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Fundamentals of unimolecular ion dissociation
Tandem mass spectrometry (MS/MS) is one of the most important analytical methods that has plenty of applications in chemistry and biology. This technique involves the selection of a precursor ion, and then its activation, usually by collision or photon absorption, to perform dissociation, and produce fragment ions. In addition to its analytical applications, tandem mass spectrometry can provide a profound insight into the fundamental properties of ions in the gas phase. Here, before explaining the employed activation techniques in this thesis, first, basic theories of unimolecular dissociation will be discussed. In this regard, Lindeman mechanism, Hinshelwood theory, RRK theory and RRKM/QET theory will be described concisely. In addition, since the transition state theory was the basis of the formulation of RRKM/QET theory, it will be explained, as well.
In the gas-phase reactions that follow first order kinetics, it seems that only one species is involved. However there is an unclear point: how do reactant ions obtain enough energy to dissociate? At the beginning, it was thought that this energy is only provided by the absorption of radiation from the surrounding. Then, in 1922, Lindemann proposed a mechanism based on collisional activation, which was further developed by Hinshelwood.79,80 This theory, which is called the Lindemann mechanism, is suitable to describe thermal unimolecular reactions, and it involves both bi- and unimolecular steps. According to this theory, the precursor ion, AB+, collides with the target gas, M, to acquire energy and undergo excitation:
Then, the excited molecular ion, [AB+]*, undergoes unimolecular dissociation to form the products (A+ and B). In Equation 1.1, k1 is the rate constant for the activation to any vibrational level (no discrimination between various vibrational levels) above the critical energy (E0) that is the minimum required energy for decomposition; k-1 is the rate constant for the deactivation from any energy level above the critical energy to any energy level below it (again no discrimination between various energy levels is considered). This 23 deactivation rate constant is often taken to be equal to λZ, in which Z is the number of collisions and λ is collision efficiency. In Equation 1.2, k2 is the rate constant for dissociation of the excited molecular ion. Here, the main hypothesis is that after the activated precursor ions achieve by collision the required energy for dissociation, they do not necessarily have to react. After excitation, they have enough time to lose their energy either by deactivation via another collision, or by unimolecular dissociation. Thus, the presence of this time lag is the basis of this theory.
According to the steady state approximation, the rates of the production and consumption of intermediate, [AB+]*, are equal; therefore, its concentration will not change during the reaction. By applying this approximation to the excited molecular ion, we can obtain an overall rate law (Equation 1.3): [ +]∗ = 0 = [ +][ ] − −1 [ +]∗[ ] − [ +]∗ (1.3)
Now, it will be useful to consider the behavior of the reaction rate at two limiting cases: high and low pressure of the target gas.
High pressure limit
At a high pressure of collision gas, there are a large number of collisions. Therefore, the probability of collisional de-activation of [AB+]* is higher than unimolecular dissociation, i.e. k-1[AB+]*[M] >> k2[AB+]* and the dissociation step is the rate limiting step. Under this 24 condition, k2 term in the denominator of the overall reaction rate can be neglected and the rate law.
This is a first order rate law with the overall rate constant of k1k2/k-1. In this high pressure limit which is also called rapid energy exchange (REX) limit, the population of ions has a Maxwell-Boltzmann distribution of internal energies. It should be noted that the Lindemann theory does not take into account photon absorption and emission, and only considers activation by collision.
Low pressure limit
At a low pressure of the target gas, there are only a few collisions. In this situation, the rate of the unimolecular dissociation will be higher than collisional de-activation, i.e. k2[AB+]* >> k-1[AB+]*[M]. Therefore, k-1[M] can be neglected in the denominator of the overall reaction rate.
According to this equation, at the low pressure limit, reaction kinetics is of second order, and bi-molecular activation is the rate determining step.
The pressure at which the rate constant arrives to half of its value at the high pressure limit can be calculated as follows: = 1 ℎ ℎ (1.10) 2 1 2[ ]1/2 1 1 2 = (1.11) [ ] + 2 −1 1/2 2 −1 [ ] = 2 = ℎ ℎ (1.12)
The equation for the overall rate constant can be also rewritten to obtain: 1 = 1 + 1 (1.13)
Therefore, we can obtain k1 and khigh pressure values from the slope and the intercept of the linear plot of 1/koverall versus 1/[M].
From the rate constant expression at high pressure, the value of k2 can be calculated: ℎ ℎ= 1 2 = 2 ( − 0 ) (1.16)
Since k2 is constant, the temperature dependence of khigh pressure is expected to show Arrhenius behavior, and the plot of log khigh pressure versus 1/T should be linear.
To summarize, the general description of the collisional activation process proposed by the Lindemann theory properly predicts the decrease of the rate constant at low pressures. In addition, it can correctly explain the fact that the reaction rate constant is changing from first order at the high pressure limit to second order at the low pressure condition. However, it has some drawbacks, for example: as the molecules get bigger, discrepancies between experimental results and the theoretical calculations increase. Furthermore, the plot of 1/koverall versus 1/[M] diverges from linearity in the experimental data, indicating that it is not well-described by this theory.
Hinshelwood improved the Lindemann theory by taking into account vibrational degrees of freedom in the activation process.80 According to the Hinshelwood theory, during collisional activation, energy is accumulated in the vibrational modes (s) of the molecular ion. After each activation, there is a step of redistribution of internal energy among all vibrational modes. This process repeats itself until the time arrives when the geometrical arrangement of atoms in the excited molecular ion becomes suitable for the reaction.
According to Equation 1.20, we need the values of k-1, k2, E0 and s to calculate koverall. To obtain them, at the beginning, we could approximate the value of k2 from the pressure in which the rate constant arrives to half of its value at the high pressure limit. Then, this primary k2 value can be used for the first approximate value of k1. In the next step, k2, E0, and s values can be found by successive iterations of this calculation, until arriving at the point where the best fit of the experimental data and the theoretical ones is obtained.
The rate constant at the high pressure limit can be obtained by: = 1 2 (1.21) ℎ ℎ −1 = 1 ( 0 ) −1 ( − 0 ) (1.22) ℎ ℎ 2( −1)!
Here, the calculation of k2, E0 and s will be iterated until the best fit is obtained. In spite of the fact that k2 is constant, the above equation predicts a non-linear plot for ln (khigh pressure) versus 1/T, for which the linear plots are observed!
To summarize, compared to the Lindemann theory, Hinshelwood theory results in an improved fitting between experiment and theory (koverall equation) by taking into account the number of normal modes (s). Nonetheless, it still suffers from some limitations; among them, the most important one is that the observed curvature in the plots of 1/koverall versus 1/[M] is poorly accounted for.
Rice, Ramsperger,81,82 and Kassel83 introduced a statistical theory for the calculation of the unimolecular dissociation rate suggesting that the reaction rate depends also on the vibrational energy of the excited molecular ion. According to their theory (RRK theory), in order for dissociation to take place, a minimum of energy should be localized in special vibrational modes, and the rate constant is proportional to the probability of this localization. In other words, the unimolecular reaction rate depends on the vibrational modes of the excited molecular ion, no matter how it is activated.
They stated that the precursor ion contains s identical harmonic oscillators, among them, one is considered as the critical oscillator (ν) where dissociation reaction takes place. In addition, they assumed that the rate of the intramolecular distribution of excess energy is faster than the unimolecular dissociation rate (ergodic assumption), and energy flows freely among all of the oscillators. Then, they defined the transition state as the configuration in which the energy of the critical oscillator is higher than the critical energy (bond energy, E0).
By considering the vibrations as discrete energy levels, a quantum mechanical expression for the probability of having the transition state configuration can be calculated using Equation 1.23.
Here, n is the total number of vibrational quanta, and m is the number of quanta localized in the critical oscillator. By considering the classical limit in which the number of quanta (n) is very large compared to the number of oscillators (s).
By multiplying the above probability with the frequency factor (ν) which is the vibrational frequency of the critical oscillator, we could obtain the dissociation rate constant.
This equation shows that as energy increases, the rate constant goes up, and as the number of oscillators increases, it goes down. To obtain an acceptable agreement between experimental data and the theoretical ones, a smaller value for the number of oscillators, s, has to be used.
To summarize, the main feature of RRK theory is that when a molecular ion gets energy, this energy is rapidly distributed over all the vibrational modes of the molecule. This process 29 continues until the point where a sufficient amount of energy quanta is transferred into the critical oscillator, then dissociation reaction takes place. Although this theory is a significant improvement compared to the previous theories, its results have some discrepancies with experiments. These originate from the assumption of s identical oscillators having the same frequency. Nevertheless, this theory can be successfully used for the calculation of binding energies.
Transition state theory
Transition state (TS) theory explains bi-molecular reaction rates. This theory, also called activated complex theory, was developed simultaneously by Eyring,84 Evans,85 and Polanyi85 in 1935. According to this theory, there is a specific geometrical configuration of all the atoms in the reacting system which is called transition state or activated complex. During the reaction process, this configuration must first be obtained, before the reaction continues thereby converting this complex to the products. In the reaction coordinate (lowest potential energy pathway between reactants and products), the potential energy surface of the transition state is higher than both reactants and products. TS theory assumes that there is a quasi-equilibrium between reactants and the activated complex in which kB is the Boltzmann constant and h is Planck’s constant. Also, K# relates to the Gibbs free energy difference between the reactants and the transition state (G#) by Equation 1.35: #= (− ∆ # ) (1.35)
Therefore, the rate constant is expressed by: = # = ∆ # (− ) (1.36) 2 ℎ G# relates to the enthalpy (H#) and entropy (S#) of activation by: ∆ # = ∆ # − ∆ # (1.37)
Therefore, the rate constant equation may be written as: ∆ # ∆ # = ( ) (− ) ( ) (1.38) ℎ
This entire equation should be multiplied by the transmission coefficient (ƙ) to take into account the probability of conversion of the activated complex to the products. Therefore, we arrive to this final expression for the overall rate constant: ∆ # ∆ # = ƙ ( ) (− ) ( ) (1.39)
This equation can be used for calculation of Gibbs free energy, enthalpy and entropy of activation. To summarize, instead of discussing about the collisional activation and deactivation, TS theory speaks about the presence of an activated complex in the reaction coordinate. Therefore, the reaction rate is directly the rate at which reactants arrive and pass through the transition state. This theory along with the RRK theory were then used to formulate RRKM theory to describe unimolecular reaction rate constants.
The RRKM/QET theory
In the early 1950s, Marcus and Rice86,87 proposed a new theory for the formulation of unimolecular dissociation rate constant based on transition state theory84,85 and RRK theory,81–83 which was named RRKM theory. At the same time, Wallentein, Wahrhaftig, Rosenstock and Eyring introduced quasi equilibrium theory88 (QET), mainly for application in mass spectrometric systems. Both of these theories resulted in the same equation for the calculation of the reaction rate as a function of energy. Derivation of the RRKM/QET equation is presented elsewhere;81,83,86,87 here we briefly discuss about physical repercussions of that, and some assumptions of both theories.
According to the RRKM theory, the reaction scheme is as follows: here, [AB+]* is the energized molecular ion (E > E0), and [AB+]# is the activated complex (transition state). Actually, the difference between these two species is in their configuration, with the latter having the suitable one for the reaction. Therefore, this theory uses the concept of transition state, and then statistical mechanics to calculate the probability that an energized molecule has the proper configuration as that of the activated complex. In addition, instead of considering a uniform frequency for all of the oscillators of the system, it deals with the real frequencies of both [AB+]* and [AB+]#.
The RRKM/QET rate constant equation for a molecule having an internal energy of E and critical energy of E0, is as follows: here, σ is the reaction degeneracy (number of the equivalent pathways of the reaction), N#(E-E0) is the sum of states at the transition state from 0 to E-E0, h is Planck’s constant and ρ(E) is the density of states of the precursor ion at the internal energy of E. But, what are the concepts of sum and density of states? Considering a molecule with s vibrational degrees of freedom (neglecting the rotational energy), and an internal energy of E, the sum of the states means the number of ways that we could distribute the energy among all the s oscillators in a way that the total energy is equal to or less than E.89 As the energy increases, the sum of states also increases because at higher energies, the number of ways of distributing the energy between the oscillators is higher. Now, the density of states at energy E is defined as the number of vibrational arrangements with energy content between E and E + δE. The sum of states is a number without dimensions, and since the density is the number of states per energy interval, its unit is E-1.89
To have a better understanding about the RRKM/QET equation, consider a precursor ion with a total internal energy of E (Figure 1-1). At the transition state, a part of this energy should be localized on the critical oscillator that is equal to the bond dissociation energy, E0.
Now, the remaining energy that could be distributed among all other oscillators is E-E0. So, this is one state (the first term of the following equation). The other possibility is that, we allocate a part of the E-E0 energy to the translational energy of the fragments, εi. Therefore, in this case, the total remaining energy that could be distributed between the vibrational modes is E-E0-εi, the second and subsequent terms.
Thus, sum of states can be obtained by adding up all the terms in the numerators of the above equation.89 Therefore, the maximum available energy for the oscillators other than the critical oscillator (E-E0) is when the fragments do not have any translational energy, and the minimum energy is when fragments have the highest translational energy (E-E0-εi=0). So, we sum up all the states between these two limits (0 to E-E0). From the above discussion, the minimum dissociation rate will be when the remaining energy for the oscillators is 0, then N(0)=1, because there is only one state corresponding to this energy (all the oscillators having zero energy). Consequently, the minimum rate constant .
which depends neither on the structure of the transition state, nor on the vibrational frequencies.89
In the RRKM/QET equation (Equation 1.43), both the sum and the density of states increase with internal energy. However, since N#(E-E0) increases faster, the rate constant will strongly depend on the internal energy. To calculate the rate constant, in addition to the critical energy, one also needs to determine the sum and the density of states. This can be done by a direct counting of states which itself needs the vibrational frequencies of the reactant and the transition state. However, as it has been shown previously,90,91 RRKM/QET calculations are not sensitive to the vibrational frequencies, and when there is not a detailed knowledge of the transition state, quantitative studies can still be performed. On the contrary, they depend on the entropy of the activation (S#). Therefore, two main parameters in the calculation of the rate constant will be E0 andS#.
Table of contents :
Spectrométrie de masse en tandem
Ionisation par électronébulisation
Chapter 1: Fundamentals of unimolecular ion dissociation
Transition state theory
The RRKM/QET theory
Chapter 2: A general introduction to the utilized fragmentation techniques
Blackbody infrared radiative dissociation (BIRD)
Low-energy collision induced dissociation (low-energy CID)
Higher-energy collision dissociation (HCD)
Chapter 3: Low-energy CID, CID and HCD mass spectrometry for structural elucidation of saccharides and clarification of their dissolution mechani
Results and discussion
Chapter 4: Investigation of hemicryptophane host-guest binding energies using high-pressure collision induced dissociation in combination with RR
Results and Discussion
Chapter 5: Investigating binding energies of host-guest complexes in the gas-phase using low-energy collision induced dissociation
Results and discussion
Chapter 6: Investigating binding energies of host-guest complexes using higher-energy collision dissociation in the gas-phase
Results and discussion
Chapter 7: Dissociation energetics of lithium-cationized -cyclodextrin and maltoheptaose studied by low-energy collision induced dissociation
Results and discussion
List of Figures
List of Tables