Calculate number of isomers of C5H10, which does not decolorise cold alkaline KMnO4 solution.

The correct answer is 7.

To find the number of isomers of C₅H₁₀ that do not decolorize cold alkaline KMnO₄ solution, we can follow these step-by-step logical deductions:

1. Understanding the Reaction with Cold Alkaline KMnO₄ (Baeyer's Reagent)
Cold alkaline KMnO₄ solution is a well-known reagent used to test for unsaturation (carbon-carbon double or triple bonds).
Alkenes undergo syn-hydroxylation with this reagent, decolorizing the purple KMnO₄ solution and forming a brown precipitate of MnO₂.
Since the isomers of C₅H₁₀ do not decolorize this solution, they must not contain any carbon-carbon double bonds. Therefore, these isomers must be cyclic saturated hydrocarbons (cycloalkanes).

2. Calculating the Double Bond Equivalent (DBE)
For a molecular formula of C_nH_2n, the degree of unsaturation or Double Bond Equivalent (DBE) is calculated as:
$\mathrm{DBE}=C+1-\frac{H}{2}$
Substituting C = 5 and H = 10 into the formula:
$\mathrm{DBE}=5+1-\frac{10}{2}=6-5=1$
A DBE of 1 indicates the presence of either one double bond or one ring. Since double bonds are ruled out by the chemical test, all possible isomers must contain exactly one ring (cycloalkanes).

3. Systematically Listing all Cycloalkane Isomers of C₅H₁₀
We can categorize the isomers based on the ring size:

A. Five-Membered Ring:
1. Cyclopentane (1 isomer)

B. Four-Membered Ring:
2. Methylcyclobutane (1 isomer)

C. Three-Membered Ring:
3. Ethylcyclopropane (1 isomer)
4. 1,1-Dimethylcyclopropane (1 isomer)
5. 1,2-Dimethylcyclopropane: This compound contains two chiral carbon centers, giving rise to stereoisomers:
- cis-1,2-Dimethylcyclopropane: It has a plane of symmetry, making it an optically inactive meso compound. (1 isomer)
- trans-1,2-Dimethylcyclopropane: It lacks a plane of symmetry and exists as a pair of non-superimposable mirror images (enantiomers), namely the (1R, 2R) and (1S, 2S) configurations. (2 isomers)

4. Total Count of Isomers
Adding up all the counted structural and stereoisomers:
Total isomers = 1 (cyclopentane) + 1 (methylcyclobutane) + 1 (ethylcyclopropane) + 1 (1,1-dimethylcyclopropane) + 1 (cis-1,2-dimethylcyclopropane) + 2 (trans-1,2-dimethylcyclopropane)
Total = 7 isomers.