Chapter 24, Problem 52P

The Rosin-Rammler-Bennet (RRB) equation is used to describe size distribution in fine dust. $F\left(x\right)$ Represents the cumulative mass of dust particles of diameter x and smaller. x' and n' are constants equal to $30\mu \text{m}$ and 1.44, respectively. The mass density distribution $F\left(x\right)$ or the mass of dust particles of a diameter x is found by taking the derivative of the cumulative distribution

$F\left(x\right)=1-e^{-{\left(x/x\text{'}\right)}^{n\text{'}}}$ $f\left(x\right)=\frac{dF\left(x\right)}{dx}$

(a) Numerically calculate the mass density distribution $f\left(x\right)$ and graph both $f\left(x\right)$ and the cumulative distribution $F\left(x\right)$.

(b) Using your results from part (a ), calculate the mode size of the mass density distribution-that is, the size at which the derivative of $f\left(x\right)$ is equal to zero.

(c) Find the surface area per mass of the dust $S_m\left({\text{cm}}^{\text{2}}\text{/g}\right)$ using

$S_m=\frac{6}{\rho }{\displaystyle \underset{d_{\min }}{\overset{\infty }{\int }}\frac{f\left(x\right)}{x}dx}$

The equation is valid only for spherical particles. Assume a density $\rho =1\text{g}{\text{cm}}^{-3}$ and a minimum diameter of dust included in the distribution $d_{\min }\text{of 1 }\mu \text{m}$.