Time-of-flight Mass Spectrometry - Theory

Theory

The potential energy of a charged particle in an electric field is related to the charge of the particle and to the strength of the electric field:

(1)

where E_p is potential energy, q is the charge of the particle, and U is the electric potential difference (also known as voltage).

When the charged particle is accelerated into time-of-flight tube by the voltage U, its potential energy is converted to kinetic energy. The kinetic energy of any mass is:

(2)

In effect, the potential energy is converted to kinetic energy, meaning that equations (1) and (2) are equal

(3)

(4)

The velocity of the charged particle after acceleration will not change since it moves in a field-free time-of-flight tube. The velocity of the particle can be determined in a time-of-flight tube since the length of the path (d) of the flight of the ion is known and the time of the flight of the ion (t) can be measured using a transient digitizer or time to digital converter.

Thus,

(5)

and we substitute the value of v in (5) into (4).

(6)

Rearranging (6) so that the flight time is expressed by everything else:

(7)

Taking the square root of the time

(8)

These factors for the time of flight have been grouped purposely. contains constants that in principle do not change when a set of ions are analyzed in a single pulse of acceleration. (8) can thus be given as:

(9)

where k is a proportionality constant representing factors related to the instrument settings and characteristics.

(9) reveals more clearly that the time of flight of the ion varies with the square root of its mass-to-charge ratio (m/q).

Consider a real world example of a MALDI time-of-flight mass spectrometer instrument which is used to produce a mass spectrum of the tryptic peptides of a protein. Suppose the mass of one tryptic peptide is 1000 daltons (Da). The kind of ionization of peptides produced by MALDI is typically +1 ions, so q = e in both cases. Suppose the instrument is set to accelerate the ions in a U = 15,000 volts (15 kilovolt or 15 kV) potential. And suppose the length of the flight tube is 1.5 meters (typical). All the factors necessary to calculate the time of flight of the ions are now known for (8), which is evaluated first of the ion of mass 1000 Da:

(10)

Note that the mass had to be converted from daltons (Da) to kilograms (kg) to make it possible to evaluate the equation in the proper units. The final value should be in seconds:

which is about 28 microseconds. If there were a singly charged tryptic peptide ion with 4000 Da mass, and it is four times larger than the 1000 Da mass, it would take twice the time, or about 56 microseconds to traverse the flight tube, since time is proportional to the square root of the mass-to-charge ratio.