Not this nah 😭 Solve for standard deviation during mixing

From the previous calculation, the standard deviation at the start of mixing is $\sigma_0 = 0.2179$.
For a perfectly mixed system, the standard deviation is $\sigma_p = 0$.

The mixing index ($M$) is a dimensionless value that quantifies the degree of mixing, ranging from 0 (completely unmixed) to 1 (perfectly mixed). The standard deviation during mixing ($\sigma$) is the actual measured standard deviation of samples taken at a specific point in time during the mixing process. These are related by the formula:

Substitute $\sigma_0 = 0.2179$ and $\sigma_p = 0$ into the formula:

Rearrange the equation to isolate $\sigma$:

To obtain a specific numerical value for the standard deviation during mixing ($\sigma$), a value for the mixing index ($M$) must be provided, or experimental data (e.g., sample concentrations during mixing) from which $\sigma$ can be calculated. Without this additional information, $\sigma$ can only be expressed in terms of $M$. Similarly, to find a numerical value for $M$, a value for $\sigma$ is needed.

The standard deviation during mixing is $\boxed{\sigma = 0.2179 (1 - M)}$.
The mixing index is $\boxed{M = \frac{0.2179 - \sigma}{0.2179}}$.