Question

If the optical rotation of a new compound is measured and found to have a specific rotation of \(+40\), how can you tell if the actual rotation is not really \(+40\) plus some multiple of \(+360\) ? In other words, how can you tell if the rotation is not actually a value such as \(+400\) or \(+760 ?\)

Short answer

Answer: The actual rotation of the compound is indeed +40 degrees and not +400, +760, or any other multiple of 360 degrees added to it.

Step-by-step solution

Identify the given optical rotation

The specific rotation of the compound is given as \(+40\) degrees. This is the observed rotation of the compound.

Check for multiples of 360 degrees

To check if the actual rotation is the given rotation plus some multiples of \(360\) degrees, compute the following: $$ \text{Actual Rotation} = \text{Given Rotation} + n \times 360, $$ where \(n\) is an integer (positive, negative or zero).

Calculate the actual rotation range

To find the range in which the actual rotation lies, we have to check for possible integer values of \(n\). Since optical rotation is measured between \(0\) and \(360\) degrees, we need to check if the observed rotation falls within this range when it is added to a multiple of \(360\) degrees. Consider the cases: - For \(n = 0\), \(\text{Actual Rotation} = 40 + 0 \times 360 = 40\) - For \(n = 1\), \(\text{Actual Rotation} = 40 + 1 \times 360 = 400\) - For \(n = -1\), \(\text{Actual Rotation} = 40 - 1 \times 360 = -320\)

Identify the correct actual rotation

From the calculations in Step 3, we can see that the actual rotation is \(40\) degrees when \(n = 0\). This value lies within the range of \(0\) to \(360\) degrees, and adding a multiple of \(360\) degrees to this rotation will place it outside of the desired range. Therefore, the actual rotation of the compound is indeed \(+40\) degrees and not \(+400\), \(+760\), or any other multiple of \(360\) degrees added to it.