The specific rotation of two glucose anomers are `alpha = + 110^(@)` and `beta = 19^@` and for the constant equilibrium mixtures is `+52.7^@`. Calculate the percentage compositions of the anomers in the equilibrium mixture.

Explanation

To solve the problem of calculating the percentage compositions of the two glucose anomers (alpha and beta) in the equilibrium mixture, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let the percentage composition of the alpha anomer be \( X \% \).
- Therefore, the percentage composition of the beta anomer will be \( (100 - X) \% \).

2. **Use the Formula for Optical Rotation:**
- The specific rotation of the equilibrium mixture can be expressed as:
\[ [\alpha]_{mix} = \frac{X \cdot [\alpha]_{alpha} + (100 - X) \cdot [\alpha]_{beta}}{100} \]
- Where:
- \( [\alpha]_{alpha} = +110^\circ \) (specific rotation of alpha anomer)
- \( [\alpha]_{beta} = +19^\circ \) (specific rotation of beta anomer)
- \( [\alpha]_{mix} = +52.7^\circ \) (specific rotation of the equilibrium mixture)

3. **Set Up the Equation:**
- Substitute the known values into the equation:
\[ 52.7 = \frac{X \cdot 110 + (100 - X) \cdot 19}{100} \]

4. **Multiply Through by 100 to Eliminate the Denominator:**
\[ 5270 = 110X + 1900 - 19X \]

5. **Combine Like Terms:**
- Combine the \( X \) terms:
\[ 5270 = 91X + 1900 \]

6. **Isolate \( X \):**
- Subtract 1900 from both sides:
\[ 5270 - 1900 = 91X \]
\[ 3370 = 91X \]

7. **Solve for \( X \):**
- Divide both sides by 91:
\[ X = \frac{3370}{91} \approx 37.2 \]

8. **Calculate the Percentage of Beta:**
- Now, calculate the percentage of the beta anomer:
\[ \text{Percentage of beta} = 100 - X = 100 - 37.2 = 62.8 \]

### Final Answer:
- The percentage composition of the alpha anomer is approximately **37.2%**.
- The percentage composition of the beta anomer is approximately **62.8%**.