Final answer:
To calculate the radius of a gold atom in a face-centered cubic crystal structure, you can use the formula: Edge length (a) = (4r)/√2. By rearranging the formula and substituting the given values, you can find that the radius of a gold atom is approximately 0.707 cm.
Step-by-step explanation:
Gold has a face-centered cubic (FCC) crystal structure, which means that the atoms are arranged in a cubic formation with atoms at each corner and in the center of each face.
To calculate the radius of a gold atom, we can use the formula:
Edge length (a) = (4r)/√2
Density (d) = Mass (m)/Volume (V)
= 19.3 g/cm³
Using the density equation, we can rearrange it to find the volume of the unit cell:
Volume (V) = Mass (m)/Density (d)
= 1 cm³
From the formula of FCC structure, we know that there are 4 atoms per unit cell.
So the volume of a gold atom can be expressed as:
Volume of atom = Volume of unit cell / 4
= 1 cm³ / 4
Now, we can calculate the radius (r) using the formula:
Radius (r) = (√2) * (Edge length (a))/4
= (√2) * (4r)/√2)/4
= r
Thus, the radius of a gold atom is equal to its edge length (a).
Therefore, we have:
Edge length (a) = (√2) * radius (r)
Let's substitute the value of the edge length (a) as 1 cm into the equation:
(√2) * radius (r) = 1 cm
radius (r) = 1 cm / (√2)
= 0.707 cm.