(+) mandelic acid has a specific rotation of `158^(@)` What would be the observed specific rotation of a mixture of 25% (-) mandelic acid and 75% (+) mandelic acid ?

Options

A. `+118.5^(@)`
B. `-118.5^(@)`
C. `-79^(@)`
D. `+79^(@)`

Answer

To find the observed specific rotation of a mixture of mandelic acid, we can use the formula for calculating the specific rotation of a mixture of two enantiomers. The specific rotation of a mixture can be calculated using the following formula:

\[ [\alpha]_{observed} = \frac{(D - L)}{(D + L)} \times [\alpha]_{pure} \]

Where:
- \(D\) is the percentage of the dextrorotatory enantiomer (+) mandelic acid.
- \(L\) is the percentage of the levorotatory enantiomer (-) mandelic acid.
- \([\alpha]_{pure}\) is the specific rotation of the pure dextrorotatory enantiomer.

### Step 1: Identify the values
From the question:
- \(D = 75\%\) (for (+) mandelic acid)
- \(L = 25\%\) (for (-) mandelic acid)
- \([\alpha]_{pure} = +158^\circ\)

### Step 2: Substitute the values into the formula
Now, we can substitute the values into the formula:

\[ [\alpha]_{observed} = \frac{(75 - 25)}{(75 + 25)} \times 158 \]

### Step 3: Calculate the numerator and denominator
Calculate the numerator and denominator:

- Numerator: \(75 - 25 = 50\)
- Denominator: \(75 + 25 = 100\)

### Step 4: Simplify the fraction
Now, we can simplify the fraction:

\[ \frac{50}{100} = 0.5 \]

### Step 5: Calculate the observed specific rotation
Now, multiply this by the specific rotation of the pure enantiomer:

\[ [\alpha]_{observed} = 0.5 \times 158 = 79^\circ \]

### Conclusion
The observed specific rotation of the mixture is:

\[ [\alpha]_{observed} = +79^\circ \]

### Final Answer
The observed specific rotation of the mixture of 25% (-) mandelic acid and 75% (+) mandelic acid is \(+79^\circ\).

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