The density of solid Ar (Ar=40 g/mole) is 1.68 g/ml at 40 K. if the argon atom is assumed to be a sphere of radius `1.50xx10^(-8)cm`, then % of solid Ar is apparently empty space?

Options

A. (a) `35.64`
B. (b) `64.36`
C. (c) `74`
D. (d) None of these

Answer

To find the percentage of solid Argon (Ar) that is apparently empty space, we can follow these steps:

### Step 1: Determine the mass of Argon in 1 ml
Given the density of solid Argon is 1.68 g/ml, we can calculate the mass of Argon in 1 ml:
\[ \text{Mass} = \text{Density} \times \text{Volume} = 1.68 \, \text{g/ml} \times 1 \, \text{ml} = 1.68 \, \text{g} \]

### Step 2: Calculate the number of moles of Argon
Using the molar mass of Argon (40 g/mol), we can find the number of moles:
\[ \text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}} = \frac{1.68 \, \text{g}}{40 \, \text{g/mol}} = 0.042 \, \text{mol} \]

### Step 3: Calculate the number of Argon atoms
Using Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol), we calculate the total number of Argon atoms:
\[ \text{Number of atoms} = \text{Number of moles} \times \text{Avogadro's number} = 0.042 \, \text{mol} \times 6.022 \times 10^{23} \, \text{atoms/mol} \approx 2.53 \times 10^{22} \, \text{atoms} \]

### Step 4: Calculate the volume occupied by one Argon atom
Assuming each Argon atom is a sphere with a radius of \(1.50 \times 10^{-8} \, \text{cm}\), we can calculate the volume of one Argon atom using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (1.50 \times 10^{-8} \, \text{cm})^3 \approx 1.41 \times 10^{-23} \, \text{cm}^3 \]

### Step 5: Calculate the total volume occupied by Argon atoms
Now, we can find the total volume occupied by all Argon atoms:
\[ \text{Total volume occupied} = \text{Number of atoms} \times \text{Volume of one atom} \approx 2.53 \times 10^{22} \, \text{atoms} \times 1.41 \times 10^{-23} \, \text{cm}^3 \approx 0.357 \, \text{cm}^3 \]
Since \(1 \, \text{cm}^3 = 1 \, \text{ml}\), this is approximately \(0.357 \, \text{ml}\).

### Step 6: Calculate the empty space
The empty space can be calculated by subtracting the volume occupied by Argon atoms from the total volume:
\[ \text{Empty space} = \text{Total volume} - \text{Total volume occupied} = 1 \, \text{ml} - 0.357 \, \text{ml} = 0.643 \, \text{ml} \]

### Step 7: Calculate the percentage of empty space
Finally, we can calculate the percentage of empty space in solid Argon:
\[ \text{Percentage of empty space} = \left(\frac{\text{Empty space}}{\text{Total volume}}\right) \times 100 = \left(\frac{0.643 \, \text{ml}}{1 \, \text{ml}}\right) \times 100 \approx 64.3\% \]

### Final Answer
The percentage of solid Argon that is apparently empty space is approximately **64.3%**.

---