Stereoisomers of D Configuration Aldohexose in Pyranose Form

To determine the total number of stereoisomers in the pyranose form of the given aldohexose in its $\text{D}$-configuration, we analyze the structure step-by-step:

1. Structure Analysis of the Open-Chain Form

The given aldohexose structure is:

$$\begin{array}{c} \text{C1: CHO} \\ | \\ \text{C2: CH}_2 \\ | \\ \text{C3: CHOH} \\ | \\ \text{C4: CHOH} \\ | \\ \text{C5: CHOH} \\ | \\ \text{C6: CH}_2\text{OH} \end{array}$$

2. Pyranose Ring Formation

A pyranose ring is a 6-membered ring consisting of 5 carbon atoms and 1 oxygen atom. It is formed by the intramolecular nucleophilic attack of the hydroxyl group ($-\text{OH}$) attached to C5 onto the aldehyde carbonyl carbon (C1).

The resulting ring backbone is: $\text{C1} - \text{C2} - \text{C3} - \text{C4} - \text{C5} - \text{O} - \text{C1}$

3. Stereocenters in the Pyranose Form

Let us evaluate each carbon atom in the cyclic pyranose form to identify the active chiral centers and their degrees of freedom:

1. $\text{C1}$ (Anomeric Carbon): Upon cyclic hemiacetal formation, $\text{C1}$ becomes a new asymmetric center (anomeric carbon) bonded to four distinct groups ($-\text{H}$, $-\text{OH}$, $-\text{O}-$, $-\text{CH}_2-$). It can exist in two distinct configurations ($\alpha$ and $\beta$ anomers). $\text{Number of configurations at C1} = 2$

2. $\text{C2}$: Contains two hydrogen atoms ($-\text{CH}_2-$), making it achiral. $\text{Number of configurations at C2} = 1$

3. $\text{C3}$: Attached to four different groups ($-\text{H}$, $-\text{OH}$, $-\text{CH}_2-$, $-\text{CHOH}-$), so it is a chiral center with two possible configurations ($R$ or $S$). $\text{Number of configurations at C3} = 2$

4. $\text{C4}$: Attached to four different groups ($-\text{H}$, $-\text{OH}$, $-\text{CHOH}-$, $-\text{CH}(\text{CH}_2\text{OH})-$), so it is a chiral center with two possible configurations ($R$ or $S$). $\text{Number of configurations at C4} = 2$

5. $\text{C5}$: This is the stereocenter that defines the $\text{D/L}$ configuration of the carbohydrate. Since the problem explicitly specifies that the molecule is in its $\text{D}$-configuration, the spatial arrangement at $\text{C5}$ is fixed. $\text{Number of configurations at C5} = 1$

6. $\text{C6}$: Contains a $-\text{CH}_2\text{OH}$ group, making it achiral. $\text{Number of configurations at C6} = 1$

4. Total Count of Stereoisomers

The total number of stereoisomers for the $\text{D}$-configuration in the pyranose form is given by the product of the possible configurations at each chiral center:

$\text{Total Stereoisomers} = 2_{\text{(at C1)}} \times 2_{\text{(at C3)}} \times 2_{\text{(at C4)}} \times 1_{\text{(at C5)}} = 2^3 = 8$