Stereoisomeric Relationships of D Erythrose Isomers

To determine the correct relationships of structures P, Q, R, and S with D-erythrose, we determine the absolute configurations ($(R/S)$ system) of the chiral centers ($\text{C2}$ and $\text{C3}$) for D-erythrose and each compound.

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1. Configuration of D-Erythrose

The Fischer projection of D-erythrose is:

$$\begin{array}{c} \text{CHO (C1)} \\ | \\ \text{H} - \text{C2} - \text{OH} \\ | \\ \text{H} - \text{C3} - \text{OH} \\ | \\ \text{CH}_2\text{OH (C4)} \end{array}$$

Configuration at C2:

• Priority of groups attached to C2:
  1. $-\text{OH}$ (Priority 1)
  2. $-\text{CHO}$ (Priority 2, attached to $(\text{O}, \text{O}, \text{H})$)
  3. $-\text{C3}$ (Priority 3, attached to $(\text{O}, \text{C}, \text{H})$)
  4. $-\text{H}$ (Priority 4)
• Determining $(R/S)$:
  • Group 1 ($-\text{OH}$) is on the Right.
  • Group 2 ($-\text{CHO}$) is at the Top.
  • Group 3 ($-\text{C3}$) is at the Bottom.
  • Direction $1 \rightarrow 2 \rightarrow 3$ goes counter-clockwise.
  • Since the lowest priority group ($-\text{H}$) is on a horizontal bond, counter-clockwise corresponds to the $(2R)$ configuration.

Configuration at C3:

• Priority of groups attached to C3:
  1. $-\text{OH}$ (Priority 1)
  2. $-\text{C2}$ (Priority 2, attached to $(\text{O}, \text{C}, \text{H})$)
  3. $-\text{CH}_2\text{OH}$ (Priority 3, attached to $(\text{O}, \text{H}, \text{H})$)
  4. $-\text{H}$ (Priority 4)
• Determining $(R/S)$:
  • Group 1 ($-\text{OH}$) is on the Right.
  • Group 2 ($-\text{C2}$) is at the Top.
  • Group 3 ($-\text{CH}_2\text{OH}$) is at the Bottom.
  • Direction $1 \rightarrow 2 \rightarrow 3$ goes counter-clockwise.
  • With $-\text{H}$ on a horizontal bond, counter-clockwise corresponds to the $(3R)$ configuration.

Thus, D-erythrose has the $(2R, 3R)$ configuration.

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2. Analysis of Structures P, Q, R, and S

Structure P:

• C2 Center: The $-\text{OH}$ group is on a dashed bond (pointing away) and $-\text{H}$ is on a wedged bond. Evaluating priorities gives the $(2R)$ configuration.
• C3 Center: The $-\text{OH}$ group is on a wedged bond and $-\text{H}$ is on a dashed bond. Evaluating priorities gives the $(3R)$ configuration.
• Configuration of P: $(2R, 3R)$, which is Identical to D-erythrose. $\mathbf{P \rightarrow 2}$

Structure Q:

• C2 Center: The configuration is inverted relative to P, yielding $(2S)$.
• C3 Center: The configuration is retained relative to P, yielding $(3R)$.
• Configuration of Q: $(2S, 3R)$, which has one inverted stereocenter. Therefore, it is a Diastereomer of D-erythrose. $\mathbf{Q \rightarrow 1}$

Structure R:

• C2 Center: The configuration is retained relative to P, yielding $(2R)$.
• C3 Center: The configuration is inverted relative to P, yielding $(3S)$.
• Configuration of R: $(2R, 3S)$, which has one inverted stereocenter. Therefore, it is a Diastereomer of D-erythrose. $\mathbf{R \rightarrow 1}$

Structure S:

• C2 Center: The configuration is inverted relative to P, yielding $(2S)$.
• C3 Center: The configuration is inverted relative to P, yielding $(3S)$.
• Configuration of S: $(2S, 3S)$, which is the non-superimposable mirror image of $(2R, 3R)$. Therefore, it is the Enantiomer of D-erythrose. $\mathbf{S \rightarrow 3}$

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Conclusion

$P \rightarrow 2, \quad Q \rightarrow 1, \quad R \rightarrow 1, \quad S \rightarrow 3$

Thus, the correct option is C.