(+) tartaric acid has a specific rotation of +12 unit when measured in 12cm polarimeter tube and 2g/ml concentration at given temperature and light. When it is diluted to half the concentration, length of tube and other parameters being same, then the specific rotation will be:

Options

A. `+6 " unit"`
B. `+12 " unit"`
C. `-6 " unit"`
D. `+24 " unit"`

Answer

To solve the problem regarding the specific rotation of (+) tartaric acid when diluted to half its concentration, we will follow these steps:

### Step 1: Understand the formula for specific rotation
The specific rotation \([α]\) is defined by the formula:
\[ [α] = \frac{α}{C \cdot L} \]
where:
- \(α\) = observed rotation (in degrees)
- \(C\) = concentration (in g/mL)
- \(L\) = length of the polarimeter tube (in dm)

### Step 2: Identify the initial conditions
From the problem statement:
- The specific rotation \([α]\) is given as +12 units.
- The concentration \(C\) is 2 g/mL.
- The length \(L\) of the polarimeter tube is 12 cm (which is 1.2 dm).

### Step 3: Calculate the observed rotation \(α\)
We can rearrange the specific rotation formula to find the observed rotation \(α\):
\[ α = [α] \cdot C \cdot L \]
Substituting the values:
\[ α = 12 \cdot 2 \cdot 1.2 \]
Calculating this gives:
\[ α = 12 \cdot 2 = 24 \]
\[ α = 24 \cdot 1.2 = 28.8 \text{ degrees} \]

### Step 4: Determine the new concentration
When the concentration is diluted to half, the new concentration \(C'\) becomes:
\[ C' = \frac{2}{2} = 1 \text{ g/mL} \]

### Step 5: Calculate the new specific rotation \([α']\)
Using the same formula for specific rotation with the new concentration:
\[ [α'] = \frac{α}{C' \cdot L} \]
Substituting the values:
\[ [α'] = \frac{28.8}{1 \cdot 1.2} \]
Calculating this gives:
\[ [α'] = \frac{28.8}{1.2} = 24 \text{ units} \]

### Final Answer
The specific rotation when the concentration is halved is +24 units.

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