1. Molecular formula and degree of unsaturation

For $C_4H_8O$,

$\text{DBE} = \frac{2C+2-H}{2} = \frac{2(4)+2-8}{2} = \frac{2}{2} = 1$

So there is one degree of unsaturation. Since the compounds asked are cyclic ethers, this one unsaturation is due to one ring.

We must count all cyclic ethers with formula $C_4H_8O$, including structural isomers + stereoisomers.

2. Possible cyclic ether ring sizes

Total atoms besides hydrogen are $4$ carbons and $1$ oxygen. The ring can be:

• 5-membered ring: $4C + 1O$ in ring
• 4-membered ring: $3$ atoms in ring? Actually for a cyclic ether with one O, a 4-membered ring contains $3C + 1O$, leaving one carbon as substituent
• 3-membered ring: $2C + 1O$, leaving two carbons as substituent(s)

Now enumerate systematically.

3. Five-membered ring isomer

(i) Tetrahydrofuran

A 5-membered cyclic ether with one oxygen:

$\text{tetrahydrofuran}$

This has formula $C_4H_8O$.

Number of isomers here = 1.

4. Four-membered ring isomers (oxetanes)

Ring = oxetane ($3C + 1O$), plus one extra carbon as a methyl substituent.

Because positions adjacent to oxygen are equivalent, there are only two structural possibilities:

(ii) 2-methyloxetane

Substituent on carbon adjacent to O.

This carbon becomes a stereogenic center because it is attached to:

• $CH_3$
• $H$
• one side of ring through $O$
• other side of ring through $CH_2$

These four groups are different, so it exists as two enantiomers.

Count = 2.

(iii) 3-methyloxetane

Substituent on carbon opposite to O.

No stereocenter is present here.

Count = 1.

So total from oxetanes =

$2+1=3$

Running total =

$1+3=4$

5. Three-membered ring isomers (epoxides / oxiranes)

Ring = oxirane ($2C + 1O$), and we still need two carbons outside ring.

These two carbons can appear as either:

• one ethyl substituent on one ring carbon, or
• two methyl substituents on ring carbon(s)

Now enumerate.

(iv) 2-ethyloxirane

One ethyl substituent on one carbon of oxirane.

That substituted carbon is attached to:

• $C_2H_5$
• $H$
• $O$
• the other ring carbon

Hence it is a stereogenic center. So this gives 2 enantiomers.

Count = 2.

(v) 2,2-dimethyloxirane

Both methyl groups on the same ring carbon.

No stereocenter.

Count = 1.

(vi) 2,3-dimethyloxirane

One methyl on each ring carbon.

This has stereoisomerism:

• cis-2,3-dimethyloxirane
• trans-2,3-dimethyloxirane

Now check chirality:

• The cis form has a plane of symmetry, so it is meso / achiral: 1 isomer.
• The trans form is chiral and exists as a pair of enantiomers: 2 isomers.

So total for 2,3-dimethyloxirane =

$1+2=3$

Thus total from epoxides =

$2+1+3=6$

Running total =

$4+6=10$

6. Final count

Total number of isomers of cyclic ethers with formula $C_4H_8O$:

$1 + 3 + 6 = 10$

7. Comparison with stored answer

Stored correct answer = 10.

Our derived answer = 10.

So the stored answer is correct.