A hypothetical metal has a simple cubic crystal structure. If its atomic weight is 70.6

g/mole and the atomic radius is 0.128 nm, compute its density.

In a simple cubic crystal there is 1 atom/unit cell so the mass of 1 atom is

70.6 g/mol/6.022E23 atoms/mol= 1.172E-22 g/atom
In the crystal a = 2r = 2*0.128E-9 nm = 2*1.28E-8 cm = 2.56E-8 cm
Volume of 1 atom = a^3 = (2.56E-8 cm)^3 = 1.68E-23 cc
density = mass/volume = 1.172E-22/1.68E-23 = 6.98 g/cc

To compute the density of the metal, we need to determine the volume of a single unit cell and divide it by the mass of that unit cell.

The simple cubic crystal structure has one atom per unit cell, and the volume of a simple cubic unit cell is given by:
V = a^3
where V is the volume, and a is the length of one side of the unit cell.

Since the atomic radius is given as 0.128 nm, the length of one side of the unit cell is equal to twice the atomic radius:
a = 2 * atomic radius

Given that the atomic radius (r) is 0.128 nm, we can substitute it into the formula to find the length of one side of the unit cell:
a = 2 * 0.128 nm

Next, we need to convert the atomic radius from nanometers (nm) to meters (m) to ensure consistent units:
a = 2 * 0.128 nm = 0.256 nm

Since 1 nm is equal to 1 x 10^-9 m, we can convert the length from nanometers to meters:
a = 0.256 nm * 1 x 10^-9 m/nm = 2.56 x 10^-10 m

Now, we can calculate the volume of a single unit cell:
V = a^3 = (2.56 x 10^-10 m)^3 = 16.79 x 10^-30 m^3

Next, we need to calculate the mass of the unit cell.
The atomic weight given is 70.6 g/mole, which represents the molar mass of the metal. To find the mass of one unit cell, we need to convert the molar mass to grams per unit cell.
The molar mass is equal to the mass of 1 mole, which contains Avogadro's number (6.022 x 10^23) of atoms.
Therefore, the mass of one unit cell is given by:
mass = atomic weight / Avogadro's number

Substituting the given values:
mass = 70.6 g/mole / (6.022 x 10^23 atoms/mole)

Now we can calculate the density of the metal:
density = mass / volume

Substituting the calculated values:
density = (70.6 g/mole / (6.022 x 10^23 atoms/mole)) / (16.79 x 10^-30 m^3)

By performing the calculations, the density of the metal in units of grams per cubic meter (g/m^3) can be determined.

To compute the density of the metal with a simple cubic crystal structure, we need to know its atomic weight and atomic radius. The atomic weight given is 70.6 g/mole, and the atomic radius is 0.128 nm.

Density (ρ) is defined as the mass per unit volume. In this case, we need to find the mass of one unit cell and the volume of that unit cell.

To determine the mass of one unit cell, we need to calculate the number of atoms in a simple cubic structure. In a simple cubic structure, there is one atom at each corner of the unit cell.

1. Determine the volume of the unit cell:
In a simple cubic structure, the volume of the unit cell is given by:
Volume = (edge length)^3
However, the edge length is twice the atomic radius, so we have:
Volume = (2 * Atomic radius)^3

2. Determine the mass of one atom:
The atomic weight is given as 70.6 g/mole, which means the mass of one mole of atoms is 70.6 g. Therefore, the mass of one atom (m) can be calculated as:
Mass of one atom (m) = Atomic weight / Avogadro's number

3. Determine the mass of one unit cell:
Since there is one atom at each corner of the unit cell, the mass of one unit cell (M) can be determined by multiplying the mass of one atom by the total number of atoms in one unit cell:
Mass of one unit cell (M) = m * Total number of atoms in one unit cell

4. Finally, calculate the density:
Density (ρ) = Mass of one unit cell / Volume of one unit cell

Now, let's plug in the given values and calculate the density:

Atomic weight (W) = 70.6 g/mole
Atomic radius (r) = 0.128 nm

1. Calculate the volume of the unit cell:
Volume = (2 * 0.128 nm)^3 (Remember to convert nm to cm)

2. Calculate the mass of one atom:
m = W / Avogadro's number

3. Calculate the mass of one unit cell:
Total number of atoms in one unit cell = 1 (corner atom)
M = m * Total number of atoms in one unit cell

4. Calculate the density:
Density = M / Volume

Follow these steps to find the density of the hypothetical metal with a simple cubic crystal structure.