Plonium crystallizes with a simple cubic structure. It has a density of 9.3 g/cm³, a radius of 167 pm, and a molar mass of 209 g/mol. Use these data to calculate Avogadro's number (the number of atoms in one mole).

검증된 단계별 안내

Identify the type of crystal structure: Plutonium crystallizes in a simple cubic structure, which means there is 1 atom per unit cell.

Calculate the volume of the unit cell using the atomic radius. Since the structure is simple cubic, the edge length (a) of the cube is twice the atomic radius:

a = 2 \(\times\) r

. Convert the radius from picometers (pm) to centimeters (cm) before calculating the volume.

Calculate the volume of the unit cell using the formula for the volume of a cube:

V = a^3

Use the density formula to find the mass of one unit cell:

\(\rho\) = \(\frac{\text{mass of unit cell}\)}{\(\text{volume of unit cell}\)}

. Rearranged, the mass of the unit cell is

\(\text{mass of unit cell}\) = \(\rho\) \(\times\) V

Relate the mass of the unit cell to the number of atoms and molar mass to find Avogadro's number. Since there is 1 atom per unit cell, the mass of one atom is the mass of the unit cell. Use the formula

N_A = \(\frac{\text{molar mass}\)}{\(\text{mass of one atom}\)}

to calculate Avogadro's number.

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Crystal Structure and Unit Cell

The crystal structure defines how atoms are arranged in a solid. A simple cubic structure has atoms at each corner of a cube, with one atom per unit cell effectively. Understanding the unit cell's geometry and the number of atoms it contains is essential to relate atomic dimensions to macroscopic properties like density.

Density and Molar Mass Relationship

Density is mass per unit volume and connects the microscopic scale (atoms) to the macroscopic scale (bulk material). Using the molar mass and density, one can find the volume occupied by one mole of atoms, which helps in calculating the number of atoms per unit volume and ultimately Avogadro's number.

Avogadro's Number Calculation

Avogadro's number is the number of atoms in one mole of a substance. By combining the unit cell volume, atomic radius, density, and molar mass, one can derive Avogadro's number experimentally. This involves calculating the volume of the unit cell, the number of atoms per cell, and relating these to the molar volume.

Plonium crystallizes with a simple cubic structure. It has a density of 9.3 g/cm3, a radius of 167 pm, and a molar mass of 209 g/mol. Use these data to calculate Avogadro's number (the number of atoms in one mole).