Total Cyclic Structural and Stereoisomers of Formula C5H10

To determine the total number of cyclic structural as well as stereoisomers for the molecular formula $\text{C}_5\text{H}_{10}$, we first calculate the degree of unsaturation (Double Bond Equivalent, $\text{DBE}$):

$\text{DBE} = C + 1 - \frac{H}{2} = 5 + 1 - \frac{10}{2} = 1$

Since the isomers must be cyclic, the DBE of $1$ corresponds entirely to a single ring (with no additional double bonds).

We categorize the possible cyclic isomers based on ring size:

1. Five-membered ring (Cyclopentane ring)

• Cyclopentane:
  • It has no stereocenters.
  • Number of isomers = $1$

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2. Four-membered ring (Cyclobutane ring)

• Methylcyclobutane:
  • The ring has a single substituent; it does not possess any chiral centers or geometric isomerism.
  • Number of isomers = $1$

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3. Three-membered ring (Cyclopropane ring)

The remaining two carbon atoms can be attached to the three-membered ring in the following ways:

• Ethylcyclopropane:

  • The substituted carbon is not a stereocenter.
  • Number of isomers = $1$

• 1,1-Dimethylcyclopropane:

  • Both methyl groups are attached to the same carbon, which has a plane of symmetry; no stereoisomerism exists.
  • Number of isomers = $1$

• 1,2-Dimethylcyclopropane:

  • The two methyl groups are attached to adjacent carbon atoms, leading to stereoisomerism:
    1. $\text{cis}$-1,2-dimethylcyclopropane: Possesses an internal plane of symmetry ($\sigma$), making it an achiral meso compound $\rightarrow$ $1$ isomer
    2. $\text{trans}$-1,2-dimethylcyclopropane: Lacks an internal plane of symmetry or center of inversion; it is chiral and exists as a pair of non-superimposable mirror images (enantiomers), namely $(1R, 2R)$ and $(1S, 2S)$ $\rightarrow$ $2$ isomers
  • Total isomers for 1,2-dimethylcyclopropane = $1 + 2 = \mathbf{3}$

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Total Count:

Summing all the cyclic structural isomers and stereoisomers:
$\text{Total isomers} = 1 + 1 + 1 + 1 + 3 = 7$