Number of closely packed colloidal particles in an aggregate

In summary, the question is asking for the number of colloidal particles in an aggregate with a radius of 20 CP particle diameters and a fractal dimension of 2.00, given that a close packed colloidal aggregate with the same radius and type of particles contains 4400 particles. However, the volume and fractal dimension of the aggregate must also be taken into account in order to find the correct answer.
  • #1
maximus123
50
0
Hello everyone

This is sort of a geometry problem. I'm sure it has an easy answer but it just won't come to me. Here's my problem

If a close packed colloidal aggregate contains 4400 monodisperse spherical colloidal particles (CP), how many colloidal particles would there be in a colloidal aggregate (or ‘floc’) of fractal dimension 2.00 with the same radius and made up of the same type of colloidal particles. Assume that the radius of both types of colloidal aggregate is 20 CP particle diameters.
A close packed colloidal aggregate of smaller spherical colloidal particles can be thought of as small spheres within a sphere.

I have the relationship

[itex]N(r)\propto r^{D_f}[/itex]

The number of colloidal particles within a radius is proportional to the radius to the power of the fractal dimension. For a close packed colloidal aggregate [itex]D_f=3[/itex] so


[itex]\frac{N_1(r)}{N_2(r)}=\frac{r^3}{r^2}\\
\frac{4400}{N_2(r)}=r
[/itex]

So if I'm right so far then all I need is the radius of the aggregate. This is what I tried


[itex]\frac{V_a}{V_c}=4400
[/itex]

where [itex]V_a[/itex] is the volume of the aggregate and [itex]V_c[/itex] is the volume of the colloidal particle, so


[itex]\frac{V_a}{V_c}=\frac{\frac{4}{3}\pi r_a^3}{\frac{4}{3}\pi r_c^3}=4400\\
\frac{\frac{4}{3}\pi \left(20\sigma\right)^3}{\frac{4}{3}\pi \left(\frac{1}{2}\sigma\right)^3}=4400
[/itex]
where I'm using [itex]\sigma[/itex] to mean the diameter of the colloidal particle. This last line of calculation is clearly wrong (or at least of no use) as everything will cancel and I'll be left with some incorrect statement such as [itex]64000=4400[/itex].

Can anyone suggest how I might find the radius of the aggregate?

Thank you
 
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  • #2
for your question. The calculation you have done so far is correct, but there are a few things to consider in order to find the radius of the aggregate.

First, the volume of a close packed colloidal aggregate is not the same as the volume of a single colloidal particle. This is because the particles in the aggregate are arranged in a specific way, creating void spaces between them. This means that the volume of the aggregate is actually larger than the volume of the same number of individual particles.

Second, the fractal dimension of an aggregate is related to how the particles are arranged within it. A fractal dimension of 2.00 indicates that the particles are arranged in a two-dimensional structure, which means they are not as tightly packed as in a three-dimensional close packed structure. This will affect the number of particles within a given radius.

To find the radius of the aggregate, you will need to take into account the volume of the aggregate and the fractal dimension. You can use the formula you have already derived, but you will need to adjust it to account for the differences in volume and fractal dimension.

I hope this helps. Let me know if you have any further questions.
 

What is the definition of "Number of closely packed colloidal particles in an aggregate"?

The number of closely packed colloidal particles in an aggregate refers to the number of individual particles that are closely packed together to form a larger structure or aggregate. These particles can be of various sizes and shapes, but they are all held together by intermolecular forces.

Why is the number of closely packed colloidal particles in an aggregate important?

The number of closely packed colloidal particles in an aggregate is important because it affects the physical and chemical properties of the aggregate. For example, the number of particles can affect the strength, stability, and reactivity of the aggregate. It can also impact how the aggregate interacts with its surroundings and other particles.

How is the number of closely packed colloidal particles in an aggregate determined?

The number of closely packed colloidal particles in an aggregate can be determined using various techniques such as microscopy, scattering methods, and sedimentation analysis. These methods involve counting the number of particles within a given volume or area and then calculating the average number of particles per unit volume.

What factors can affect the number of closely packed colloidal particles in an aggregate?

The number of closely packed colloidal particles in an aggregate can be affected by various factors such as the size and shape of the individual particles, the concentration and type of dispersing medium, and the presence of any additives or impurities. Temperature, pressure, and pH can also influence the number of particles in an aggregate.

How does the number of closely packed colloidal particles in an aggregate relate to the overall stability of the aggregate?

The number of closely packed colloidal particles in an aggregate is directly related to its stability. Generally, the more particles that are closely packed together, the more stable the aggregate will be. This is because the intermolecular forces between particles increase as the number of particles increases, making it more difficult for the aggregate to break apart or disperse.

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