Question

When a lactose is dissolved in water, what would be the final solubility per 100 g water after mutarotation and solubilization of \(\alpha\)-lactose

• A. 7 g
• B. 18.2 g
• C. 30.8 g
• D. 50 g

Hint:

Remember the solubility values of lactose at \(20^\circ\text{C}\):
- Initial solubility (\(\alpha\)-form) = \(7\text{ g}\)
- Final solubility (equilibrium mixture) = \(18.2\text{ g}\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Lactose exists in two isomeric forms, \(\alpha\) and \(\beta\), which can convert into each other in solution through a process called mutarotation until a stable equilibrium is reached.
Key Formula or Approach:
At room temperature (\(20^\circ\text{C}\)):
- The initial solubility of \(\alpha\)-lactose is low, about \(7\text{ g / 100 g}\) of water.
- Upon dissolution, \(\alpha\)-lactose mutarotates into the more soluble \(\beta\)-lactose until a stable \(\beta/\alpha\) ratio of \(1.58\) is achieved at equilibrium:
\[ \text{Equilibrium Ratio} \left(\frac{\beta}{\alpha}\right) \approx 1.58 \]

Step 2: Detailed Explanation:
1. When \(\alpha\)-lactose is added to water, \(7\text{ g}\) dissolves immediately to saturate the solution with \(\alpha\)-lactose.
2. Over time, some of this dissolved \(\alpha\)-lactose converts to \(\beta\)-lactose via mutarotation.
3. This conversion reduces the concentration of dissolved \(\alpha\)-lactose below its saturation limit (\(7\text{ g}\)), allowing more undissolved \(\alpha\)-lactose to dissolve.
4. This cycle of dissolution and mutarotation continues until the dissolved \(\alpha\)-lactose is at its saturation limit (\(7\text{ g}\)) and is in equilibrium with the newly formed \(\beta\)-lactose.
5. Using the equilibrium ratio of \(1.58\), the amount of dissolved \(\beta\)-lactose is:
\[ 7\text{ g} \times 1.58 \approx 11.1\text{ g} \]
6. The final total solubility of lactose at equilibrium is the sum of both forms:
\[ 7\text{ g} \ (\alpha\text{-lactose}) + 11.2\text{ g} \ (\beta\text{-lactose}) \approx 18.2\text{ g per 100 g of water} \]

Step 3: Final Answer
The final solubility of lactose at equilibrium is 18.2 g per 100 g of water.