The specific rotation of equilibrium mixture of the three forms of glucose is:

52.5

The correct answer is option 2. 52.5.

Specific rotation (\([α]\)) is a measure of how much a chiral substance rotates plane-polarized light. It depends on the substance's concentration, the path length of the light through the sample, and the wavelength of the light used. For a pure substance, specific rotation is a characteristic property.

Glucose exists in several forms, but the primary ones we consider are:

α-D-Glucose: This isomer has a specific rotation of +112.2°.

β-D-Glucose: This isomer has a specific rotation of +18.7°.

D-Glucose: Often used to refer to the mixture of α and β forms.

In an aqueous solution, glucose exists in an equilibrium between the α-D-glucose and β-D-glucose forms. The equilibrium ratio of these two forms changes with temperature but typically settles at about 36% α-D-glucose and 64% β-D-glucose at room temperature.

To find the specific rotation of the equilibrium mixture, you need to consider the weighted average of the specific rotations of α-D-glucose and β-D-glucose based on their proportions in the mixture.

Let us approximate the specific rotation of the equilibrium mixture:

α-D-Glucose: +112.2° (36% of the mixture)

β-D-Glucose: +18.7° (64% of the mixture)

The specific rotation of the equilibrium mixture is calculated as:

\([α]_{\text{mixture}} = (0.36 \times 112.2) + (0.64 \times 18.7)\)

\([α]_{\text{mixture}} = 40.392 + 11.984 = 52.376°\)

This is very close to the commonly cited value of +52.5° for the equilibrium mixture of glucose in solution.

Summary: The specific rotation of the equilibrium mixture of α-D-glucose and β-D-glucose is approximately 52.5°, which reflects the weighted average rotation of the two forms in solution.