Step 1: Understanding the Concept:
Covalent chemical bonds are classified based on the atomic orbital overlap patterns that create them. A Sigma ($\sigma$) bond is formed by the direct head-on (axial) overlap of atomic orbitals along the internuclear axis; every single covalent bond contains exactly one sigma bond. A Pi ($\pi$) bond is formed by the lateral side-by-side overlap of unhybridized parallel $p$-orbitals above and below the bonding axis.
Step 2: Key Formula or Approach:
To count bonds accurately, write out the explicit structural formula of the molecule to locate every single or multiple link:
- A Single Bond ($\text{A}-\text{B}$) consists of exactly: $1\sigma$ bond.
- A Double Bond ($\text{A}=\text{B}$) consists of exactly: $1\sigma$ bond + $1\pi$ bond.
- A Triple Bond ($\text{A}\equiv\text{B}$) consists of exactly: $1\sigma$ bond + $2\pi$ bonds.
Step 3: Detailed Explanation:
Let's analyze the molecular structure of ethene ($\text{C}_2\text{H}_4$):
Ethene is the simplest alkene, featuring two carbon atoms linked together by a double bond, with each carbon atom also bonding to two separate hydrogen atoms. Writing out its complete Lewis structural geometry reveals:
$$ \text{H}-\text{C}(\text{H})=\text{C}(\text{H})-\text{H} $$
Let's inventory every single bond type present in this structure:
1. Carbon-Hydrogen ($\text{C}-\text{H}$) links: There are $4$ separate single $\text{C}-\text{H}$ bonds in total ($2$ on each carbon atom). Each single link contributes exactly $1\sigma$ bond:
$$ 4 \times 1\sigma = 4\sigma \text{ bonds} $$
2. Carbon-Carbon ($\text{C}=\text{C}$) link: The two central carbon atoms are joined by a double bond. A double bond always consists of $1\sigma$ bond and $1\pi$ bond:
$$ 1 \times 1\sigma = 1\sigma \text{ bond} $$
$$ 1 \times 1\pi = 1\pi \text{ bond} $$
Now, add the individual values together to find the net bond count for the entire molecule:
- Total Sigma ($\sigma$) bonds = $4\sigma \text{ (from C-H)} + 1\sigma \text{ (from C=C)} = 5\sigma$
- Total Pi ($\pi$) bonds = $1\pi \text{ (from C=C)}$
The final count is $5\sigma$ and $1\pi$ bonds, which matches option (B).
Step 4: Final Answer:
The number of sigma ($\sigma$) and pi ($\pi$) bonds in ethene are 5$\sigma$ and 1$\pi$ respectively.