For absolutely pure systems, the answer is yes as best that we can measure today (see notes below). Copper is copper is copper is copper is ... regardless of how finely ground it is or how big it is.

Due respect is noted to the "quantum mechanical differences" or "binding energy state differences" of surface atoms versus bulk atoms BUT, given the measurement precision of today's instruments, the mass differences being proposed are absolutely immeasurable. In any case, they would also require that both Cu systems be perfectly pure and in a perfect vacuum. We can not get there today (and we will never get there in the future). And then let's not even start on a discussion of how grain boundaries or defects give quantum or binding energy state differences to change the masses of the atoms. We CANNOT measure such changes, even if we could create the perfect systems that are needed to have them in the first place.

Also in reality, no substance is perfectly pure. In such a case, your question becomes totally ambiguous. I can give you either answer (yes or no) because I have no knowledge of the purity of what you have before you. I can answer with a better precision only after you define the purity of your system with greater precision.

So, when your question is taken simply as a high school level chemistry question, the answer to it is yes. When it is taken as anything else, the answer to it is I don't know ... you tell me.

If we assume 100% purity in both cases, they should, though the densities and several other properties will be different. This also assumes that each sample has only one isotope of Cu (or would have an equal average atomic weight).

In the real world, they will not have an equal number of atoms, as the nanoparticles will have significant surface area for oxidation and will have leftover contaminants from their production. The bulk will also have impurities, so the number of atoms between each sample will never be identical.

Yes, sure! They may, or may not, have the same density. But since the weight is defined by the number of atoms, and since the atoms do not change (considering the same isotopes), that means "Equal weight => same number of atoms".

technically, yes. in my opinion, however, depending on the environments where you evaluate your materials. if in ambient conditions, the Cu nanoparticles may adsorb more gaseous molecules relative to bulk materials. in vacuum conditions and in 100 % purity, yes for sure.

It is reported, “Materials at the nanoscale, because of quantum mechanics effects, they behave differently from bulk materials. Indeed the emerging properties are due to the quantity variation in number of atoms i.e. in nanoparticles, number of atoms is less than their bulk matter where millions of atoms together in” .

If both bulk and nano have equal number of atoms; how these different behaviors will be created?

At the surface,atoms are not contrained by other atoms on all sides, as they are in a bulk crystal. In nanoparticles there are proportionally more atoms at the surface than in the bulk. This means that the properties a material on the nanoscale can be dominated by the contributions from the surface atoms, and therefore be different to the properties of the bulk material. It is not the number of atoms that are different, it is the number of atoms that are in the bulk-like state that are different when you compare nanoparticles and bulk systems.

While beaker full of nanoparticles contains the same number of atoms as a bulk piece with the same mass, each individual nanoparticle contains only a tiny fraction of this number. This is why nanoparticles can behave so very differently.

As for the exact mechanism behind the change in behaviour, that depends on the properties you're talking about. If you give us more information, we may be able to help.

In theory yes they should have equal number, but then practically it is impossible since one has to make sure there is absolutely no adsorbed impurities, no gas in the space between nanoparticles, chemical composition is exactly the same in bulk and on surface of every particle. It is impractical to ensure all these.

But definitely a good thought experiment

Volume can be different but weight has to be same depending upon the synthesis method, there can be method introduced impurities.

If both Samples are 100% pure and in Vacuum they have the same mass ergo same amount of Atoms. Except if one of theme is moving near the speed of light.

For absolutely pure systems, the answer is yes as best that we can measure today (see notes below). Copper is copper is copper is copper is ... regardless of how finely ground it is or how big it is.

Due respect is noted to the "quantum mechanical differences" or "binding energy state differences" of surface atoms versus bulk atoms BUT, given the measurement precision of today's instruments, the mass differences being proposed are absolutely immeasurable. In any case, they would also require that both Cu systems be perfectly pure and in a perfect vacuum. We can not get there today (and we will never get there in the future). And then let's not even start on a discussion of how grain boundaries or defects give quantum or binding energy state differences to change the masses of the atoms. We CANNOT measure such changes, even if we could create the perfect systems that are needed to have them in the first place.

Also in reality, no substance is perfectly pure. In such a case, your question becomes totally ambiguous. I can give you either answer (yes or no) because I have no knowledge of the purity of what you have before you. I can answer with a better precision only after you define the purity of your system with greater precision.

So, when your question is taken simply as a high school level chemistry question, the answer to it is yes. When it is taken as anything else, the answer to it is I don't know ... you tell me.

I would say "no", at least from practical standpoint of view: Cu nanoparticles are heavily oxidized at the surface, so that you may get a significant percentage of the Cu atoms in oxidized form. Depending on size of the nanoparticles, you may get 25% of the Cu atoms as oxides!