Estimating the concentration of nanoparticles from the particle size data

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Purpose: The purpose of this exercise is to estimate the molar concentration of nanoparticles based on the particle diameter measured by TEM.

Learning Outcomes:

At the end of this assignment you will be able to

Calculate the molar concentration of any nanoparticle preparation using the Feret’s diameter estimated through TEM image analysis.

Assignment:

The first step in estimating the molar concentration of nanoparticles is to calculate the average number of gold atoms per nanoparticle. In the specific case of gold, Liu and coworkers (11) determined the following relationship between the average number of gold atoms (N) per nanoparticle and the particle diameter (D):

\[\mathrm{N = \dfrac{π \left(19.3 \dfrac{g}{cm^3}\right) \mathit{D}^3}{6\left(197 \dfrac{g}{mol}\right)}} \label{eq. 1}\]

This equation assumes a spherical shape and a uniform face-centered cubic (fcc) structure. In equation \(\ref{eq. 1}\), 19.3 g/cm³ is the density for fcc gold and 197 g/mol is the gold atomic mass.

Q23. According to data from Table 2, nanoparticles synthesized with a 2:1 citrate to tetrachloroauric acid ratio and pH 5.4 have a Feret’s diameter of 21.7 nm. What is the value of N?

The next step is to calculate the molar concentration of a nanoparticle solution. This can be done by dividing the total number of gold atoms (N_total) equivalent to the amount of tetrachloroauric acid added to the reaction volume by the average number of gold atoms (N).

\[\mathrm{Concentration\: (mol/L) = \dfrac{N_{Total}}{N \times V \times N_A}} \label{eq. 2}\]

where V is the reaction volume in liters and N_A is Avogadro’s number. For example, for the 2:1 citrate to tetrachloroauric acid ratio, assume you reacted all the gold contained in 50.0 mL of 0.25 mM HAuCl₄.

Q24. What is the total number of gold atoms (N_Total) in 50.0 mL of a 0.25 mM solution of HAuCl₄?

Q25. What is the molar concentration of nanoparticles in this solution?

Now you can estimate the nanoparticle concentration in each of your experimental preparations. Complete the data in Table 4 by estimating the nanoparticle concentration for each experimental combination. The concentration value obtained in the case of reacting 2:1 citrate to tetrachloroauric acid at pH 5.4 is provided as an example of order of magnitude to be expected.

Table 4. Summary of experimental results – use this table to report nanoparticle concentration.

Citrate/Au	pH	Particle Feret’s Diameter	Nanoparticle concentration (M)
2.0:1	4.2	20.00 ± 0.50
	5.4	21.73 ± 0.40	3.16 x 10^-9
	7.0	21.22 ± 1.00
4.0:1	4.2	23.30 ± 0.78
	5.4	26.52 ± 1.57
	7.0	26.76 ± 0.99
7.0:1	4.2	31.87 ± 2.45
	5.4	33.91 ± 4.25
	7.0	No particles formed