The average distance between molecules of an ideal gas at STP is approximately of the order of

Options

A. 1nm
B. 100 nm
C. 100 cm
D. `1 mu m`

Explanation

To find the average distance between molecules of an ideal gas at standard temperature and pressure (STP), we can follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = Pressure (in Pascals)
- \( V \) = Volume (in cubic meters)
- \( n \) = Number of moles
- \( R \) = Universal gas constant (\( 8.314 \, \text{J/(mol K)} \))
- \( T \) = Temperature (in Kelvin)

### Step 2: Set the Conditions for STP
At standard temperature and pressure (STP):
- Pressure, \( P = 1 \, \text{atm} = 10^5 \, \text{Pa} \)
- Temperature, \( T = 273 \, \text{K} \)
- Volume, \( V = 1 \, \text{m}^3 \)

### Step 3: Calculate the Number of Moles
Using the ideal gas law, we can rearrange the equation to find the number of moles \( n \):
\[ n = \frac{PV}{RT} \]

Substituting the values:
\[ n = \frac{(10^5 \, \text{Pa})(1 \, \text{m}^3)}{(8.314 \, \text{J/(mol K)})(273 \, \text{K})} \]

Calculating the denominator:
\[ 8.314 \times 273 \approx 2270.62 \]

Now substituting back:
\[ n \approx \frac{10^5}{2270.62} \approx 44 \, \text{moles} \]

### Step 4: Calculate the Number of Molecules
To find the total number of molecules, we multiply the number of moles by Avogadro's number (\( N_A \)):
\[ N = n \times N_A \]
Where \( N_A \approx 6.02 \times 10^{23} \, \text{molecules/mol} \).

Substituting the values:
\[ N \approx 44 \times 6.02 \times 10^{23} \approx 2.65 \times 10^{25} \, \text{molecules} \]

### Step 5: Calculate the Average Distance Between Molecules
The average distance \( d \) between molecules can be approximated using the formula:
\[ d \approx \left( \frac{1}{N} \right)^{1/3} \]

Substituting the value of \( N \):
\[ d \approx \left( \frac{1}{2.65 \times 10^{25}} \right)^{1/3} \]

Calculating:
\[ d \approx (3.77 \times 10^{-26})^{1/3} \approx 1.56 \times 10^{-9} \, \text{m} \]

### Step 6: Convert to Nanometers
Converting meters to nanometers:
\[ d \approx 1.56 \, \text{nm} \]

### Final Answer
The average distance between molecules of an ideal gas at STP is approximately of the order of \( 1 \, \text{nm} \).

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