Which of the following reaction ends in finite time ?

Options

A. zero order
B. `1^(st)` order
C. `2^(nd)` order
D. `3^(rd)` order

Explanation

To determine which of the following reactions ends in finite time, we need to analyze the different types of reactions provided: zero-order, first-order, second-order, and third-order reactions.

### Step-by-Step Solution:

1. **Understanding Reaction Orders**:
- **Zero-Order Reaction**: The rate of reaction is constant and does not depend on the concentration of the reactant. The rate law can be expressed as:
\[ \text{Rate} = k \]
- **First-Order Reaction**: The rate of reaction is directly proportional to the concentration of one reactant. The rate law is:
\[ \text{Rate} = k[A] \]
- **Second-Order Reaction**: The rate of reaction depends on the square of the concentration of one reactant or the product of the concentrations of two reactants. The rate law can be:
\[ \text{Rate} = k[A]^2 \quad \text{or} \quad \text{Rate} = k[A][B] \]
- **Third-Order Reaction**: The rate of reaction depends on the concentration of three reactants or the cube of one reactant. The rate law can be:
\[ \text{Rate} = k[A]^3 \quad \text{or} \quad \text{Rate} = k[A]^2[B] \quad \text{or} \quad \text{Rate} = k[A][B][C] \]

2. **Deriving the Integrated Rate Laws**:
- **Zero-Order Reaction**:
\[ [A] = [A_0] - kt \]
This equation shows that the concentration of A decreases linearly with time. When [A] reaches zero, the reaction stops, which occurs in a finite time.

- **First-Order Reaction**:
\[ [A] = [A_0] e^{-kt} \]
The concentration of A approaches zero asymptotically as time increases, meaning it never actually reaches zero in finite time.

- **Second-Order Reaction**:
\[ \frac{1}{[A]} = \frac{1}{[A_0]} + kt \]
Similar to the first-order reaction, the concentration of A approaches zero but never actually reaches it in finite time.

- **Third-Order Reaction**:
\[ \frac{1}{[A]^2} = \frac{1}{[A_0]^2} + kt \]
Again, the concentration approaches zero but does not reach it in finite time.

3. **Conclusion**:
- Among the four types of reactions analyzed, only the **zero-order reaction** reaches zero concentration in a finite amount of time. Therefore, the correct answer is that the zero-order reaction ends in finite time.

### Final Answer:
The zero-order reaction is the only reaction that ends in finite time.