from the ionization
event. [i] Write an expression for s’ as a function of initial velocity (vo) and
delay time (td) and use this expression to calculate values for column C in the
spreadsheet.
c. As it passes through the first grid, an ion will have a kinetic energy (U1) that is
the sum of its initial energy (Uo) and the energy derived from acceleration (Ua1).
Write expressions for [i]U1 in terms of Uo and Ua1 [i] Uo in terms initial velocity
(vo) and [ii] Ua1 in terms of the electric field in this first region (E1), s’, q, and
m. [iii] the velocity of the ion as it passes through the first grid (v1) in terms of
U1. Use these expressions to calculate values for column D.
d. Force can be written as the product of mass (m) and acceleration (a) or as the
product of electric field (E) and charge (q). [i] Based on this, write a general
expression for a in terms of E, q, and m. Denote the units of each variable. Use
this expression to solve for the acceleration that occurs in region 1 (a1) and use
that expression to find values for Column E.
e. A body undergoing constant acceleration, a, for a time, t, will reach a final
velocity, vf, that is equal to the initial velocity, vo, plus the product, ta. [i] Use
this fact to write an expression for the time spent in region 1 (T1) after the
acceleration event in terms of vo , v1 and a1. Use this expression to solve for
values in Column F.
f. Next, we must determine the amount of time an ion will spend traveling through
the region between V1 and V2 (Region 2). As in Region 1, the energy that an
ion has (U2) as it crosses V2 will be a sum of the energy it had upon entering
Region 2 (U1) and the energy it gained by acceleration in Region 2 (Ua2). [i]
Write an expression for the velocity of an ion at V2 as a function of Uo, U1, and
Ua2. [ii] Write an expression for Ua2 in terms of the electric field in region 2
(E2). Use these expressions (and previous expressions for Uo and U1) to solve
for the ion velocities at the exit of Region 2 (v2). (Column G)
g. Use methods similar to parts c and d to complete columns H and I of the
spreadsheet.
h. Upon exiting Region 2, an ion enters the field-free drift region of the TOFMS.
It will undergo no more acceleration. Its velocity thus remains constant. [i]
Write an expression for the time an ion spends in the drift region in terms of the
length of the drift region (D) and its velocity upon exiting Region 2 (v2). Use
this expression to solve for the time spent in the drift region (TD).
i. An ions total flight time (TTotal) is the sum of time spent in Region 1, Region 2,
and the Drift Region. Use this fact to complete Column K. (For comparison,
Vestal reports times around 51 microsec. Your value should be close to this.)
j. You now have a working model of the PerSeptive Biosystems Voyager RP
TOFMS. As a final step in the problem, we must determine the best delay time,
which is the delay time that minimizes the distribution in ion times of flight.
Vary delay time in your spreadsheet and generate a plot of flight times as a
function of delay time. This can be done by copying calculated values from
cells K35-K41 to the corresponding cells in rows 45-51.
k. Print your results, including all cells of the spreadsheet and the generated plot.
Which delay time is optimum?