Calculate the dihedral angle (θ) for the conformations shown.
(c)

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Step 1: Understand the concept of dihedral angle (θ). The dihedral angle is the angle between two planes formed by four atoms or groups in a molecule. In this case, the angle θ is between the bromine (Br) and methyl (CH₃) groups in the staggered conformation of the molecule.

Step 2: Analyze the Newman projection provided in the image. The front carbon atom has three substituents: hydrogen (H), bromine (Br), and hydrogen (H). The back carbon atom has three substituents: hydrogen (H), methyl (CH₃), and hydrogen (H). The dihedral angle is measured between the Br group on the front carbon and the CH₃ group on the back carbon.

Step 3: Identify the relative positions of the substituents. In the staggered conformation shown, the Br group and CH₃ group are separated by a specific angle. This angle is determined by the spatial arrangement of the groups in the Newman projection.

Step 4: Recall the standard dihedral angles for staggered conformations. In staggered conformations, the angles between substituents are typically 60°, 120°, or 180°. The image shows the Br and CH₃ groups separated by one-third of a full rotation, which corresponds to a dihedral angle of 120°.

Step 5: Confirm the dihedral angle by visual inspection. The Br group and CH₃ group are positioned such that they are not directly opposite (180°) or adjacent (60°), but rather separated by 120°. This matches the standard staggered conformation angle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Dihedral Angle

The dihedral angle, often denoted as θ, is the angle between two intersecting planes defined by four atoms in a molecule. It is crucial in understanding the spatial arrangement of atoms and how they influence molecular conformation. In the context of conformations, the dihedral angle helps determine the stability and reactivity of different molecular shapes.

Conformational Analysis

Conformational analysis is the study of the different spatial arrangements of atoms in a molecule that can be interconverted by rotation around single bonds. This analysis is essential for understanding the energy differences between various conformations, which can affect the physical and chemical properties of the compound. It often involves visualizing and calculating angles like the dihedral angle.

Steric Hindrance

Steric hindrance refers to the repulsion between atoms that occurs when they are brought close together, affecting the stability of molecular conformations. In the context of dihedral angles, steric hindrance can influence the preferred angles between substituents, impacting the overall conformation of the molecule. Understanding steric effects is vital for predicting the most stable conformations and their reactivity.

Calculate the dihedral angle (θ) for the conformations shown.(c)