Question

The three vibrational frequencies in \(\mathrm{H}_{2} \mathrm{O}\left(1595,3657, \text { and } 3756 \mathrm{cm}^{-1}\right)\) are all much larger than the corresponding frequencies in \(\mathrm{D}_{2} \mathrm{O}(1178,1571,\) and \(2788 \mathrm{cm}^{-1}\) ). This follows from the fact that vibrational frequency is given by the square root of a (mass-independent) quantity, which relates to the curvature of the energy surface at the minima, divided by a quantity that depends on the masses of the atoms involved in the motion. As discussed in Section \(26.8 .4,\) vibrational frequencies enter into both terms required to relate the energy obtained from a quantum chemical calculation (stationary nuclei at 0 K) to the enthalpy obtained experimentally (vibrating nuclei at finite temperature), as well as the entropy required to relate enthalpies to free energies. For the present purpose, focus is entirely on the so-called zero point energy term, that is, the energy required to account for the latent vibrational energy of a molecule at \(0 \mathrm{K}\) The zero point energy is given simply as the sum over individual vibrational energies (frequencies). Thus, the zero point energy for a molecule in which isotopic substitution has resulted in an increase in mass will be reduced from that in the unsubstituted molecule: A direct consequence of this is that enthalpies of bond dissociation for isotopically substituted molecules (light to heavy) are smaller than those for unsubstituted molecules. a. Perform B3LYP/6-31G* calculations on HCl and on its dissociation products, chlorine atom and hydrogen atom. Following geometry optimization on HCl, calculate the vibrational frequency for both HCl and DCl and evaluate the zero point energy for each. In terms of a percentage of the total bond dissociation energy, what is the change noted in going from HCl to DCl? \(\mathrm{d}_{1}\) -Methylene chloride can react with chlorine atoms in either of two ways: by hydrogen abstraction (producing HCl) or by deuterium abstraction (producing DCl): Which pathway is favored on the basis of thermodynamics and which is favored on the basis of kinetics? b. Obtain the equilibrium geometry for dichloromethyl radical using the B3LYP/6-31G* model. Also obtain vibrational frequencies for both the unsubstituted and the deuterium-substituted radical and calculate zero point energies for the two abstraction pathways (you already have zero point energies for HCl and DCl). Which pathway is favored on the basis of thermodynamics? What would you expect the (thermodynamic) product ratio to be at room temperature? c. Obtain the transition state for hydrogen abstraction from methylene chloride using the B3LYP/6-31G* model. A reasonable guess is shown here: Calculate vibrational frequencies for the two possible structures with one deuterium and evaluate the zero point energies for these two structures. (For the purpose of zero point energy calculation, ignore the imaginary frequency corresponding to the reaction coordinate.) Which pathway is favored on the basis of kinetics? Is it the same or different from the thermodynamic pathway? What would you expect the (kinetic) product ratio to be at room temperature?

Short answer

In this exercise, we performed B3LYP/6-31G* calculations on HCl, DCl, and the dichloromethyl radical to obtain optimized geometries, vibrational frequencies, and zero-point energies. We compared the bond dissociation energies between HCl and DCl and evaluated the thermodynamic and kinetic favorability of different reaction pathways involving hydrogen and deuterium abstraction. The thermodynamically favored pathway and product ratio at room temperature were determined using the obtained zero point energies, and the favored pathway based on kinetics was determined by analyzing the transition state structures and their zero-point energies.

Step-by-step solution

Perform B3LYP/6-31G* Calculations for HCl and DCl

To begin, perform B3LYP/6-31G* calculations on HCl and its dissociation products. This will allow you to obtain optimized geometries and frequencies for both HCl and DCl.

Evaluate Zero-Point Energies

Using the vibrational frequencies obtained from the B3LYP/6-31G* calculations, evaluate the zero-point energies for HCl and DCl. This can be done by summing over the individual vibrational energies.

Calculate Bond Dissociation Energies and Compare

Determine the bond dissociation energies for HCl and DCl, and compare them as a percentage of the total bond dissociation energy. This will show the change in energy when going from HCl to DCl. b. Determine Favored Pathway based on Thermodynamics

Obtain Equilibrium Geometry for Dichloromethyl Radical

Perform B3LYP/6-31G* calculations to obtain the equilibrium geometry for the unsubstituted and deuterium-substituted dichloromethyl radical. This will be used for further analysis of the abstraction pathways.

Calculate Zero-Point Energies of Abstraction Pathways

Using the vibrational frequencies obtained for both the unsubstituted and deuterium-substituted radicals, calculate the zero-point energies. Alongside the previously calculated zero-point energies for HCl and DCl, this will help determine the favored pathway based on thermodynamics.

Thermodynamic Product Ratio

Using the calculated values, determine which reaction pathway is thermodynamically favored and calculate the expected thermodynamic product ratio at room temperature. c. Determine Favored Pathway based on Kinetics

Obtain Transition State

Perform B3LYP/6-31G* calculations to obtain the transition state for hydrogen abstraction from methylene chloride. You might want to use a reasonable guess shown in the exercise.

Calculate Zero-Point Energies for Transition State Structures

Calculate the vibrational frequencies for each of the two possible structures (with one deuterium) using the B3LYP/6-31G* calculations. Evaluate their zero-point energies, ignoring the imaginary frequency corresponding to the reaction coordinate.

Kinetic Product Ratio

Determine which reaction pathway is favored based on kinetics using the zero-point energies and compare it to the thermodynamically favored pathway. Calculate the expected kinetic product ratio at room temperature.

Key concepts

Zero Point Energy

Zero Point Energy (ZPE) refers to the minimum amount of energy that a quantum mechanical system will possess, even when at rest (0 K). Think of it as the lowest rung on an energy ladder that molecules can't step off. In the case of vibrations in molecules, ZPE is the energy due to vibrational motion that persists even when a molecule is cooled to absolute zero.

To calculate it, one sums the energies of all the individual vibrational modes of a molecule. The formula for a single vibrational mode is \( \frac{1}{2}hv \) where \( h \) is Planck's constant and \( v \) is the vibrational frequency. Thus, heavier isotopes tend to lower the ZPE because their vibrational frequencies are less due to greater mass, as seen with the comparison of \( \mathrm{H}_2\mathrm{O} \) and \( \mathrm{D}_2\mathrm{O} \).

Bond Dissociation Energy

Bond Dissociation Energy (BDE) is the energy required to break a bond in a molecule, producing isolated atoms. It's a critical concept for understanding the stability of molecules and the energy involved in chemical reactions.

Since the ZPE is part of the total energy of a molecule, when calculating BDE, the ZPE must be considered. As isotopic substitution increases the mass and lowers the ZPE, the BDE is also affected—typically reduced, because less energy is required to dissociate the heavier isotope-laden molecule. This is particularly relevant in reactions involving isotopes, as seen in the example of \( \mathrm{HCl} \) versus \( \mathrm{DCl} \) where the change of vibrational frequencies due to isotopic substitution affects the BDE.

Isotopic Substitution Thermodynamics

Isotopic Substitution Thermodynamics involves understanding how the replacement of an atom by its isotope affects the properties of a molecule. This substitution can change the mass, vibrational frequencies, and thus the ZPE.

The effects are quite significant when it comes to thermodynamic properties like enthalpy and entropy. Since vibrational frequencies are lower for heavier isotopes, ZPE decreases, and the substance becomes more thermodynamically stable. This concept can help predict the outcome of reactions—whether a reaction favors the lighter or heavier isotope, like in the thermodynamics of \( \mathrm{HCl} \) and \( \mathrm{DCl} \) where bond energies differ due to isotopic substitution.

B3LYP/6-31G* Calculations

B3LYP/6-31G* calculations are a computational method used in quantum chemistry to predict molecular structures, energies, and other properties. B3LYP refers to a hybrid functional, which combines Hartree-Fock methods with density functional theory (DFT), while 6-31G* represents a particular basis set describing the electrons in the system. This level of theory is commonly used for its balance between computational cost and accuracy.

In practice, B3LYP/6-31G* calculations optimize molecular geometries and predict vibrational frequencies—essential in determining the zero-point energies of molecules, as evidenced by the problems involving ZPE calculations for \( \mathrm{HCl} \) and \( \mathrm{DCl} \).

Transition State Analysis

Transition State Analysis focuses on identifying the configuration of a molecule at the highest energy point along the reaction pathway—the transition state. This transient state cannot be isolated, but its properties can be inferred through computational simulations such as B3LYP/6-31G* calculations.

By analyzing the transition state, chemists can predict the rate of a chemical reaction—higher energy barriers correspond to slower reactions. Transition states are crucial for understanding reaction kinetics, including predicting which pathway a reaction may prefer at room temperature, as in the determination of either hydrogen or deuterium abstraction from methylene chloride.

Enthalpy and Entropy in Physical Chemistry

Enthalpy and entropy are fundamental concepts in thermodynamics, representing heat content and the measure of disorder, respectively, in a physical system. Enthalpy changes indicate the heat released or absorbed during a reaction, while entropy measures the dispersal of energy. Both properties are influenced by vibrational frequencies.

When a reaction occurs, there is a change in enthalpy (\(\Delta H\)) and entropy (\(\Delta S\)), which interact to determine the free energy change (\(\Delta G\)). This determines whether a reaction is thermodynamically favorable. Calculating these properties can guide the prediction of reaction outcomes, including whether a reaction is endothermic or exothermic and the balance between order and chaos in a system's energy distribution.

Quantum Chemical Calculations

Quantum Chemical Calculations involve the use of quantum mechanics to determine the electronic structure of molecules. These calculations predict a wide range of properties, such as energy levels, molecular geometries, and reaction pathways. By solving Schrödinger's equation for a system, quantum chemistry offers insights into both the stationary and dynamic aspects of molecular behavior.

Methods such as the aformentioned B3LYP/6-31G* involve approximations because exact solutions are often not feasible for complex molecules. Nevertheless, these calculations provide invaluable information on the fundamental forces governing molecular interactions, helping to explain observations and predict new phenomena.

Kinetic vs Thermodynamic Control

Kinetic versus Thermodynamic Control refers to the divergent paths a reaction can take based on different reaction conditions. Kinetic control often involves lower temperatures and rapid reaction times, steering the reaction towards products that form faster, even if they are not the most stable. In contrast, thermodynamic control favors the most stable products, given sufficient time and higher temperatures, allowing the system to reach the lowest energy state.

Understanding this can predict which product will prevail under certain conditions. For instance, in reactions like those involving methylene chloride and chlorine atoms, the choice between the formation of HCl or DCl is influenced by whether the reaction is kinetically or thermodynamically controlled, and this choice determines the final product distribution at room temperature.