Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors
Derive: Mun
\( \mu_n = \frac{\sum j R_j}{\sum R_j} = \frac{\sum j R_j}{I_0} \)
\( = \frac{\sum I_{0j} (1 - \phi) \phi^{j-1}}{I_0} \)
\( = \sum_{j=1}^{\infty} j (1 - \phi) \phi^{j-1} = \sum_{j=1}^{\infty} j \phi^{j-1} - \sum_{j=1}^{\infty} j \phi^j \)
\( = 1 + \sum_{j=2}^{\infty} j \phi^{j-1} - \sum_{j=1}^{\infty} j \phi^j \)
Let \( (j-1) = \ell \) \ then \ \( j = \ell + 1 \)
\( \mu_n = 1 + \sum_{\ell=1}^{\infty} (\ell + 1) \phi^\ell - \sum_{j=1}^{\infty} j \phi^j \)
\( = 1 + \sum_{\ell=1}^{\infty} \phi^\ell + \sum_{j=1}^{\infty} \ell \phi^\ell - \sum_{j=1}^{\infty} j \phi^j \)
\( = 1 + \sum \phi^j = \frac{1}{(1 - \phi)} \)