Which of the following statements are correct about the following reaction :
`R-overset(O)overset(||)(C)-OH+R'OHoverset(H^(o+))hArrRCOOR'+H_(2)O`
(a)Rate `= K[RCOOH][R' OH][H^(+)]`. First order in acid, alcohol, and `H^(oplus)` , third order overall.
(b)Rate : `K[RCOOH][R' OH]`. First order in acid and alcohol , second order overall.
(c)Slow and rate-determining step `(R.D.S)` is the addition of alcohol to the protonated acid.
(d)Slow and rate-determining step `(R.D.S)` is the addition of acid to the protonated alcohol.

Options

A. Rate `= K[RCOOH][R' OH][H^(+)]`. First order in acid, alcohol, and `H^(oplus)` , third order overall.
B. Rate : `K[RCOOH][R' OH]`. First order in acid and alcohol , second order overall.
C. Slow and rate-determining step `(R.D.S)` is the addition of alcohol to the protonated acid.
D. Slow and rate-determining step `(R.D.S)` is the addition of acid to the protonated alcohol.

Answer

To analyze the given reaction and determine which statements are correct, we can break down the process step by step.

### Step 1: Understanding the Reaction
The reaction involves a carboxylic acid (RCOOH) reacting with an alcohol (R'OH) in the presence of an acid (H⁺) to form an ester (RCOOR') and water (H₂O). The presence of H⁺ increases the electrophilicity of the carbonyl carbon in the carboxylic acid, making it more susceptible to nucleophilic attack by the alcohol.

### Step 2: Mechanism of the Reaction
1. **Protonation of the Carboxylic Acid**: The oxygen in the carboxylic acid (RCOOH) takes up a proton (H⁺) from the acidic medium, leading to the formation of a protonated carboxylic acid (RCOOH₂⁺).
2. **Nucleophilic Attack**: The alcohol (R'OH) acts as a nucleophile and attacks the electrophilic carbon of the protonated carboxylic acid. This step is the slow and rate-determining step (R.D.S) of the reaction.
3. **Formation of the Ester**: After the nucleophilic attack, a tetrahedral intermediate is formed, which eventually leads to the elimination of water and the formation of the ester (RCOOR').

### Step 3: Analyzing Each Statement
- **Statement (a)**: Rate = K[RCOOH][R'OH][H⁺]. First order in acid, alcohol, and H⁺, third order overall.
- **Analysis**: This statement is correct. The rate-determining step involves all three reactants, thus making the reaction first order with respect to each and third order overall.

- **Statement (b)**: Rate = K[RCOOH][R'OH]. First order in acid and alcohol, second order overall.
- **Analysis**: This statement is incorrect. It does not consider the H⁺ concentration, which is crucial in the rate-determining step.

- **Statement (c)**: Slow and rate-determining step (R.D.S) is the addition of alcohol to the protonated acid.
- **Analysis**: This statement is correct. The slow step involves the nucleophilic attack of the alcohol on the protonated acid.

- **Statement (d)**: Slow and rate-determining step (R.D.S) is the addition of acid to the protonated alcohol.
- **Analysis**: This statement is incorrect. The protonation occurs on the acid, not the alcohol, and the slow step involves the alcohol attacking the protonated acid.

### Conclusion
The correct statements are:
- (a) and (c).