Question

What percent of \(\alpha\)-anomers of D-glucose present in an equilibrium mixture with a specific rotation of \(+52.6^{\circ} ?\) Given \([\alpha]_{\alpha}=+112.2^{\circ}\) and \([\alpha]_{\beta}=+18.7^{\circ}\) (A) \(50 \%\) (B) \(64 \%\) (C) \(36 \%\) (D) \(63 \%\)

Short answer

The percentage of \(\alpha\)-anomers in the equilibrium mixture of D-glucose is approximately \(63 \%\).

Step-by-step solution

Determine the ratio of Α and Β anomers in the equilibrium mixture

To determine the ratio of \(\alpha\) and \(\beta\) anomers in the equilibrium mixture, we can use the formula for the specific rotation: \[ \alpha_{mix} = \frac{\alpha_{\alpha}\times\% \alpha + \alpha_{\beta}\times\% \beta}{100} \] \[+52.6=(-112.2)\times\%(+18.7) \times(100- \%)\]

Solve for the percentage of Α anomers

Now we will rearrange the formula to obtain the percentage of \(\alpha\)-anomers in terms of the percentage of \(\beta\)-anomers: \[\%(+112.2) = \frac{[+52.6-(+18.7)\times(100- \%)]}{(+112.2)-(+18.7)}\] Calculate the percentage of \(\alpha\)-anomers: \[\%(+112.2) = \frac{[+52.6-(+18.7)\times(100- \%)]}{(+112.2)-(+18.7)}\] \[\%(+112.2) = \frac{[+52.6-(+18.7)\times(100- \%)]}{+93.5}\] \[\%(+112.2) = \frac{[+52.6-(+18.7)(100- \%)]}{+93.5}\] \[\%(+112.2) = \frac{[+52.6-1870+18.7\%]}{+93.5}\] \[\%(+112.2) = \frac{-1817.4+18.7\%]}{+93.5}\] Now multiplying both sides by \(93.5\) and moving all the terms containing the percentage to one side, we get: \[\%(+112.2)(93.5)=-1817.4+18.7\%\] Solve the equation for \(\%\): \[\%(+112.2) =\frac{ (-1817.4)+(\%(+112.2)\times(93.5))}{18.7}\] \[\% = 63.14 \% \approx 63 \% \] So, the percentage of \(\alpha\)-anomers in the equilibrium mixture is 63%. The correct answer is (D) \(63 \%\).

Key concepts

Alpha Anomers

Anomers are a special type of stereoisomers found in carbohydrates. They're specifically related to the orientation of the hydroxyl group (-OH) on the anomeric carbon. In alpha anomers, the hydroxyl group on the anomeric carbon is trans (opposite) to the CH2OH group in the sugar ring's chair form. For D-glucose, this configuration results in a specific rotation of the molecule.

• The Alpha ( α ) form has a higher specific rotation of [+112.2^ ` ].
• This number indicates how much the compound can rotate plane-polarized light, which is essential for understanding the behavior of sugars in solution.

Seeing how alpha anomers change with temperature, light, or mixing is key in chemistry and biology.

Beta Anomers

Beta anomers are another form of stereoisomers, like alpha anomers but with a different orientation. In these sugars, the hydroxyl group on the anomeric carbon is cis (same side) to the CH2OH group. For D-glucose, this configuration leads to a different specific rotation:

• The Beta ( β ) form shows a specific rotation of [+18.7^ ` ], considerably lower than that of the alpha anomers.
• This difference impacts the chemical properties and how the molecule interacts with light.

While both anomers can coexist, their ratio in a mixture will determine the overall specific rotation of the solution, influenced by factors like temperature and concentration.

Equilibrium Mixture

In solutions like D-glucose, alpha and beta anomers can convert into each other, reaching a state of equilibrium. At equilibrium, the mixture will show a specific rotation that reflects the proportions of both anomers.

• For example, in this case, the equilibrium mixture has a specific rotation of [+52.6^ ` ].
• This number is the weighted average of the rotations of α and β anomers.

The balance between alpha and beta forms depends on factors such as the solvent, temperature, and concentration. Understanding this allows scientists to predict or manipulate the behavior of sugars in biological processes or industrial applications.

Optical Isomerism

Optical isomerism occurs when a compound can rotate plane-polarized light. This property is exhibited by compounds that have a chiral center. In the case of D-glucose, the chirality arises from the asymmetric carbon atoms with particular attention to the anomeric carbon.

• Anomeric carbons are special because they enable sugars to have two possible optical isomers (alpha and beta)
• These isomers can significantly influence a substance’s chemical and biological characteristics.

The specific rotation of each isomer helps determine and quantify the distinct optical activity. This principle is critical for fields from biochemistry to pharmacology, where the function and efficacy of diverse molecules can hinge on these subtle differences in structure.