When inversion of sucrose is studied at pH = 5, the half-life period is always found to be 500 minutes irrespective of any initial concentration but when it is studied at pH = 6, the half-life period is found to be 50 minutes. Derive the rate law expression for the inversion of sucrose.

Explanation

To derive the rate law expression for the inversion of sucrose, we can follow these steps:

### Step 1: Understand the Reaction
The inversion of sucrose involves the hydrolysis of sucrose (C12H22O11) into glucose and fructose. This reaction can be represented as:
\[ \text{C}_{12}\text{H}_{22}\text{O}_{11} + \text{H}_2\text{O} \rightarrow \text{C}_6\text{H}_{12}\text{O}_6 + \text{C}_6\text{H}_{12}\text{O}_6 \]
This reaction is important because it changes the optical rotation of the solution, which is why it's referred to as "inversion."

### Step 2: Analyze the Half-Life Data
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