What you want to do is not strictly possible with these particles, which are polydiserse. The most pragmatic way to do it anyway would be to find the literature extinction coefficient for 5 nm gold particles (there are tables) and measure the absorbance in the UV-vis spectrum at the wavelength for which you have the coefficient, usually done at about 520 nm. Lambert-Beer's law will then give you directly the molar concentration of the particles but this will be an approximation since many of the particles are not of 5 nm diameter and hence will have a different extinction coefficient.

OD=log(e)*sigma*(N/V)*d,

sigma - extinction (absorption, in your case) cross-section (use some model numbers for Q, sigma=Q*S, Q=0.33 - efficiency for 5 nm particles (double check this from literature but I think it's correct), calculate S - cross-section)

(N/V) - number of nanoparticles (N) in volume (V),

d- lateral size of the measuring volume in the direction of propagation( usually, the length of the cuvette)

Make sure you use correct units for V and d!

Once you get N, the rest is easy. Knowing the density of gold, the molar mass etc. allows you to express N in whatever units you want.

NOTE: Q=0.25 (in air) and Q=0.33 (in water) - I've just checked it!

One of possible references: M. Quinten, "Optical properties of Nanoparticle Systems: Mie and Beyond", Wiley-VCH, 2011

The concentration of nanoparticles is most often expressed in number of particle/ml.

Once you know that, use Avogadro's number to convert it to molarity (6 x 10^23 particles /liter is 1M).

thank you very much for the answers Mr. S.Grabtchak, Eliza H

Actually I have a Gold-PEG nanoparticles ....

The description of the company is:

OD: 50

diameter 5 nm

Core Size 3 - 7 nm

1 mL in glass bottle

dispersion in H2O

I want the solution in nano -mole for in-vitro study

How I can convert in nano-mole?

regards and thanks

What you want to do is not strictly possible with these particles, which are polydiserse. The most pragmatic way to do it anyway would be to find the literature extinction coefficient for 5 nm gold particles (there are tables) and measure the absorbance in the UV-vis spectrum at the wavelength for which you have the coefficient, usually done at about 520 nm. Lambert-Beer's law will then give you directly the molar concentration of the particles but this will be an approximation since many of the particles are not of 5 nm diameter and hence will have a different extinction coefficient.

Thanks M Brust Its really informative and helpful

thanks and regards

To determine the molar gold concentration in a gold nanoparticle solution I also recommend to use UV-Vis measurement but at a wavelength of 400 nm. At this spectral region, organic molecules and the surface plasmon resonance have only minor influence. The absorbance is mainly due to interband transitions of gold and therefore can serve as a more robust measure of the gold concentration.

Still, particle size, surface functionalization etc. do influence the absorbance at 400 nm making it inexact to give a general extinction coefficient for all gold nanoparticle. Instead, we have carried out a widespread study to validate the determination of gold concentration via UV-Vis absorbance at 400 nm and can give a distribution of extinction coefficients that you can use for your purpose.

http://pubs.acs.org/doi/abs/10.1021/ac502053s

You could also digest the nanoparticles and express the "quantity" relative to gold ion concentration (using AAS) . This would then avoid the issue of variable particle sizes.

You can even generate a calibration curve and then determine the exact concentration of NPs that your cells uptake relative to starting concentration (%). 

Protocols widely available for nanoparticle digestion and ion quantification.

I think this link is useful, 

http://www.instanano.com/characterization/theoretical/concentration-calculation-from-uv-vis-absorbance/