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Gold Nanoparticles (AuNP) have been found many applications
in chemistry and biochemistry, for example, as labels for bioanalysis or
as payload carriers for biotherapeutics. Recently, investigations of the
potential risks of the applications of AuNP and its products to human
health and to the environment are also dynamic. It is very critical to
know the mass or the number of nanoparticles in a sample in order to
understand these biological and environmental effects. This requires
analytical methods that focus on the average size of the particles,
number of particles, and the total mass of particles in a liquid or solid
sample. |

Gold content in solutions can be measured by methods
such as electrochemical methods, Inductively Coupled Plasma-
Mass Spectrometry (ICP-MS), and Flame Atomic Absorption
Spectrophotometry (FAAS). These methods measure the amount of
total gold element in a solution after the chemical digestion of gold,
usually by aqua regia or HBr/Br_{2} solutions. In 2008, National Institute
of Standards and Technology (NIST) released AuNP standards with
known gold concentration and average particles diameter of 10, 30,
and 60 nm [1]. The gold content was measured by using inductivelycoupled
plasma optical emission spectrometry (ICP-OES) after
digestion. On the other hand, the average diameter of the AuNP was
measured by Atomic Force Microscopy (AFM), Scanning Electron
Microscopy (SEM), Transmission Electron Microscopy (TEM),
Differential Mobility Analysis, Dynamic Light Scattering, and Small-
Angle X-ray Scattering. In several recent reports the average diameter,
total gold content, and number of AuNP have been characterized
in natural water systems and uptake in cell lines. In these reports, a
combination of TEM and ICP-MS, ICP-OES, or atomic absorption
Spectrophotometry (AAS) was used for the AuNP characterization and
determination. |

Given the diameter of the AuNP and the total amount (in mole,
mol/L, number of gold atoms, or gram) of gold, it seems not very hard
to calculate the number of AuNP in a suspension: |

Absolute monodisperse AuNP suspension does not exist. The diameter of the AuNPs in a synthesized suspension varies and usually follows a discrete log-normal distribution. Therefore, the diameter of the AuNP that has been used for the calculations refers to the mean value of the AuNPs in a suspension. The mean value is usually obtained by measuring the diameters of tens or hundreds AuNPs in their TEM image. The mean diameter was calculated by the individual diameters of measured AuNPs. Considering that there are many different statistical ways to calculate the mean value of a set of data, an accurate statistical mean should be used to calculate the number of AuNPs in a suspension and a nominal value that can be used to describe the size of AuNPs. |

In this statistical discussion, AuNP population refers to the
number of all the particles. Population equals to N means there are |

N nanoparticle in a suspension. Given a suspension of AuNP has
population of N; each individual AuNP can be identified as P_{i}, where i
= 1 ~ N. Assume all the AuNPs are geometrically spherical. Particle P_{i}
has a radii of r_{i}. |

The arithmetic average, or the numerical average, radius of the AuNPs, should be: |

(1) |

In this case, all the AuNPs have the same weight contributed to the average. The weight is assigned to be unity. |

When the weight of AuNP P_{i} equals to the radius (r_{i}) of the AuNP,
the weighted average can be called radius average, which can be
calculated as: |

(2) |

When the weight of AuNP P_{i} equals to the cross-section area (πr_{i}^{2} ) of
the AuNP, the weighted average can be called cross-section-area (CSA)
average, which can be calculated as: |

(3) |

When the weight of AuNP P_{i} equals to the surface area ( 4πr^{2}_{i} ) of the
AuNP, the weighted average can be called surface-area (SA) average,
which can be calculated as: |

(4) |

When the weight of NP Pi equals to the volume ( 4 /3 •πr^{3}_{i} ) of the NP,
the weighted average can be called volume (V) average, which can be
calculated as: |

(5) |

For example, we have a set of 21 AuNPs. Each of these AuNP has
different radius. We assuming r_{1} = 4.00 nm, while r_{i+1} – r_{i} = 0.10 nm.
Then r_{21} is 6.00 nm. Following the above equations, the arithmetic and
weighted average radius are calculated and summarized in table 1. |

Weighted mean radii of AuNP should be obtained and used depending on the measurement methods. In these methods, the probabilities of each particle are picked and measured are identical or different. The probability may proportional to the radius, crosssection area, surface area, or volume, which are the weights of the mean. Currently, TEM is the prevailing method of AuNP radius measurement. When TEM is used, the probability of each AuNP is picked and measured is the same either by software or by operator. Therefore, the arithmetic mean values should be obtained by a number average of the measured radii of all AuNPs. However, depend on how these measured radii of AuNPs are used, statistical moment means should be considered when calculating the mean value. The moment mean radius of AuNPs can be calculated by equation: |

(6) |

Therefore, the arithmetic mean, , corresponds to [1,0]; while
the weighted means, _{V} _{SA} _{CSA} , and _{r} correspond to [4,3], [3,2],
and [2,1], respectively. With the concept of moment mean, the
number of AuNPs can be calculated more accurately. In experiments,
the total amount (mass or volume) of gold can be measured by ICPMS,
ICP-OES, electrochemistry, or AAS. The mean radius of AuNPs
can be measured by TEM. The number of the AuNP can be calculated
by dividing the total volume by the mean volume. For a spherical
particle, the volume equals 4/3
Πr^{3} . Here, the radius r should not be the
arithmetic mean of the radii measured by TEM; rather, it should be
the number-volumetric mean of the measured radii. In the context of
moment mean, the number-volumetric mean is [3,0]: |

(7) |

Therefore, the mean volume of AuNPs should be calculated by the [3,0] value of all the measured AuNP radii, instead of the arithmetic mean of the measured radii. |

The size distribution of synthesized AuNPs in dispersions is usually
a log-normal distribution that produces a “skewed” distribution curve.A geometric standard deviation (s_{g}) should be calculated and provided.
Then the number-volume mean radius can be calculated by the Hatch-
Choate equation [2]. |

(8) |

In summary, in this essay, different ways to calculate the mean radii of AuNPs from experimental data are discussed. The calculations are based on statistical concepts of arithmetic mean, weighted means, and moment means. Previously, the mean values and the distributions of particles’ size have been discussed intensively in statistics and engineering. As the application of AuNP becomes more and more popular, the characterization of the mean size of AuNP and the number of AuNP in the areas of chemistry, biochemistry, physiology, environmental science, and food science will be more and more frequent. We believe these different ways of calculation should be noticed when experimental data are processed. |

## References |

Table 1 |

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