The rate of reaction is doubled for every `10^(@)C` rise in temperature. The increase in rate as result of an increase in temperature from `10^(@)C` to `100^(@)C` is how many times of the original rate?

Options

A. 112
B. 512
C. 400
D. 256

Answer

To solve the problem, we need to determine how many times the rate of reaction increases when the temperature is raised from \(10^\circ C\) to \(100^\circ C\), given that the rate of reaction doubles for every \(10^\circ C\) rise in temperature.

### Step-by-Step Solution:

1. **Identify the Temperature Increase**:
The temperature increases from \(10^\circ C\) to \(100^\circ C\). The total increase in temperature is:
\[ 100^\circ C - 10^\circ C = 90^\circ C \]

2. **Determine the Number of 10 Degree Increments**:
Since the rate doubles for every \(10^\circ C\) increase, we need to find out how many \(10^\circ C\) increments fit into \(90^\circ C\):
\[ \frac{90^\circ C}{10^\circ C} = 9 \]
This means there are 9 increments of \(10^\circ C\).

3. **Calculate the Rate Increase**:
If the rate doubles with each increment, we can express the final rate after \(9\) increments as:
\[ R = R_0 \times 2^n \]
where \(R_0\) is the original rate and \(n\) is the number of increments. Here, \(n = 9\):
\[ R = R_0 \times 2^9 \]

4. **Calculate \(2^9\)**:
Now, we calculate \(2^9\):
\[ 2^9 = 512 \]

5. **Final Rate Expression**:
Therefore, the rate at \(100^\circ C\) is:
\[ R = R_0 \times 512 \]

6. **Conclusion**:
The increase in rate as a result of the temperature increase from \(10^\circ C\) to \(100^\circ C\) is \(512\) times the original rate.

### Final Answer:
The increase in rate is **512 times** the original rate.

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