Complete this table by computing the following quantities: $\backslash$Delta Ni$/\backslash$Delta Dpi, $\backslash$Delta Ni$/$N$\backslash$Delta Dpi,

$\backslash$Delta Si$/\backslash$Delta Dpi, $\backslash$Delta Si$/$S$\backslash$Delta Dpi, $\backslash$Delta Mi$/\backslash$Delta Dpi, $\backslash$Delta Mi$/$M$\backslash$Delta Dpi,$\backslash$Delta Ni$/\backslash$Delta logDpi$,\backslash$Delta Ni$/$N$\backslash$Delta logDpi$,\backslash$Delta Si$/\backslash$Delta logDpi$,$

$\backslash$Delta Si$/$S$\backslash$Delta logDpi$,\backslash$Delta Mi$/\backslash$Delta logDpi$,\backslash$Delta Mi$/$M$\backslash$Delta logDpi$,$ where M $=$ particle mass.

b$.$ Plot $\backslash$Delta Ni$/\backslash$Delta logDpi$,\backslash$Delta Si$/\backslash$Delta logDpi$,\backslash$Delta Mi$/\backslash$Delta logDpi as histograms.

c$.$ Determine the geometric mean diameter and geometric standard deviation of the

lognormal distribution to which these data adhere and plot the continuous distributions on

the three plots from part $($b$).$

d$.$ For this lognormal distribution, develop an analytical expression that relates fined by the

mass median diameter to the count median diameter $($i$.$e$.,$ your result from part $($c$))$ as a

function of the parameters that define this distribution. What is the mass median diameter

of this particle size distribution?