The density of krypton (face centered cubic cell) is `3.19g//cm^(3)`. What is the radius of the atom? `("Atomic mass": Kr=84)`

Options

A. 0.198nm
B. 0.221nm
C. 0.206nm
D. 0.225nm

Answer

To find the radius of a krypton atom in a face-centered cubic (FCC) unit cell given its density and atomic mass, we can follow these steps:

### Step 1: Understand the relationship between density, atomic mass, and unit cell volume
The density (D) of a substance is given by the formula:
\[ D = \frac{Z \cdot M}{V \cdot N_A} \]
where:
- \( Z \) = number of atoms per unit cell (for FCC, \( Z = 4 \))
- \( M \) = molar mass of the substance (for krypton, \( M = 84 \, g/mol \))
- \( V \) = volume of the unit cell (which is \( a^3 \), where \( a \) is the edge length of the cube)
- \( N_A \) = Avogadro's number (\( N_A = 6.022 \times 10^{23} \, mol^{-1} \))

### Step 2: Rearranging the formula to find the volume of the unit cell
From the density formula, we can rearrange it to find \( a^3 \):
\[ a^3 = \frac{Z \cdot M}{D \cdot N_A} \]

### Step 3: Substitute the known values
Substituting the known values into the equation:
- \( Z = 4 \)
- \( M = 84 \, g/mol \)
- \( D = 3.19 \, g/cm^3 \)
- \( N_A = 6.022 \times 10^{23} \, mol^{-1} \)

Now, substituting these values:
\[ a^3 = \frac{4 \cdot 84}{3.19 \cdot 6.022 \times 10^{23}} \]

### Step 4: Calculate \( a^3 \)
Calculating the numerator:
\[ 4 \cdot 84 = 336 \]

Calculating the denominator:
\[ 3.19 \cdot 6.022 \times 10^{23} \approx 1.920 \times 10^{24} \]

Now, substituting these into the equation:
\[ a^3 = \frac{336}{1.920 \times 10^{24}} \approx 1.75 \times 10^{-22} \, cm^3 \]

### Step 5: Calculate the edge length \( a \)
To find \( a \), take the cube root of \( a^3 \):
\[ a = (1.75 \times 10^{-22})^{1/3} \approx 5.57 \times 10^{-8} \, cm \]

### Step 6: Relate edge length \( a \) to atomic radius \( r \)
In an FCC unit cell, the relationship between the edge length \( a \) and the atomic radius \( r \) is given by:
\[ a = 4r/\sqrt{2} \]
Thus,
\[ r = \frac{\sqrt{2} \cdot a}{4} \]

### Step 7: Substitute \( a \) into the radius formula
Substituting the value of \( a \):
\[ r = \frac{\sqrt{2} \cdot (5.57 \times 10^{-8})}{4} \]

### Step 8: Calculate the radius \( r \)
Calculating this gives:
\[ r \approx \frac{1.414 \cdot 5.57 \times 10^{-8}}{4} \approx 1.98 \times 10^{-8} \, cm \]

### Step 9: Convert to angstroms (optional)
To convert to angstroms (1 angstrom = \( 10^{-8} \, cm \)):
\[ r \approx 19.8 \, pm \]

### Final Answer
The radius of the krypton atom is approximately \( 1.98 \times 10^{-8} \, cm \) or \( 19.8 \, pm \).

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