Size-Based Prediction of MnP AbundanceΒΆ

Across water, soil, sediment, sludge, and air, there is a strong inverse power-law relationship:

\[ n(x) \propto x^{-\alpha} \]

Where:

  • \(x\) is the characteristic size of the particles (e.g., geometric mean of the bin).

  • \(n(x)\) is the number of MP particles of size \(x\) (in \(\mu m\))

  • the exponent \(\alpha\) is a positive constant typically between 1 and 2

The raw power-law distribution is:

\[ n(x) = b \cdot x^{-\alpha} \]

Taking logarithms of both sides yields a linear form:

\[ \log(n) = -\alpha \log(x) + \log(b) \]

Based on this study, particles are only observed in a limited size range. Therefore, both \(x\) and \(n(x)\) should be normalized to the bin width:

\[ \hat n = \frac{n}{x_{UB} - x_{LB}} \]
\[ \hat x = \sqrt{x_{UB} \cdot x_{LB}} \]

Where:

  • \(n\): number of particles in the bin

  • \(x_{UB}, x_{LB}\): upper and lower size limits of the bin,

  • \(\hat n\): density of particles in a specific size range

  • \(\hat x\): geometric mean of the bin size

After these corrections, the relationship becomes:

\[ \log (\hat n) = -\alpha \log (\hat x) + \log (b) \tag{1} \]

Exponentiating both sides:

\[ \hat n = b \cdot \hat x^{-\alpha} \]

Or equivalently:

\[ n = b \cdot (x_{UB} - x_{LB}) \cdot \left( \sqrt{x_{UB} \cdot x_{LB}} \right)^{-\alpha} \tag{2} \]

Suppose the number of particles \(\hat n_{\text{ref}}\) in a reference bin \([x_{LB.\text{ref}}, x_{UB.\text{ref}}]\) is known, to predict the number of particles \(\hat n_{\text{pred}}\) in another bin \([x_{LB.\text{pred}}, x_{UB.\text{pred}}]\) with the same constants \(\alpha\) and \(b\), apply Equation (2) to both bins

\[ n_{\text{pred}} = b \cdot (x_{UB.\text{pred}} - x_{LB.\text{pred}}) \cdot \left( \sqrt{x_{UB.\text{pred}} \cdot x_{LB.\text{pred}}} \right)^{-\alpha} \]
\[ n_{\text{ref}} = b \cdot (x_{UB.\text{ref}} - x_{LB.\text{ref}}) \cdot \left( \sqrt{x_{UB.\text{ref}} \cdot x_{LB.\text{ref}}} \right)^{-\alpha} \]

Take the ratio:

\[ \frac{n_{\text{pred}}}{n_{\text{ref}}} = \frac{(x_{UB.\text{pred}} - x_{LB.\text{pred}}) \cdot \left( \sqrt{x_{UB.\text{pred}} \cdot x_{LB.\text{pred}}} \right)^{-\alpha} } {(x_{UB.\text{ref}} - x_{LB.\text{ref}}) \cdot \left( \sqrt{x_{UB.\text{ref}} \cdot x_{LB.\text{ref}}} \right)^{-\alpha} } \]
\[ \therefore n_{\text{pred}} = n_{\text{ref}} \cdot \left( \frac{x_{UB.\text{pred}} - x_{LB.\text{pred}}}{x_{UB.\text{ref}} - x_{LB.\text{ref}}} \right) \cdot \left( \frac{x_{UB.\text{pred}} \cdot x_{LB.\text{pred}}}{x_{UB.\text{ref}} \cdot x_{LB.\text{ref}}} \right)^{- \alpha / 2} \]

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