The radius of gyration (Rg) is directly proportional to the molecular weight Ma, where ‘a’ is a constant. The value of ‘a’ for a spherical molecule is

The radius of gyration ($R_g$) is a structural parameter that gives insight into the distribution of mass around an axis in a molecule. For a spherical molecule, the relationship can be defined mathematically, and the value of the constant $a$ can be derived through its physical significance.

1. ### Break Down the Problem

• Identify the relationship between $R_g$ and $M_a$.
• Determine the constant $a$ specifically for spherical molecules.

2. ### Relevant Concepts

The radius of gyration for a spherical object is defined by the equation: $R_g = k \cdot M_a^a$ Here, $R_g$ is the radius of gyration, $M_a$ is the molecular weight, and $k$ is a proportionality constant. For different shapes of molecules, the 'a' would adjust accordingly.

3. ### Analysis and Detail

For a spherical molecule, the distribution of mass is uniform. The theoretical value of $a$ for such molecules, based on the principles of polymer physics and dimensional analysis, is typically:

• For spherical molecules, $a$ is generally found to be $\frac{1}{3}$ in relevant studies and applications.

4. ### Verify and Summarize

The theoretical derivation and empirical observations have consistently supported that $a = \frac{1}{3}$ for spherical molecules. This indicates that the radius of gyration scales with the molecular weight raised to the power of one-third.

Final Answer

For a spherical molecule, the value of $a$ is $\frac{1}{3}$.