Question

If \(x=\) Total number of plane of symmetry in eclipsed conformation of ethane and \(y=\) Total number of two fold axis of symmetry \(\left(\mathrm{C}_{2}\right)\) in methane. Then value of \((x+y)\) is: (A) 4 (B) 7 (C) 6 (D) 5

Short answer

The short answer is: In the eclipsed conformation of ethane, there are 3 planes of symmetry, and in methane, there are 3 two-fold axes of symmetry. Therefore, the value of (x+y) is 6. The correct option is (C) 6.

Step-by-step solution

Total planes of symmetry (x) in eclipsed conformation of ethane

To find the total number of planes of symmetry in the eclipsed conformation of ethane, we must visualize its molecular structure: Ethane (C₂H₆) consists of two carbon atoms forming a single bond, with six hydrogen atoms attached to the carbons. In the eclipsed conformation, the two carbon atoms and their respective hydrogen atoms are aligned, making the molecule symmetric. Drawing the planes of symmetry, it is observed that there are three planes of symmetry when we bisect through the carbon-carbon bond and with each plane containing one hydrogen bond from each carbon atom. Therefore, x = 3.

Total two-fold axes of symmetry (y) in methane

Now we need to find the total number of two-fold axes of symmetry (y) in methane. Methane (CH₄) consists of one carbon atom bonded to four hydrogen atoms in a tetrahedral arrangement. In order to find two-fold axes of symmetry, we must find lines on which, upon rotating the molecule by 180° (about that line), the molecular arrangement will remain the same. In methane, we have three two-fold axes of symmetry, each passing through one pair of hydrogen atoms, and the central carbon atom. Therefore, y = 3.

Calculating the total value of (x+y)

Now that we have the values of x and y, we can simply add them together: (x+y) = 3 (total planes of symmetry in ethane) + 3 (total two-fold axes of symmetry in methane) = 6 So, the correct answer is (C) 6.

Key concepts

Plane of Symmetry

The concept of symmetry is highly important in understanding molecular structures. A plane of symmetry, sometimes referred to as a mirror plane, is an imaginary plane that divides a molecule into two halves that are mirror images of each other. In other words, if you were to place a mirror on this plane, the molecule on one side would be reflected to show the exact same molecule on the other side.

Identifying planes of symmetry involves examining the spatial arrangement of atoms in a molecule. A molecule with a plane of symmetry must be identical on both sides of this plane. This concept is integral in identifying chirality; molecules without a plane of symmetry are chiral, which means they have distinct 'right-handed' and 'left-handed' forms, known as enantiomers. On the other hand, molecules with one or more planes of symmetry are achiral and do not exhibit this handedness.

Eclipsed Conformation of Ethane

When studying the structure of ethane, its conformations—ways in which the molecule rotates about the carbon-carbon bond—are key to understanding its symmetry. Ethane (C₂H₆) can exist in different conformations, but one of the most notable is the eclipsed conformation. In this arrangement, the hydrogen atoms connected to one carbon atom are aligned directly with the hydrogen atoms connected to the second carbon.

Despite appearing symmetrical, the eclipsed conformation is not the most stable due to the increased repulsion between the electron clouds of adjacent hydrogen atoms. However, this conformation does have symmetry elements, in particular, it has planes of symmetry that can be drawn through the carbon-carbon bond and align with the hydrogen atoms. In the exercise mentioned, we identified three such planes.

Two-Fold Axis of Symmetry

A two-fold axis of symmetry, also known as a C₂ axis, is an imaginary line that passes through the center of a molecule around which the molecule can be rotated by 180° to yield an indistinguishable arrangement of atoms from its original configuration. It's called two-fold because completing two 180° rotations (which amounts to a full 360° turn) brings the molecule back to its original orientation.

A molecule such as methane, with its tetrahedral shape, exhibits this two-fold axis. The two-fold axes pass through the molecule, bisecting bonds or travelling between pairs of atoms. Methane, specifically, has three such axes, each one passing through the midpoint of a pair of hydrogen atoms and the central carbon atom.

Methane Molecular Structure

Methane is the simplest hydrocarbon, composed of a single carbon atom bound to four hydrogen atoms. The 3D molecular structure of methane is tetrahedral. This means that the hydrogen atoms are positioned at the corners of a regular tetrahedron with the carbon atom at the center. This shape is highly symmetrical and results in equal bond angles of approximately 109.5° between any two bonds.

The tetrahedral arrangement allows for multiple symmetry elements including the three two-fold axes of symmetry we discussed before. Due to this symmetry, methane is non-polar – it has no dipole moment – as the bond dipoles from the carbon to the hydrogens cancel each other out. Understanding the structure of methane is not only important for grasping molecular symmetry but also for predicting its chemical behavior and reactivity.