The given information states that a 90% pure 2-aryl propionic acid solution has an initial specific rotation ([alpha]d) of 135. When this solution is treated with a base at room temperature for one hour, the [alpha]d changes to 120. After three hours, the optical purity of the solution is reduced to 40%. We need to determine the optical purity of the solution after one hour and its [alpha]d after three hours.

1. Initial Specific Rotation of the Solution:
The initial specific rotation ([alpha]d) of the 90% pure 2-aryl propionic acid solution is given as 135.

2. Change in Specific Rotation After One Hour:
After treating the solution with a base at room temperature for one hour, the [alpha]d changes to 120. This change indicates that some of the optically active compounds have been converted to their enantiomers or have been racemized.

3. Calculation of Optical Purity After One Hour:
To calculate the optical purity after one hour, we need to compare the change in [alpha]d with the initial [alpha]d. The formula to calculate optical purity is:

Optical Purity (%) = ([alpha]d_observed - [alpha]d_enantiomer) / ([alpha]d_pure - [alpha]d_enantiomer) * 100

Assuming the enantiomer has a specific rotation of 0, we can calculate the optical purity after one hour:

Optical Purity = (120 - 0) / (135 - 0) * 100 = 88.89%

Therefore, the optical purity of the solution after one hour is approximately 88.89%.

4. Reduction in Optical Purity After Three Hours:
The given information states that the optical purity is reduced to 40% after three hours. This reduction indicates that further racemization or conversion of the optically active compounds has occurred.

5. Calculation of [alpha]d After Three Hours:
To calculate the [alpha]d after three hours, we need to consider the change in optical purity and the initial [alpha]d. The formula to calculate the observed [alpha]d is:

[alpha]d_observed = (Optical Purity * [alpha]d_pure) / 100

Substituting the given values:

[alpha]d_observed = (40 * 135) / 100 = 54

Therefore, the [alpha]d of the solution after three hours is 54.

In summary:
- After one hour, the optical purity of the solution is approximately 88.89%.
- After three hours, the [alpha]d of the solution is 54.