Gold Nanoparticles The product shown i Figure P 3.30...

Gold Nanoparticles The product shown i Figure $P 3.30$ contains gold nanoparticles in water. The manufacturer claims that drinking this beverage improves human health. How many gold atoms are in a gold nanoparticle with a diameter of $2.00 \mathrm{nm}(d=19.3 \mathrm{g} / \mathrm{mL}$ $\left.1 \mathrm{amu}=1.66054 \times 10^{-24} \mathrm{g}\right) ?$. (IMAGE CAN'T COPY)

Key Concepts

Nanoparticles

Nanoparticles are materials with dimensions on the nanometer scale, typically between 1 and 100 nanometers. Their small size imparts unique physical and chemical properties compared to bulk materials. In this context, understanding nanoparticles involves examining how their size, shape, and composition determine their behavior and applications.

Spherical Volume Calculation

When dealing with particles that are approximately spherical, the volume of a nanoparticle can be calculated using the formula V = (4/3)?r^3, where r is the radius of the sphere. This geometric relationship is crucial for relating the particle’s physical dimensions to its overall volume, which then links to other properties such as mass.

Density and Mass Relationship

Density is defined as mass per unit volume and serves as a bridge between the geometric volume of an object and its mass. By knowing the density and computing the volume, one can determine the mass of a nanoparticle. This approach is essential in materials science for relating macroscopic material properties to the mass of individual particles.

Atomic Mass and Unit Conversion

Every element has a characteristic atomic mass, typically expressed in atomic mass units (amu), which can be converted to grams using a conversion factor (1 amu = 1.66054×10^-24 g). This conversion is critical when calculating the mass of a single atom in the context of bulk materials or nanoparticles.

Atom Counting by Mass Division

To determine the number of atoms in a nanoparticle, the mass of the nanoparticle (found using its volume and density) is divided by the mass of a single atom (obtained via atomic mass and conversion factors). This calculation links the continuous measure of mass to the discrete count of atoms, bridging macroscopic and atomic scales.

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