The rate law for a reaction between the substances A and B is given by
Rate = `k[A]^(n)[B]^(m)`
On doubling the concentration of A and halving the concentration of B, the ratio of the new rate to the earlier rate of the reaction will be as:

Options

A. `m+n`
B. `n-m`
C. `2^((n-m))`
D. `2^(m+n)`

Answer

To solve the problem, we need to analyze the change in the rate of the reaction when the concentrations of substances A and B are altered. Let's break it down step by step.

### Step-by-Step Solution:

1. **Write the Initial Rate Law**:
The rate of the reaction is given by the equation:
\[ \text{Rate} = k[A]^n[B]^m \]
where \( k \) is the rate constant, \( [A] \) is the concentration of substance A, and \( [B] \) is the concentration of substance B.

2. **Define Initial Concentrations**:
Let the initial concentrations of A and B be \( [A] \) and \( [B] \) respectively. Thus, the initial rate \( R_1 \) can be expressed as:
\[ R_1 = k[A]^n[B]^m \]

3. **Change the Concentrations**:
According to the problem, the concentration of A is doubled and the concentration of B is halved:
- New concentration of A: \( [A]_{\text{new}} = 2[A] \)
- New concentration of B: \( [B]_{\text{new}} = \frac{1}{2}[B] \)

4. **Write the New Rate**:
The new rate \( R_2 \) with the changed concentrations can be expressed as:
\[ R_2 = k[2A]^n\left[\frac{1}{2}B\right]^m \]

5. **Substitute the New Concentrations**:
Substitute the new concentrations into the rate equation:
\[ R_2 = k(2[A])^n\left(\frac{1}{2}[B]\right)^m \]
This simplifies to:
\[ R_2 = k \cdot 2^n[A]^n \cdot \frac{1}{2^m}[B]^m \]

6. **Combine the Terms**:
Combine the terms to express \( R_2 \) in terms of \( R_1 \):
\[ R_2 = k[A]^n[B]^m \cdot \frac{2^n}{2^m} \]
Thus, we can write:
\[ R_2 = R_1 \cdot \frac{2^n}{2^m} \]

7. **Find the Ratio of New Rate to Initial Rate**:
The ratio of the new rate \( R_2 \) to the initial rate \( R_1 \) is:
\[ \frac{R_2}{R_1} = \frac{2^n}{2^m} = 2^{n - m} \]

### Final Answer:
The ratio of the new rate to the earlier rate of the reaction is:
\[ \frac{R_2}{R_1} = 2^{n - m} \]