The diameter of an oxygen molecule is 3 `Ã…` The ratio of molecular volume to the actual volume occupied by the oxygen gas at STP is

Options

A. `2xx10^(-4)`
B. `1xx10^(-4)`
C. `1.5xx10^(-4)`
D. `4xx10^(-4)`

Explanation

To find the ratio of the molecular volume to the actual volume occupied by oxygen gas at STP, we can follow these steps:

### Step 1: Calculate the volume of a single oxygen molecule.
The volume \( V \) of a sphere (which we can approximate the oxygen molecule as) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Given the diameter of an oxygen molecule is \( 3 \, \text{Å} \), we can find the radius:
\[ r = \frac{\text{diameter}}{2} = \frac{3 \, \text{Å}}{2} = 1.5 \, \text{Å} \]

### Step 2: Convert the radius to centimeters.
Since \( 1 \, \text{Å} = 10^{-8} \, \text{cm} \), we convert the radius:
\[ r = 1.5 \, \text{Å} = 1.5 \times 10^{-8} \, \text{cm} \]

### Step 3: Calculate the volume of one oxygen molecule in cubic centimeters.
Substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (1.5 \times 10^{-8})^3 \]
Calculating \( (1.5 \times 10^{-8})^3 \):
\[ (1.5)^3 = 3.375 \quad \text{and} \quad (10^{-8})^3 = 10^{-24} \]
Thus,
\[ (1.5 \times 10^{-8})^3 = 3.375 \times 10^{-24} \, \text{cm}^3 \]
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \pi (3.375 \times 10^{-24}) \approx 4.52 \times 10^{-24} \, \text{cm}^3 \]

### Step 4: Calculate the volume occupied by 1 mole of oxygen gas at STP.
At STP (Standard Temperature and Pressure), 1 mole of any ideal gas occupies approximately \( 22.4 \, \text{L} \) or \( 22400 \, \text{cm}^3 \).

### Step 5: Calculate the number of molecules in 1 mole of oxygen.
Using Avogadro's number, \( N_A \approx 6.022 \times 10^{23} \) molecules/mole.

### Step 6: Calculate the total molecular volume for 1 mole of oxygen.
Total molecular volume for 1 mole of oxygen:
\[ \text{Total molecular volume} = V \times N_A \approx (4.52 \times 10^{-24} \, \text{cm}^3) \times (6.022 \times 10^{23}) \approx 2.72 \times 10^{-1} \, \text{cm}^3 \]

### Step 7: Calculate the ratio of molecular volume to actual volume.
Now, we find the ratio of the molecular volume to the actual volume occupied by oxygen gas at STP:
\[ \text{Ratio} = \frac{\text{Total molecular volume}}{\text{Actual volume at STP}} = \frac{2.72 \times 10^{-1} \, \text{cm}^3}{22400 \, \text{cm}^3} \approx 1.21 \times 10^{-5} \]

### Final Answer
The ratio of the molecular volume to the actual volume occupied by oxygen gas at STP is approximately \( 1.21 \times 10^{-5} \).

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