A cube of gold that is 1.00 cm on a side has a mass of 19.3 g. A single gold atom has a mass of 197.0 u. (a) How many gold atoms are in the cube?

검증된 단계별 안내

Calculate the volume of the gold cube using the formula for the volume of a cube: \( V = \text{side}^3 \).

Determine the density of gold using the mass and volume of the cube: \( \text{Density} = \frac{\text{mass}}{\text{volume}} \).

Convert the mass of a single gold atom from atomic mass units (u) to grams using the conversion factor: 1 u = 1.66053906660 \(\times\) 10^{-24} \(\text{ g}\).

Calculate the number of gold atoms in the cube by dividing the total mass of the cube by the mass of a single gold atom in grams.

Use Avogadro's number (6.022 \(\times\) 10^{23} \(\text{ atoms/mol}\)) to verify the number of atoms if needed.

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주요 개념

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Density and Mass

Density is defined as mass per unit volume and is a crucial property of materials. In this case, the density of gold can be calculated using the mass of the cube (19.3 g) and its volume (1.00 cm³). Understanding how to relate mass and volume through density is essential for determining the number of atoms in the cube.

Molar Mass and Avogadro's Number

The molar mass of a substance is the mass of one mole of its entities, typically expressed in grams per mole. For gold, the molar mass is approximately 197.0 g/mol. Avogadro's number, 6.022 x 10²³, represents the number of atoms in one mole of a substance, allowing us to convert between mass and the number of atoms.

Calculating Number of Atoms

To find the number of atoms in a sample, one can use the formula: number of atoms = (mass of the sample / molar mass) x Avogadro's number. This calculation combines the concepts of mass, molar mass, and Avogadro's number to determine how many individual atoms are present in a given mass of a substance.