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. 1. `2xx10^(-4)`
B. 2. `1xx10^(-4)`
C. 3. `1.5xx10^(-4)`
D. 4. `4xx10^(-4)`

Explanation

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

### Step 1: Convert Diameter to Radius
Given the diameter of an oxygen molecule is 3 Å (angstroms), we first need to find the radius (R).

\[ R = \frac{D}{2} = \frac{3 \, \text{Å}}{2} = 1.5 \, \text{Å} \]

To convert this into meters:
\[ 1 \, \text{Å} = 10^{-10} \, \text{m} \implies R = 1.5 \times 10^{-10} \, \text{m} \]

To convert this into centimeters:
\[ R = 1.5 \times 10^{-10} \, \text{m} = 1.5 \times 10^{-8} \, \text{cm} \]

### Step 2: Calculate the Molecular Volume
The molecular volume (Vm) of one oxygen molecule can be calculated using the formula for the volume of a sphere:

\[ V = \frac{4}{3} \pi R^3 \]

Substituting the value of R:
\[ V = \frac{4}{3} \pi (1.5 \times 10^{-8} \, \text{cm})^3 \]

Calculating \(R^3\):
\[ (1.5 \times 10^{-8})^3 = 3.375 \times 10^{-24} \, \text{cm}^3 \]

Now substituting this back into the volume formula:
\[ V = \frac{4}{3} \pi (3.375 \times 10^{-24}) \approx 4.52 \times 10^{-23} \, \text{cm}^3 \]

### Step 3: Calculate the Total Molecular Volume for One Mole
Using Avogadro's number (\(N_A = 6.023 \times 10^{23}\)):
\[ V_m = V \times N_A = (4.52 \times 10^{-23} \, \text{cm}^3) \times (6.023 \times 10^{23}) \approx 8.51 \, \text{cm}^3 \]

### Step 4: Determine the Actual Volume at STP
At STP (Standard Temperature and Pressure), the volume occupied by one mole of an ideal gas is approximately:
\[ V_{STP} = 22400 \, \text{cm}^3 \]

### Step 5: Calculate the Ratio of Molecular Volume to Actual Volume
Now we can find the ratio of the molecular volume to the actual volume at STP:
\[ \text{Ratio} = \frac{V_m}{V_{STP}} = \frac{8.51 \, \text{cm}^3}{22400 \, \text{cm}^3} \]

Calculating this gives:
\[ \text{Ratio} \approx 3.8 \times 10^{-4} \]

### Final Answer
The ratio of molecular volume to the actual volume occupied by the oxygen gas at STP is approximately \(4 \times 10^{-4}\).

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