Question

Torsional strain resulting from eclipsed \(\mathrm{C}-\mathrm{H}\) bonds is approximately \(4.2 \mathrm{~kJ}(1.0\) \(\mathrm{kcal}) / \mathrm{mol}\), and that for eclipsed \(\mathrm{C}-\mathrm{H}\) and \(\mathrm{C}-\mathrm{CH}_{3}\) bonds is approximately \(6.3 \mathrm{~kJ}\) \((1.5 \mathrm{kcal}) / \mathrm{mol}\). Given this information, sketch a graph of energy versus dihedral angle for propane.

Short answer

Question: Sketch a graph of energy versus dihedral angle for propane using the provided torsional strain values (C-H eclipsed: \(4.2 \mathrm{~kJ/mol}\), C-H/C-CH3 eclipsed: \(6.3 \mathrm{~kJ/mol}\)). Answer: To sketch the energy versus dihedral angle graph for propane, plot points for the energy levels at different dihedral angles: (0°, 4.2 kJ/mol), (60°, 0), and (120°, 6.3 kJ/mol). Continue plotting points, considering the symmetry of propane and its conformations. Connect the points with a smooth curve, showing a periodic pattern of energy peaks and troughs corresponding to the eclipsed and staggered conformations of propane.

Step-by-step solution

Understand the dihedral angles and conformations of propane

In propane, there are two main conformations to consider - staggered and eclipsed. The dihedral angle is a measure of the angle between the hydrogen atoms or groups in these conformations. When the dihedral angle is 0°, the conformation is eclipsed, meaning the atoms are directly in line with each other. When the dihedral angle is 60°, the conformation is staggered, meaning the atoms are at a maximum distance from each other.

Calculate energy levels for various dihedral angles

Given the torsional strain values, we can calculate the energy levels for each conformation of propane: 1. At a dihedral angle of 0° (eclipsed conformation with C-H and C-H bonds): Energy = \(4.2 \mathrm{~kJ/mol}\) 2. At a dihedral angle of 60° (staggered conformation): Energy = 0, as there is no torsional strain in the staggered conformation. 3. At a dihedral angle of 120° (eclipsed conformation with C-H and C-CH3 bonds): Energy = \(6.3 \mathrm{~kJ/mol}\)

Sketch the energy versus dihedral angle graph

To sketch the graph, we will plot energy on the y-axis and dihedral angle on the x-axis. Using the energy levels calculated in step 2, we can plot the points on the graph and draw a smooth curve representing the energy levels for all dihedral angles from 0° to 360°. 1. Plot point (0°, 4.2 kJ/mol) 2. Plot point (60°, 0) 3. Plot point (120°, 6.3 kJ/mol) 4. Continue plotting points, considering the symmetry of propane and its conformations. 5. Draw a smooth curve connecting the points, representing the energy levels for all dihedral angles. The resultant graph will show a periodic pattern of energy peaks and troughs, corresponding to the eclipsed and staggered conformations of propane.

Key concepts

Torsional Strain

Torsional strain is a type of strain that occurs when molecules assume conformations that cause the atoms or groups of atoms to be eclipsed or overlapped as seen from a particular bond axis. This results in increased repulsion between electrons of the atoms or groups, which in turn raises the potential energy of the molecule. In organic chemistry, understanding torsional strain is crucial for predicting the stability of different molecular conformations.

Torsional strain is most evident when comparing staggered and eclipsed conformations around a single bond. In the staggered conformation, the bonds are as far apart as possible; thus, the torsional strain is minimal, if not entirely zero. However, in the eclipsed conformation, the bonds are aligned or 'eclipsed', leading to maximal torsional strain and a subsequent increase in the molecule's energy. For example, in propane, an eclipsed conformation would have higher torsional strain and energy than a staggered conformation.

Dihedral Angle

The dihedral angle in chemistry describes the angle between two planes formed by two pairs of atoms bonded to a central atom. Imagine you have two adjacent carbons in a hydrocarbon chain, with each having two hydrogens attached. The dihedral angle would be measured between any two hydrogens—one from each carbon. It's a crucial aspect of conformational analysis as it helps in determining the spatial relationship between atoms in a molecule.

For a clear understanding, envision a propeller with its blades set at different angles — the dihedral angle would be the angle between the planes of any two blades. This concept is vital when analyzing the conformations of molecules like propane, where dihedral angles significantly influence the potential energy and the overall stability of the molecule.

Propane Conformations

Propane, a three-carbon alkane, can adopt several conformations based on the rotations around its carbon-carbon bonds. The two primary conformations are staggered and eclipsed. When propane is in the staggered conformation, the H atoms on the central carbon are as far apart as possible from the H atoms on the adjacent carbons, which minimizes repulsion and, consequently, the torsional strain. The staggered form is, thus, the most stable.

Conversely, the eclipsed conformation has the H atoms on the central carbon aligned with those on the neighboring carbons, resulting in increased electron repulsion due to proximity and causing higher torsional strain. When studying propane, it's essential to visualize the molecule in three dimensions to accurately perceive these conformations and the resulting impact on its stability and reactivity.

Energy Versus Dihedral Angle Graph

An energy versus dihedral angle graph allows chemists to visualize the relationship between the conformational states of a molecule, like propane, and its potential energy. As the molecule rotates around its bonds, the dihedral angles change, and the potential energy will fluctuate accordingly. In such a graph, the x-axis represents the dihedral angle, which ranges from 0° to 360°, while the y-axis denotes the potential energy, typically expressed in kilojoules per mole (kJ/mol).

At 0°, corresponding to the eclipsed conformation, the graph will typically peak, indicating a high-energy state due to torsional strain. At 60°, the dihedral angle of the staggered conformation, the potential energy is at its lowest, reflecting a stable, low-strain state. The pattern then repeats every 120°, reflecting the symmetry in the rotations about the carbon-carbon bond. This visualization is a crucial tool in conformational analysis as it readily displays the energetically favorable and unfavorable orientations of atoms in a molecule.