If specific rotation of glucose solution is `52^@` and fructose solution `-92^@` then what will be specific rotaion of invert sugar ?

Options

A. `-20^@`
B. `+20^@`
C. `-72^@`
D. `+72^@`

Answer

To find the specific rotation of invert sugar, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Components**:
- We know that invert sugar is formed by the hydrolysis of sucrose, which yields equal amounts of glucose and fructose.
- The specific rotation of glucose is given as \( +52^\circ \).
- The specific rotation of fructose is given as \( -92^\circ \).

2. **Determine the Average Specific Rotation**:
- Since invert sugar is an equimolar mixture of glucose and fructose, we can calculate the specific rotation of invert sugar by averaging the specific rotations of glucose and fructose.
- The formula for the average specific rotation (\[ [\alpha] \]) is:
\[ [\alpha] = \frac{[\alpha]_{glucose} + [\alpha]_{fructose}}{2} \]
- Plugging in the values:
\[ [\alpha] = \frac{52^\circ + (-92^\circ)}{2} \]

3. **Perform the Calculation**:
- Simplifying the expression:
\[ [\alpha] = \frac{52^\circ - 92^\circ}{2} = \frac{-40^\circ}{2} = -20^\circ \]

4. **Conclusion**:
- Therefore, the specific rotation of invert sugar is \( -20^\circ \).

### Final Answer:
The specific rotation of invert sugar is \( -20^\circ \).

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