Let's examine each of the choices to identify which one shows a hydrogen bond.
1. Option 1: [tex]$H_3N \cdots \cdots H - O - H$[/tex]
- In this option, we have [tex]$H_3N$[/tex] (ammonia) and [tex]$H - O - H$[/tex] (water).
- A hydrogen bond typically forms between a hydrogen atom that is covalently bonded to a highly electronegative atom (like nitrogen, oxygen, or fluorine) and another electronegative atom.
- Here, nitrogen (N) is a highly electronegative atom, and hydrogen (H) is bonded to oxygen (O) in water, which is also highly electronegative.
- The interaction [tex]$H_3N \cdots \cdots H - O - H$[/tex] represents a hydrogen bond between the hydrogen of the water molecule and the nitrogen in ammonia.
2. Option 2: [tex]$H - H$[/tex]
- This represents a simple covalent bond between two hydrogen atoms.
- There is no electronegative atom involved to facilitate a hydrogen bond.
3. Option 3: [tex]$H_2O \cdots \cdots\cdot H - CH_3$[/tex]
- In this option, we have [tex]$H_2O$[/tex] (water) and [tex]$H - CH_3$[/tex] (a methyl group bonded to hydrogen).
- Although oxygen (in [tex]$H_2O$[/tex]) is electronegative, the hydrogen bonded to the carbon of the methyl group does not participate in a hydrogen bond because carbon is not electronegative enough to create the bond.
4. Option 4: [tex]$H_4C \cdots \cdots H - F$[/tex]
- This represents a hypothetical situation where methane ([tex]$CH_4$[/tex], but written here as [tex]$H_4C$[/tex]) possibly interacts with hydrogen fluoride ([tex]$HF$[/tex]). However, the representation is fundamentally incorrect as methanol ([tex]$H_4C$[/tex]) does not exist in that form and no hydrogen bond would be formed in this given configuration.
Based on the detailed analysis of each option, the choice that shows a hydrogen bond is:
Option 1: [tex]$H_3N \cdots \cdots H - O - H$[/tex].
