In the study of inversion of sucrose in presence of acid, if `r_(0),r_(1)" and "r_(oo)` represent the polarimetric readings at times 0, t and `oo` respectively, then at the `50%` inversion, which of the following relationship will hold good ?

Options

A. `r_(t)=r_(0)+r_(oo)`
B. `r_(t)=(1)/(2)(r_(0)+r_(oo))`
C. `r_(t)=r_(0)-r_(oo)`
D. `r_(t)=(1)/(2)(r_(0)-r_(oo))`

Explanation

To solve the problem regarding the inversion of sucrose in the presence of acid and the polarimetric readings at different times, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Polarimetric Readings:**
- Let \( R_0 \) be the polarimetric reading at time \( t = 0 \).
- Let \( R_t \) be the polarimetric reading at time \( t \).
- Let \( R_{\infty} \) be the polarimetric reading at time \( t = \infty \).

2. **Understand the Concept of Inversion:**
- The inversion of sucrose refers to the conversion of sucrose into glucose and fructose, which can be monitored by changes in optical rotation (polarimetric readings).
- At 50% inversion, half of the sucrose has been converted.

3. **Relate Polarimetric Readings to Concentration:**
- The change in polarimetric readings can be related to the concentration of sucrose remaining.
- The relationship can be expressed as:
\[ R_0 - R_t \propto x \]
where \( x \) is the amount of sucrose that has been inverted.

4. **Determine \( x \) at 50% Inversion:**
- At 50% inversion, the amount of sucrose inverted \( x \) is equal to half of the initial concentration \( A \):
\[ x = \frac{A}{2} \]

5. **Express the Change in Polarimetric Readings:**
- The change in readings can also be expressed as:
\[ R_0 - R_{\infty} \propto A \]
- Therefore, at 50% inversion:
\[ R_0 - R_t = \frac{A}{2} \]

6. **Substituting the Proportional Relationships:**
- From the above relationships, we can write:
\[ R_0 - R_t = \frac{1}{2}(R_0 - R_{\infty}) \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ 2(R_0 - R_t) = R_0 - R_{\infty} \]
- Expanding this leads to:
\[ 2R_0 - 2R_t = R_0 - R_{\infty} \]

8. **Solving for \( R_t \):**
- Rearranging the terms, we get:
\[ 2R_t = R_0 + R_{\infty} \]
- Thus, we find:
\[ R_t = \frac{1}{2}(R_0 + R_{\infty}) \]

### Final Result:
At 50% inversion, the relationship that holds good is:
\[ R_t = \frac{1}{2}(R_0 + R_{\infty}) \]