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The bond dissociation energies of the species \(\mathrm{NO}\) \(\mathrm{CF}^{-},\) and \(\mathrm{CF}^{+}\) follow the trend \(\mathrm{CF}^{+}>\mathrm{NO}>\mathrm{CF}^{-}\). Explain this trend using MO theory.

Short Answer

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The bond dissociation energies of the species NO, CF鈦, and CF鈦 can be explained using Molecular Orbital (MO) theory. According to MO theory, atomic orbitals combine to form bonding or antibonding molecular orbitals, which contribute to the strength of the bond. In the NO molecule, there are six electrons in bonding molecular orbitals and three electrons in antibonding molecular orbitals. For the CF鈦 ion, there are also six bonding electrons and three antibonding electrons, but due to the localization of the additional electron and its repulsion to the bonding electrons, the CF鈦 bond is weaker than the NO bond. For the CF鈦 ion, there are six bonding electrons and only two antibonding electrons, making the CF鈦 bond stronger than both the NO and CF鈦 bonds. Therefore, the observed trend in bond dissociation energies (CF鈦 > NO > CF鈦) can be explained by the net number of electrons in the bonding molecular orbitals and the effects of electron repulsion.

Step by step solution

01

Understanding Molecular Orbital (MO) Theory

MO theory is a method for explaining the electronic structure of molecules using principles of quantum mechanics. In this theory, atomic orbitals from each atom in the molecule combine or interact to form new molecular orbitals. These new orbitals can be bonding or antibonding based on the constructive or destructive overlap of the wave functions from the atomic orbitals. Bonding MOs are lower in energy and stabilize the molecule, while antibonding MOs are higher in energy and destabilize the molecule. The strength of the bond in the molecule is determined by the electron occupation in bonding and antibonding MOs.
02

Applying MO Theory to NO

In the NO molecule, the nitrogen (N) atom has the electronic configuration [He]2s虏2p鲁, and the oxygen (O) atom has the electronic configuration [He]2s虏2p鈦. Combining the atomic orbitals of N and O, we have five bonding molecular orbitals and five antibonding molecular orbitals. Filling the electrons, we get two electrons in 蟽 2s bonding, two electrons in 蟽 * 2s antibonding, four electrons in 蟺 2p bonding, and one electron in the antibonding MOs (either 蟺 * 2p or 蟽 * 2p). Hence, there are six electrons in bonding MOs and three electrons in antibonding MOs, which contribute to the overall bond strength.
03

Applying MO Theory to CF鈦

In the CF鈦 ion, the carbon (C) atom has the electronic configuration [He]2s虏2p虏, and the fluorine (F) atom has the electronic configuration [He]2s虏2p鈦. Adding an extra electron to form the CF鈦 ion results in two electrons in 蟽 2s bonding, two electrons in 蟽 * 2s antibonding, six electrons in 蟺 2p bonding, and one electron in either 蟺 * 2p or 蟽 * 2p antibonding MOs. Similar to NO, there are six electrons in bonding MOs and three electrons in antibonding MOs, but the CF鈦 bond is weaker in nature due to the localization of the extra electron and repulsion to bonding electrons.
04

Applying MO Theory to CF鈦

In the CF鈦 ion, the carbon (C) atom has the electronic configuration [He]2s虏2p虏, and the fluorine (F) atom has the electronic configuration [He]2s虏2p鈦. Removing an electron to form the CF鈦 ion results in two electrons in 蟽 2s bonding, two electrons in 蟽 * 2s antibonding, four electrons in 蟺 2p bonding, and only one electron in either 蟺 * 2p or 蟽 * 2p antibonding MOs. So, there are six electrons in bonding MOs and only two electrons in antibonding MOs, which makes the CF鈦 bond stronger than both NO and CF鈦 bonds.
05

Explaining the Trend

Now that we have applied MO theory to all three species, we can explain the trend in bond dissociation energies. CF鈦 has the highest bond dissociation energy because it has the highest net number of electrons in bonding MOs (6 - 2 = 4) compared to CF鈦 (6 - 3 = 3) and NO (6 - 3 = 3). CF鈦 has a weaker bond than NO due to the extra electron's repulsion with the bonding electrons, thus having a lower bond dissociation energy than NO. This gives us the trend: CF鈦 > NO > CF鈦, as observed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bond Dissociation Energy
Bond dissociation energy (BDE) is the amount of energy required to break a bond in a molecule into separate atoms while the molecule is in a gaseous state. BDE is a vital concept in chemistry because it provides insight into the strength of a bond. Generally speaking, the higher the bond dissociation energy, the stronger the bond.

When assessing the BDE of a molecule, we look at how bonding and antibonding molecular orbitals contribute to the molecular stability. If there are more electrons in bonding orbitals compared to antibonding orbitals, the molecule will be more stable, and thus, have a higher BDE. Factors influencing BDE include the types of atoms involved, the electron distribution, and the presence of charges in ionic species.

Using molecular orbital theory to compare BDE across species, we can make predictions about their chemical reactivity and stability. For instance, in the example of CF鈦, NO, and CF鈦, the species with the most electrons in bonding orbitals relative to antibonding orbitals (CF鈦) has the highest BDE, signifying the strongest bond. Conversely, the CF鈦 molecule has a relatively lower BDE due to the repulsion between the extra electron and the electrons in bonding orbitals, which weakens the bond.
Molecular Orbitals
Molecular orbitals are a cornerstone of molecular orbital theory, providing a framework for understanding the electronic structure of molecules. They are formed when atomic orbitals, the regions around an atom where electrons are likely to be found, combine as a molecule forms.

There are two main types of molecular orbitals 鈥 bonding and antibonding. Bonding orbitals result from the constructive interference of atomic orbital wave functions, leading to a lower energy state that promotes electrons to pair up and stabilize the molecule. Antibonding orbitals, on the other hand, occur from destructive interference, leading to a higher energy state that is typically empty or less filled in stable molecules.

The concept of molecular orbitals helps us elucidate why some molecules hold together more tightly than others. For example, the CF鈦 ion is more stable than its neutral or negatively charged counterparts because it has fewer electrons in its antibonding orbitals, which strengthens the bond between carbon and fluorine.
Electronic Structure
The electronic structure of a molecule describes the arrangement of electrons in the molecular orbitals. It encompasses the total number of electrons and their distribution among the different orbitals. This arrangement is pivotal for determining not only the reactivity and properties of a molecule but also its shape and symmetry.

To decipher the electronic structure, one must consider the molecular orbital diagram, which shows the energy levels and occupancy of the molecular orbitals. In the molecular orbital diagram for a molecule like NO or the ions CF鈦 and CF鈦, we fill the diagram according to the Aufbau principle, placing electrons in the lowest energy orbitals first.

The difference in electronic structures of ions such as CF鈦 and CF鈦 can be attributed to the gain or loss of an electron, affecting how the orbitals are filled and thereby impacting the molecule's overall stability and bond strength. For example, the removal of an electron when forming CF鈦 leads to fewer electrons in the antibonding orbitals, hence a stronger bond, compared to CF鈦, which has an additional electron causing extra repulsion.

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Most popular questions from this chapter

Arrange the following in terms of decreasing bond energy and bond length: \(\mathrm{O}_{2}^{+}, \mathrm{O}_{2}, \mathrm{O}_{2}^{-},\) and \(\mathrm{O}_{2}^{2-}\).

Calculate the bond order in each of the following species. Which of the species in parts \((a-d)\) do you expect to have the shorter bond length? a. \(\mathrm{Li}_{2}\) or \(\mathrm{Li}_{2}^{+}\) b. \(\mathrm{C}_{2}\) or \(\mathrm{C}_{2}^{+}\) \(\mathbf{c} . \mathrm{O}_{2}\) or \(\mathrm{O}_{2}^{+}\) d. \(\mathrm{F}_{2}\) or \(\mathrm{F}_{2}\)

Evaluate the energy for the two MOs generated by combining a H1s and a \(\mathrm{F} 2 p\) AO. Use Equation (23.12) and carry out the calculation for \(S_{H F}=0.075,0.18,\) and 0.40 to mimic the effect of increasing the atomic separation in the molecule. Use the parameters \(H_{11}=-13.6 \mathrm{eV}\) \\[H_{22}=-18.6 \mathrm{eV}, \text { and } H_{12}=-1.75 S_{12} \sqrt{H_{11} H_{22}}\\] Explain the trend that you observe in the results.

Using \(\zeta\) as a variational parameter in the normalized function \(\psi_{H 1 s}=1 / \sqrt{\pi}\left(\zeta / a_{0}\right)^{3 / 2} e^{-\zeta r / a_{0}}\) allows one to vary the size of the orbital. Show this by calculating the probability of finding the electron inside a sphere of radius \(a_{0}\) for different values of \(\zeta\) using the standard integral \\[\int x^{2} e^{-a x} d x=-e^{-a x}\left(\frac{2}{a^{3}}+2 \frac{x}{a^{2}}+\frac{x^{2}}{a}\right)\\] a. Obtain an expression for the probability as a function of \(\zeta\) b. Evaluate the probability for \(\zeta=1.5,2.5,\) and 3.5

Make a sketch of the highest occupied molecular orbital (HOMO) for the following species: a. \(N_{2}^{+}\) b. \(\mathrm{Li}_{2}^{+}\) c. \(\mathrm{O}_{2}^{-}\) d. \(\mathrm{H}_{2}\) e. \(C_{2}^{+}\)

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