Question

Microporous silica particles with a density of$\mathbf{2}\mathbf{.}\mathbf{2}\mathbf{g}\mathbf{/}\mathbf{mL}$and a diameter$\mathbf{10}\mathbf{\mu m}$of have a measured surface area of calculated the surface area of the$\mathbf{300}{\mathbf{m}}^{\mathbf{2}}$spherical silica as if it were simply solid particles what does this calculation tell you about the shape or porosity of the particles.

Short answer

Microporous silica particles with a density calculation tell you about the shape or porosity of the particles.

First we calculate volume of sphere:

$$\begin{gathered} \mathrm{v}=\frac{4}{3}.{\mathrm{\pi r}}^3 \\ \mathrm{v}=\frac{4}{3}.\mathrm{\pi }\left(5.{10}^{-4}\mathrm{cm}\right)3 \\ \mathrm{p}=0.27{\mathrm{m}}^2 \end{gathered}$$

Step-by-step solution

definition of volume

The amount of three-dimensional space enclosed by a closed surface is expressed as a scalar quantity.

First we calculate volume of sphere:

$$\begin{gathered} \mathrm{v}=\frac{4}{3}.{\mathrm{\pi r}}^3 \\ \mathrm{v}=\frac{4}{3}.\mathrm{\pi }\left(5.{10}^{-4}\mathrm{cm}\right)3 \\ \mathrm{p}=5.24.{10}^{-4}{\mathrm{cm}}^3 \end{gathered}$$

Then we calculate mass of one sphere:

$$\begin{gathered} \mathrm{m}=\mathrm{\rho }.\mathrm{v} \\ \mathrm{m}=2.2\mathrm{g}/\mathrm{mL}.5.24.{10}^{-10}\mathrm{mL} \\ \mathrm{m}=1.15.{10}^{-9}\mathrm{g} \end{gathered}$$

So, the number of particles in 1 g is:

role="math" localid="1654843608416" $$\begin{gathered} \mathrm{N}=\frac{1\mathrm{g}}{1.10.{10}^{-9}\mathrm{g}} \\ \mathrm{N}=8.7.{10}^8 \end{gathered}$$

Step 2:surface one particle

The surface of one particle is:

$$\begin{gathered} \mathrm{p}=4.{\mathrm{\pi r}}^2 \\ \mathrm{p}=4.\mathrm{\pi }{\left(5.{10}^{-6}\mathrm{m}\right)}^2 \\ \mathrm{p}=3.14.{10}^{-10}{\mathrm{m}}^2 \end{gathered}$$

Now we calculate surface of particles:

$$\begin{gathered} \mathrm{p}=3.10.{10}^{-10}{\mathrm{m}}^2.8.7.{10}^2 \\ \mathrm{p}=0.27{\mathrm{m}}^2 \end{gathered}$$

The measured surface area is 300m², so the particles have irregular shapes and they are porous.

Hence,

p=0.27m²