What is the correct order of the rate of alkaline hydrolysis of these esters?
`CH_(3)COOCH_(3)["rate "=r_(1)]`,
`CH_(3)COOC_(2)H_(5)["rate "=r_(2)]`,
`CH_(3)COOC_(3)H_(7)["rate "=r_(3)]`

Options

A. `r_(1) gt r_(2) gt r_(3)`
B. `r_(1) lt r_(2) lt r_(3)`
C. `r_(1) lt r_(2) gt r_(3)`
D. `r_(1) gt r_(2) lt r_(3)`

Answer

To determine the correct order of the rate of alkaline hydrolysis of the given esters, we need to analyze the structure of each ester and the effect of the alkyl groups on the carbonyl carbon's electrophilicity.

### Step-by-Step Solution:

1. **Identify the Esters**:
- The esters given are:
- \( CH_3COOCH_3 \) (Methyl acetate)
- \( CH_3COOC_2H_5 \) (Ethyl acetate)
- \( CH_3COOC_3H_7 \) (Propyl acetate)

2. **Understand Alkaline Hydrolysis**:
- Alkaline hydrolysis involves the nucleophilic attack of hydroxide ions (\( OH^- \)) on the carbonyl carbon of the ester. The rate of this reaction depends on the electrophilicity of the carbonyl carbon.

3. **Analyze the Inductive Effect**:
- The alkyl groups attached to the ester influence the electron density on the carbonyl carbon through the +I (inductive) effect. Alkyl groups are electron-donating, which can reduce the electrophilicity of the carbonyl carbon.
- The more electron-donating the alkyl group, the slower the reaction, as the carbonyl carbon becomes less positive and less susceptible to nucleophilic attack.

4. **Compare the Alkyl Groups**:
- **Methyl group (\( CH_3 \))**: This group has the least +I effect.
- **Ethyl group (\( C_2H_5 \))**: This group has a moderate +I effect.
- **Propyl group (\( C_3H_7 \))**: This group has the most significant +I effect due to its larger size and more electrons.

5. **Determine the Order of Electrophilicity**:
- Since the methyl group has the least electron-donating effect, it will make the carbonyl carbon the most electrophilic, allowing for the fastest nucleophilic attack.
- The ethyl group will be next, and the propyl group will be the least electrophilic due to its stronger +I effect.

6. **Establish the Rate Order**:
- Based on the analysis:
- \( r_1 \) (Methyl acetate) > \( r_2 \) (Ethyl acetate) > \( r_3 \) (Propyl acetate)
- Therefore, the correct order of the rate of alkaline hydrolysis is:
\[ r_1 > r_2 > r_3 \]

### Final Answer:
The correct order of the rate of alkaline hydrolysis of the esters is:
\[ r_1 > r_2 > r_3 \]