1. Identify the possible isomers of $C_4H_8$ among the options

The given options are alkenes:

• But-1-ene: $CH_2=CH-CH_2-CH_3$
• cis-But-2-ene: $CH_3-CH=CH-CH_3$
• trans-But-2-ene: $CH_3-CH=CH-CH_3$
• 2-methylpropene: $(CH_3)_2C=CH_2$

2. Recall oxidation of alkenes with hot acidic $KMnO_4$

Oxidative cleavage of the double bond occurs.

General results:

• A double-bond carbon with:
  • two hydrogens $\to CO_2$
  • one hydrogen $\to$ carboxylic acid
  • no hydrogen $\to$ ketone

The question says:

• effervescence of a gas is observed
• a ketone is formed

Effervescence of gas indicates formation of $CO_2$. So one carbon of the double bond must become $CO_2$. Formation of ketone means the other double-bond carbon must have no hydrogen.

Thus the alkene must be of type: $R_2C=CH_2$ On oxidation: $R_2C=CH_2 \xrightarrow[H^+]{KMnO_4} R_2C=O + CO_2 + H_2O$

3. Check each option

Option A: But-1-ene

Structure: $CH_2=CH-CH_2-CH_3$ Oxidative cleavage gives:

• terminal $CH_2 \to CO_2$
• the other carbon has one H $\to$ carboxylic acid So products are $CO_2$ and propanoic acid, not ketone.

Therefore, A is incorrect.

Option B: cis-But-2-ene

Structure: $CH_3-CH=CH-CH_3$ Each double-bond carbon has one H, so oxidation gives two molecules of acetic acid: $CH_3COOH + CH_3COOH$ No ketone, no $CO_2$ effervescence.

Therefore, B is incorrect.

Option C: trans-But-2-ene

Same oxidative cleavage behavior as cis-but-2-ene. Products are two molecules of acetic acid. No ketone, no $CO_2$ effervescence.

Therefore, C is incorrect.

Option D: 2-methylpropene

Structure: $(CH_3)_2C=CH_2$ On oxidative cleavage:

• terminal $CH_2 \to CO_2$
• substituted carbon with no H $\to$ ketone Thus: $(CH_3)_2C=CH_2 \xrightarrow[H^+]{KMnO_4} (CH_3)_2CO + CO_2 + H_2O$ Ketone formed is acetone, and gas evolved is $CO_2$.

Therefore, D is correct.

4. Final conclusion

The compound that gives effervescence due to $CO_2$ and forms a ketone on oxidation with acidic $KMnO_4$ is: $\boxed{\text{2-methylpropene}}$