Question

The dipole moment of 1,2 -dichloroethane is \(1.12 \mathrm{D}\) at room temperature. Show how one could use this dipole moment to calculate the proportions of the possible conformations that are expected to be present.

Short answer

To calculate the proportions of possible conformations of 1,2-dichloroethane, we first identify the two significant conformations: anti and gauche. We then calculate the dipole moments for each conformation, assuming C-C bond length of \(1.54\) Å, C-Cl bond length of \(1.75\) Å, and H-C-H bond angle of \(109.5^{\circ}\). The overall dipole moment for the anti conformation is 0 D, while the gauche conformation's dipole moment can be expressed as \(M_c = M_v + a\cos{\frac{\theta}{2}}\). Using the Boltzmann distribution for the conformations and the given overall dipole moment of 1.12 D, we can find the ratio of populations between gauche and anti conformations as \( \frac{N_{gauche}}{N_{anti}} = \frac{M}{N_{anti} * M_c} \). These proportions will depend on the specific values of bond lengths, angles, dihedral angles, and temperature.

Step-by-step solution

Identify Conformations of 1,2-Dichloroethane

1,2-dichloroethane, as an alkane with a C-C single bond, has various conformations in which the dihedral angles of the Chlorine atoms differ. In general, we look for staggered or eclipsed conformations; however, since we have substituted groups such as Cl, it is essential to consider the overlap of these groups in the conformations. In this case, we have two significant conformations to consider: 1. Anti conformation: The Cl atoms are pointed in opposite directions, and the dihedral angle between them is \(\approx180^{\circ}\) (Staggered conformation). 2. Gauche conformation: One Cl atom is at a dihedral angle of \(\approx60^{\circ}\) or \(\approx300^{\circ}\) concerning the other Cl atom.

Calculate the Dipole Moments of the Conformations

We shall now calculate the dipole moments of these conformations. We can assume that the C-C bond length is \(1.54\) Å, the C-Cl bond length is \(1.75\) Å, and the H-C-H bond angle is \(109.5^{\circ}\). 1. Anti conformation: The Cl atoms are opposite, so the overall dipole moment is 0 D. 2. Gauche conformation: In this conformation, the Cl-Cl dipole moment has both horizontal and vertical components. We can calculate the horizontal component (\(M_h\)) by using the following formula: \( M_h = 2 * \mu * \cos{\frac{\theta}{2}} \), where \(\mu\) is the C-Cl bond dipole moment and \(\theta\) is the dihedral angle between Cl atoms. Since \(\mu\) is unknown, we will express the horizontal component as \(M_h = a\cos{\frac{\theta}{2}}\). Next, we need to find the contribution of the C-H bonds. Their dipole moment (\(M_c\)) can be calculated by considering the contribution of the vertical (\(M_v\)) and horizontal (\(M_h\)) parts: \(M_c = M_v + a\cos{\frac{\theta}{2}}\).

Determine the Proportions of Conformations

Using the Boltzmann distribution for the conformations, we can determine the proportion of the conformations by calculating the ratio of their populations: \(ratio = \frac{N_{gauche}}{N_{anti}} = e^{-\frac{\Delta E}{RT}} \), where \(N_{gauche}\) and \(N_{anti}\) are populations of gauche and anti conformations, respectively, \(\Delta E\) is the difference in energy between the conformations, \(R\) is the gas constant, and \(T\) is the temperature. Since the dipole moment of the entire molecule is \(M = 1.12\text{D}\), we have that: \(M = N_{anti} * 0 + N_{gauche} * M_c\) Now, we can substitute the ratio of populations into the equation: \(M = N_{anti} * (ratio * M_c)\) Finally, we can solve the equation for \(ratio\): \(ratio = \frac{M}{N_{anti} * M_c}\) Using the given overall dipole moment of 1.12 D and the calculated values for the anti and gauche conformations, we can find the proportion of conformations expected to be present in the sample: \( \frac{N_{gauche}}{N_{anti}} = ratio \) Note that the proportion will depend on the actual values of bond lengths and angles, the dihedral angle between the Cl atoms, the dipole moment of the C-Cl bond, and the temperature. To calculate the proportions, use the provided values or any new experimental data.

Key concepts

1,2-Dichloroethane Conformations

Understanding the conformations of 1,2-dichloroethane is crucial in predicting its physical properties, such as its dipole moment. This molecule can adopt various spatial arrangements due to the rotation about the carbon-carbon single bond. Two primary conformations are significant: the anti and gauche forms. In the anti conformation, the two chlorine (Cl) atoms are positioned on opposite sides, leading to a dihedral angle close to 180 degrees. This staggered configuration minimizes electron repulsion and is typically more stable.

In the gauche conformation, the Cl atoms are closer, with a dihedral angle of roughly 60 or 300 degrees. This proximity leads to a higher electronic repulsion and is generally less stable compared to the anti conformation. These orientations have direct implications on the molecule's overall dipole moment—a vector quantity describing the separation of charge within the molecule—an essential factor in determining how 1,2-dichloroethane behaves in different environments.

Boltzmann Distribution

The Boltzmann distribution is a statistical distribution of energy states of molecules in thermal equilibrium. It is incredibly relevant in conformational analysis because it describes the probability of a molecule being found in a particular energy state at a given temperature. According to this principle, molecules at a higher energy level (like gauche conformations of 1,2-dichloroethane) will be less populated than those at a lower energy level (such as anti conformations), especially at room temperature. The ratio of the number of molecules in different conformations, like gauche to anti, can be expressed as \( ratio = \frac{N_{gauche}}{N_{anti}} = e^{-\frac{\Delta E}{RT}} \), where \( N_{gauche} \) and \( N_{anti} \) are the populations of each conformation, \( \Delta E \) is the energy difference between them, \( R \) is the gas constant, and \( T \) is temperature. This relationship shows that even small energy differences can significantly affect the population ratios at equilibrium.

Gauche and Anti Conformations

The gauche and anti conformations of 1,2-dichloroethane differ not only in stability but also in their dipole moments. Since the anti conformation has the Cl atoms positioned on opposite sides of the molecule, it effectively cancels out the dipole moments resulting in a net dipole moment of zero Debye (D). In contrast, the gauche conformation, with its Cl atoms at a closer vicinity and an angle leading to partial overlap, has a nonzero dipole moment due to the vector sum of individual dipole moments pointing in different directions.

The calculation of the dipole moment for the gauche conformation can be intricate because it involves summing vector components of both C-Cl and C-H bonds. The presence of a nonzero dipole moment in the gauche conformation contributes to the overall dipole moment of 1,2-dichloroethane in a sample, where both conformations are present in a ratio determined by their respective energies and temperatures.

Conformational Analysis

Conformational analysis is the study of the different shapes that a molecule can assume due to rotations around single bonds. It is key in understanding the physical and chemical properties of molecules like 1,2-dichloroethane. Scientists perform conformational analysis by calculating energies, dipole moments, and potential energy surface (PES) scans. Predicting the most stable conformations involves considering steric hindrance, electronic effects, and the molecule's substituents.

In the context of 1,2-dichloroethane, conformational analysis allows us to calculate the proportion of gauche and anti conformations present at equilibrium. By applying the dipole moment equations to the established Boltzmann distribution, we can determine the population ratio based on the measured overall dipole moment. This approach aids chemists and physicists in deciphering the behavior of chemical substances like 1,2-dichloroethane under various conditions and hence is instrumental in the fields of materials science and molecular engineering.