Introduction:
In this question, we are given a racemic mixture of an unknown compound 'a' with a specific rotation of 16. We are asked to determine the specific rotation of the optically pure compound of 'a', denoted as 'x'. To find this, we need to understand the concept of specific rotation, optical activity, and racemic mixtures.

Specific Rotation:
Specific rotation is a measure of the optical activity of a compound, which refers to its ability to rotate plane-polarized light. It is denoted by the symbol '[α]' and is expressed in degrees per decimeter (°/dm). Specific rotation is dependent on the concentration of the compound, the length of the sample tube, and the wavelength of light used.

Optical Activity:
Optical activity is the ability of a compound to rotate the plane of polarized light. A compound can be optically active if it contains a chiral center, which is an asymmetric carbon atom with four different substituents. The direction and magnitude of the rotation depend on the nature of the compound and the concentration.

Racemic Mixture:
A racemic mixture is a 1:1 mixture of enantiomers, which are mirror-image isomers. Enantiomers have equal and opposite specific rotations, and when combined in equal amounts, they cancel out each other's optical activity, resulting in a racemic mixture with zero net rotation.

Solution:
1. Given that the racemic mixture of compound 'a' has an observed specific rotation of 16, we can conclude that this value represents the net rotation of both enantiomers combined.
2. Since the racemic mixture is a 2:3 mixture, we can assume that one enantiomer is present in a 2:5 ratio and the other enantiomer in a 3:5 ratio.
3. Let's assume that the specific rotations of the two enantiomers are α1 and α2.
4. The observed specific rotation of the racemic mixture can be calculated using the formula:
[α]obs = (α1 * ratio1) + (α2 * ratio2)
where ratio1 and ratio2 are the respective ratios of the enantiomers in the mixture.
5. Substituting the given values, we have:
16 = (α1 * 2/5) + (α2 * 3/5)
6. Since the racemic mixture has zero net rotation, the sum of the specific rotations of the enantiomers should be zero. This leads to the equation:
α1 + α2 = 0
7. Solving the system of equations, we can find the specific rotations of the two enantiomers.
8. Let's assume α1 = x, the specific rotation of the optically pure compound 'a'.
9. Substituting this into the equation α1 + α2 = 0, we have:
x + α2 = 0
α2 = -x
10. Substituting α2 = -x into the equation 16 = (α1 * 2/5) + (α2 * 3/5), we have:
16 = (x * 2/5) + (-x * 3/5)
16 = (2x -