Unimolecular ion decomposition

Wahrhaftig diagram showing the relationship between internal energy and unimolecular ion decomposition for a hypothetical ion ABCD⁺.

A Wahrhaftig diagram (named after Austin L. Wahrhaftig) illustrates the relative contributions in unimolecular ion decomposition of direct fragmentation and fragmentation following rearrangement. The x-axis of the diagram represents the internal energy of the ion. The lower part of the diagram shows the logarithm of the rate constant k for unimolecular dissociation whereas the upper portion of the diagram indicates the probability of forming a particular product ion. The green trace in the lower part of the diagram indicates the rate of the rearrangement reaction given by

${\displaystyle {\ce {ABCD+ -> {AD+}+ BC}}}$

and the blue trace indicates the direct cleavage reaction

${\displaystyle {\ce {ABCD+ -> {AB+}+ CD}}}$

A rate constant of 10⁶ s⁻¹ is sufficiently fast for ion decomposition within the ion source of a typical mass spectrometer. Ions with rate constants less than 10⁶ s⁻¹ and greater than approximately 10⁵ s⁻¹ (lifetimes between 10⁻⁵ and 10⁻⁶ s) have a high probability of decomposing in the mass spectrometer between the ion source and the detector. These rate constants are indicated in the Wahrhaftig diagram by the log k = 5 and log k = 6 dashed lines.

Indicated on the rate constant plot are the reaction critical energy (also called the activation energy) for the formation of AD⁺, E₀(AD⁺) and AB⁺, E₀(AB⁺). These represent the minimum internal energy of ABCD⁺ required to form the respective product ions: the difference in the zero point energy of ABCD⁺ and that of the activated complex.

When the internal energy of ABCD⁺ is greater than E_m(AD⁺), the ions are metastable (indicated by m^*); this occurs near log k > 5. A metastable ion has sufficient internal energy to dissociate prior to detection.[2][3] The energy E_s(AD⁺) is defined as the internal energy of ABCD⁺ that results in an equal probability that ABCD⁺and AD⁺ leave the ion source, which occurs at near log k = 6. When the precursor ion has an internal energy equal to E_s(AB⁺), the rates of formation of AD⁺ and AB⁺ are equal.

Schematic of "tight" and "loose" transition state complexes for a hypothetical ion ABCD⁺.

Like all chemical reactions, the unimolecular decomposition of ions is subject to thermodynamic versus kinetic reaction control: the kinetic product forms faster, whereas the thermodynamic product is more stable.[4] In the decomposition of ABCD⁺, the reaction to form AD⁺ is thermodynamically favored and the reaction to form AB⁺is kinetically favored. This is because the AD⁺ reaction has favorable enthalpy and the AB⁺ has favorable entropy.

In the reaction depicted schematically in the figure, the rearrangement reaction forms a double bond B=C and a new single bond A-D, which offsets the cleavage of the A-B and C-D bonds. The formation of AB⁺ requires bond cleavage without the offsetting bond formation. However, the steric effect makes it more difficult for the molecule to achieve the rearrangement transition state and form AD⁺. The activated complex with strict steric requirements is referred to as a "tight complex" whereas the transition state without such requirements is called a "loose complex."

• Metastability
• Transition state theory
• RRKM theory
• Tandem mass spectrometry