Which of the following reaction `(s)` is `//` are of the first order ?

Options

A. The decompoistion of ammonium nitrate in an aqueous solution.
B. The inverison of cane sugar in the presence of an acid.
C. The acidic hydrolyiss of ethyl acetate.
D. All radioactive decays.

Answer

To determine which of the given reactions are first-order reactions, we will analyze each option based on the characteristics of first-order kinetics.

### Step-by-Step Solution:

1. **Understanding First-Order Reactions**:
- A reaction is classified as first-order if the rate of reaction is directly proportional to the concentration of one reactant. Mathematically, this can be expressed as:
\[ \text{Rate} = k[A] \]
- Here, \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.

2. **Analyzing the First Option: Decomposition of Ammonium Nitrate**:
- The reaction is given as:
\[ 2 \text{NH}_4\text{NO}_3 \rightarrow 2 \text{N}_2 + 3 \text{H}_2 + 2 \text{O}_2 \]
- In this case, increasing the concentration of NH4NO3 does not affect the rate of reaction. This indicates that the reaction does not depend on the concentration of the reactant, thus it is **not a first-order reaction**.

3. **Analyzing the Second Option: Inversion of Cane Sugar**:
- The reaction is:
\[ \text{C}_{12}\text{H}_{22}\text{O}_{11} + \text{H}_2\text{O} \xrightarrow{\text{acid}} \text{C}_6\text{H}_{12}\text{O}_6 + \text{C}_6\text{H}_{12}\text{O} \]
- This reaction involves the conversion of sucrose into glucose and fructose in the presence of acid. The rate of this reaction is dependent on the concentration of sucrose, indicating that it is a **first-order reaction**.

4. **Analyzing the Third Option: Acidic Hydrolysis of Ethyl Acetate**:
- The reaction is:
\[ \text{C}_2\text{H}_5\text{OCOCH}_3 + \text{H}_2\text{O} \xrightarrow{\text{acid}} \text{C}_2\text{H}_5\text{OH} + \text{CH}_3\text{COOH} \]
- Similar to the previous reaction, the rate of hydrolysis depends on the concentration of ethyl acetate. Therefore, this reaction is also classified as a **first-order reaction**.

5. **Analyzing the Fourth Option: Radioactive Decays**:
- Radioactive decay processes are inherently first-order because the rate of decay is proportional to the amount of radioactive substance present. Thus, this is also a **first-order reaction**.

### Summary of Results:
- **First-Order Reactions**:
- Inversion of cane sugar
- Acidic hydrolysis of ethyl acetate
- Radioactive decays
- **Not First-Order**:
- Decomposition of ammonium nitrate

### Final Answer:
The reactions that are of first order are:
- Inversion of cane sugar
- Acidic hydrolysis of ethyl acetate
- Radioactive decays