Bond Dissociation Energies

Abstract

The bond dissociation energy, D _e , is a critical parameter in thermodynamics, spectroscopy, and kinetics. For example, in thermodynamics, bond energies directly affect heats of formation and reaction. Dissociation energies thus also play key roles determining rates of reaction as a function of temperature. This chapter will give several examples of how spectroscopic information can reveal D _e (of ground and excited electronic states).

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The bond dissociation energy, D _e , is a critical parameter in thermodynamics, spectroscopy, and kinetics. For example, in thermodynamics, bond energies directly affect heats of formation and reaction. Dissociation energies thus also play key roles determining rates of reaction as a function of temperature. This chapter will give several examples of how spectroscopic information can reveal D _e (of ground and excited electronic states).

3.1 Birge–Sponer Method

A simple (but highly approximate) model, known as the Birge–Sponer method, can be used to directly convert spectroscopic parameters to dissociation energies [1]. It is based on a model of constant anharmonicity. If the anharmonicity is constant, the vibrational level spacing decreases to zero in the limit of dissociation.

$$\displaystyle\begin{array}{rcl} G(v)& =& \omega _{e}(v + 1/2) -\omega _{e}x_{e}(v + 1/2)^{2}{}\end{array}$$

(3.1)

$$\displaystyle\begin{array}{rcl} G(v + 1)& =& \omega _{e}(v + 3/2) -\omega _{e}x_{e}(v + 3/2)^{2}{}\end{array}$$

(3.2)

$$\displaystyle\begin{array}{rcl} \Delta G(v)& =& G(v + 1) - G(v) = -2\omega _{e}x_{e}v + (\omega _{e} - 2\omega _{e}x_{e}){}\end{array}$$

(3.3)

The expression above [Eq. (3.3)] is linear with the form

$$\displaystyle{ \Delta G(v) = av + b }$$

(3.4)

where the slope, a, is − 2ω _e x _e and the intercept, b, is $\omega _{e} - 2\omega _{e}x_{e}$. $\Delta G$, the separation between vibrational transitions, goes to zero when v = v _D , as shown in Fig. 3.1, and the corresponding dissociation energy is G(v _D ). (Note that G(v _D ) is also proportional to the integrated area under the curve of Fig. 3.1.)

Fig. 3.1

Illustration of Birge–Sponer method for finding the dissociative vibrational level, v _D , and the dissociation energy, D _e

The “real” values of $\Delta G(v)$ are not perfectly linear with quantum number v because the anharmonicity of a real molecule tends to increase near the dissociation limit. The estimation by the Birge–Sponer method thus tends to overpredict the dissociation energy, D _e .

At the dissociation limit, v = v _D and $\Delta G(v_{D}) = 0$.

$$\displaystyle{ \Delta G(v_{D}) = 0 = -2\omega _{e}x_{e}v_{D} + (\omega _{e} - 2\omega _{e}x_{e}) }$$

(3.5)

Solving for v _D yields

$$\displaystyle{\fbox{ $v_{D} = \frac{\omega _{e}} {2\omega _{e}x_{e}} - 1$} }$$

(3.6)

Substituting Eq. (3.6) into

$$\displaystyle{ D_{e} = \omega _{e}(v_{D} + 1/2) -\omega _{e}x_{e}(v_{D} + 1/2)^{2} }$$

(3.7)

gives

$$\displaystyle{\fbox{ $D_{e} = \frac{\omega _{e}^{2}} {4\omega _{e}x_{e}} -\frac{\omega _{e}x_{e}} {4} \approx \frac{\omega _{e}^{2}} {4\omega _{e}x_{e}} = \frac{\omega _{e}} {4x_{e}}$} }$$

(3.8)

Example:HCl

For HCl, ω _e  = 2990 cm⁻¹ and x _e  = 0. 0174. Thus,

$$\displaystyle{v_{D} = 27.7 \rightarrow 27\ \text{next lowest integer}}$$

and

$$\displaystyle{D_{e} = 513\,\text{kJ/mole.}}$$

A more accurate value, based on known thermochemistry, is D _e  = 427 kJ/mole. Thus, Birge–Sponer gives a good first estimate, but overpredicts D _e  by about 20 %.

3.2 Thermochemical Approach

The thermochemical approach is another method for finding D _e but can also depend indirectly on spectroscopic data. This method is based on evaluating the extent of a reaction as a function of temperature. For example, if the reaction were I$_{2} \rightarrow 2$ I, then the equilibrium constant is given by

$$\displaystyle{K_{p} = \frac{P_{\mathrm{I}}^{2}} {P_{\mathrm{I}_{2}}} }$$

and its change with temperature may be given by Van’t Hoff’s equation:

$$\displaystyle{\frac{d(\ln K_{p})} {dT} = \frac{\Delta H} {RT^{2}}\ \ }$$

where

$$\displaystyle{\Delta H =\sum v_{i}H_{i} = H_{\mathrm{prod}} - H_{\mathrm{react}} = D_{e} + 2\int _{\mathrm{I}}\hat{c}_{p}dT -\int _{\mathrm{I}_{2}}\hat{c}_{p}dT}$$

Thus, measurements of partial pressures can be used to infer K _p (T) and hence both $\Delta H$ and D _e ″. Partial pressures or species concentrations are often measured spectroscopically (e.g., by laser absorption) because these techniques provide an experimentalist with the ability to accurately measure a single species within a mixture.

3.3 Predissociation

Another way of establishing key energies is with the curve-crossing method, so named because it refers to a predissociative excited electronic state whose potential curve can cross the potential curve of the ground electronic state [1]. Two examples are shown below: HNO (nitroxyl) and N₂O (nitrous oxide).

3.3.1 HNO

Even though HNO is nonlinear (Fig. 3.2), we can plot its energy level diagram versus H–NO bond distance (i.e., for dissociation of HNO to H+NO) (Fig. 3.3).

Fig. 3.2

HNO structure

Fig. 3.3

Curve-crossing method for HNO dissociation

$$\displaystyle{\Delta E \approx \Delta H_{R},}$$

but

$$\displaystyle{\Delta H_{R} = \Delta H_{f}^{\mathrm{H}} + \Delta H_{ f}^{\mathrm{NO}} - \Delta H_{ f}^{\mathrm{HNO}}}$$

Therefore, we can solve for $\Delta H_{f}^{\mathrm{HNO}}$ from an estimate of $\Delta E$, and knowledge of $\Delta H_{f}^{\mathrm{H}}$ and $\Delta H_{f}^{\mathrm{NO}}$.

Recall:

• 1 $\frac{\mathrm{kcal}} {\mathrm{mol}} = 349.7\,\mathrm{cm^{-1}}$

• 1 cal = 4. 187 J

Note:

• 1. Dissociation (without curve-crossing) of the ground state HNO leads to NO$(^{2}\Sigma )$, rather than the lower energy state NO$(^{2}\Pi )$.

• 2. Predissociation occurs at 590 nm (17,000 cm⁻¹) in absorption spectra, corresponding to A ≈ 49 kcal/mol. This is then an upper bound on $\Delta H_{R}$.

• 3. $\Delta H_{R}(0\ \text{K}) \approx \Delta H_{f}^{\mathrm{H}} + \Delta H_{f}^{\mathrm{NO}} - \Delta H_{f}^{\mathrm{HNO}}$, and hence we can use the upper bound value of $\Delta H_{R}$, and known values for $\Delta H_{f}^{\mathrm{H}}$ and $\Delta H_{f}^{\mathrm{NO}}$ to establish a value for the heat of formation of HNO, $\Delta H_{f}^{\mathrm{HNO}}$.

• 4. Because both electronic states of HNO have the same multiplicity (singlet states with spin = 0), there is an allowed absorption spectrum (Fig. 3.4).

  Fig. 3.4

  

  Allowed absorption spectrum for HNO

• 5. For polyatomic molecules, the electronic term symbols include a tilde ($\sim$) over the initial symbol; Roman symbols are used to denote the electronic structure (e.g.,¹ A″ and¹ A′) unless the molecule is linear, in which case Greek symbols are used (e.g., $^{1}\Sigma$ or $^{3}\Pi$).

3.3.2 N₂O

Nitrous oxide is important in combustion chemistry and is linked to NO production and the greenhouse effect. N₂O is also a source of atomic oxygen in shock tube kinetics experiments (Fig. 3.5).

Fig. 3.5

N₂O structure

There are three relevant energies on the diagram in Fig. 3.6: the depth of the bound potential well, D _e ″; the energy of the curve intersection, E _act; and the difference in energy between the repulsive state products and the bottom of the ground state potential well.

Fig. 3.6

Potential energy wells for N₂O

Note:

• 1. There is a spin change between the ground electronic state of N₂O and the excited repulsive state shown. As a result, there is no strong absorption process between these electronic states.

• 2. The dissociation products may result in either O(¹ D) or O(³ P), but the latter is lower in energy, and hence more likely.

• 3. A measurement of the dissociation rate for

  $$\displaystyle{\mathrm{N_{2}O + M \rightarrow \mathrm{N_{2}} + O + M},}$$

  e.g. in a shock tube gives E _act, i.e. $k \propto \exp (-65\ \mathrm{[kcal/mole]}/RT)$.

• 4. The observed activation energy of 65 kcal/mol provides a lower bound on the dissociation energy, D _e ″.

3.4 Exercises

• 1. A banded structure is observed in the absorption spectrum of ground-state oxygen, which changes to a continuum at a wavelength corresponding to 7.047 eV. The upper electronic state of molecular oxygen dissociates into one ground state (³ P) atom and one excited (¹ D) atom; the excitation energy of the (¹ D) atom relative to the (³ P) atom is 1.967 eV. Determine D″ _o for O₂ in kcal/mole.

• 2. The zero-point energy of the ground state of O₂ is 793 cm⁻¹, and the difference in energy between the potential-energy minima of the two electronic states, T _e , of Problem 1 is 49,800 cm⁻¹. Determine D _e  for the upper and ground states of O₂ in cm⁻¹.

• 3. Partial electronic band origin data for an absorption spectrum from the ground electronic state is listed below.

  v′	v″ = 0
  1	31,800 cm−1
  2	32,400 cm−1
  3	32,800 cm−1

  • (a) Estimate the bond dissociation energy of the upper electronic state

  • (b) If G(v″ = 0) is 575 cm⁻¹, what is T _e in cm⁻¹?

• 4. Given the following band origin data (in cm⁻¹) for an electronic system:

  • (a) Determine ω _e ′, ω _e x _e ′, ω _e ″, ω _e x _e ″

  • (b) Calculate v′ _D and D′ _e using the Birge–Sponer method

  v′	v″ = 0	v″ = 1	v″ = 2	v″ = 3
  0	25,000
  1		24,700
  2	26,300
  3	26,800	25,800	24,800	23,800