The critical volume of a gas is 0.072 lit. `"mol"^(-1)`.The radius of the molecule will be, in cm

Options

A. `(3/(4pi)xx10^(-23))^(1/3)`
B. `((4pi)/3xx10^(-23))^(1/3)`
C. `((3pi)/4xx10^(-23))^(1/3)`
D. `(3/(4pi)xx10^(-8))^(1/3)`

Explanation

To find the radius of the molecule given the critical volume of a gas, we can use the relationship between critical volume (Vc) and the volume occupied by the molecules. Here's a step-by-step solution:

### Step 1: Understand the relationship between critical volume and molecular volume
The critical volume (Vc) of a gas is related to the volume occupied by the gas molecules. It can be expressed as:
\[ V_c = 3b \]
where \( b \) is the volume occupied by one mole of molecules.

### Step 2: Express \( b \) in terms of the radius of the molecule
The volume \( b \) can be expressed using the radius \( r \) of the molecules and Avogadro's number (\( N_A \)):
\[ b = \frac{16}{3\pi r^3} N_A \]
Substituting this into the equation for critical volume gives:
\[ V_c = 3 \left( \frac{16}{3\pi r^3} N_A \right) = \frac{16 N_A}{\pi r^3} \]

### Step 3: Rearrange the equation to solve for \( r^3 \)
Rearranging the equation for \( r^3 \) gives:
\[ r^3 = \frac{16 N_A}{\pi V_c} \]

### Step 4: Substitute the values
Given:
- \( V_c = 0.072 \, \text{liters/mol} \)
- Convert \( V_c \) to cm³:
\[ 0.072 \, \text{liters} = 0.072 \times 1000 \, \text{cm}^3 = 72 \, \text{cm}^3 \]
- \( N_A = 6.023 \times 10^{23} \, \text{mol}^{-1} \)

Now substitute these values into the equation:
\[ r^3 = \frac{16 \times (6.023 \times 10^{23})}{\pi \times 72} \]

### Step 5: Calculate \( r^3 \)
Calculating the right-hand side:
\[ r^3 = \frac{16 \times 6.023 \times 10^{23}}{\pi \times 72} \]

### Step 6: Calculate \( r \)
Finally, take the cube root to find \( r \):
\[ r = \left( \frac{16 \times 6.023 \times 10^{23}}{\pi \times 72} \right)^{1/3} \]

### Step 7: Convert \( r \) to cm
Since we are already calculating \( r \) in cm, we can directly compute the value.

### Final Calculation
Using a calculator, compute the numerical value of \( r \) from the above equation to find the radius of the molecule in centimeters.