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 how the substituents affect the electrophilicity of the carbonyl carbon.

### Step-by-Step Solution:

1. **Identify the Esters**: We have three esters:
- Ester 1: \( CH_3COOCH_3 \) (Methyl acetate)
- Ester 2: \( CH_3COOC_2H_5 \) (Ethyl acetate)
- Ester 3: \( CH_3COOC_3H_7 \) (Propyl acetate)

2. **Understand the Mechanism**: The hydrolysis of esters involves a nucleophilic attack by hydroxide ion (\( OH^- \)) on the electrophilic carbonyl carbon of the ester. This leads to the formation of a tetrahedral intermediate.

3. **Electrophilicity of Carbonyl Carbon**: The reactivity of the ester towards hydrolysis depends on the electrophilicity of the carbonyl carbon. The more electrophilic the carbonyl carbon, the faster the hydrolysis.

4. **Effect of Alkyl Groups**: Alkyl groups can have an electron-donating (+I) effect, which can decrease the electrophilicity of the carbonyl carbon:
- Methyl group (\( CH_3 \)) has a weak +I effect.
- Ethyl group (\( C_2H_5 \)) has a stronger +I effect than methyl.
- Propyl group (\( C_3H_7 \)) has an even stronger +I effect than ethyl.

5. **Comparing the Esters**:
- **Ester 1**: \( CH_3COOCH_3 \) has two methyl groups, which exert a weak +I effect, leading to higher electrophilicity of the carbonyl carbon.
- **Ester 2**: \( CH_3COOC_2H_5 \) has one ethyl and one methyl group. The ethyl group has a stronger +I effect than the methyl group, reducing the electrophilicity compared to Ester 1.
- **Ester 3**: \( CH_3COOC_3H_7 \) has one propyl and one methyl group. The propyl group has the strongest +I effect, leading to the lowest electrophilicity of the carbonyl carbon among the three esters.

6. **Order of Reactivity**: Based on the above analysis, the order of reactivity (rate of hydrolysis) will be:
- \( r_1 > r_2 > r_3 \)
- Therefore, the correct order of the rate of alkaline hydrolysis is:
- \( CH_3COOCH_3 (r_1) > CH_3COOC_2H_5 (r_2) > CH_3COOC_3H_7 (r_3) \)

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