A `25xx10^(-3)m^(3)` volume cylinder is filled with 1 mol of `O_2` gas at room temperature (300 K).The molecular diameter of `O_2`, and its root mean square speed, are found to be 0.3 nm and `200 m//s`, respectively.What is the average collision rate (per second) for an `O_2` molecule ?

Options

A. `~10^12`
B. `~10^11`
C. `~10^10`
D. `~10^13`

Answer

To find the average collision rate (per second) for an O₂ molecule in a cylinder, we can use the following steps:

### Step 1: Identify Given Values
- Volume of the cylinder, \( V = 25 \times 10^{-3} \, m^3 \)
- Number of moles of \( O_2 = 1 \, mol \)
- Temperature, \( T = 300 \, K \)
- Molecular diameter of \( O_2, \sigma = 0.3 \, nm = 0.3 \times 10^{-9} \, m \)
- Root mean square speed, \( V_{rms} = 200 \, m/s \)

### Step 2: Calculate the Number of Molecules
Using Avogadro's number, \( N_A = 6.022 \times 10^{23} \, molecules/mol \):
\[ N = n \times N_A = 1 \times 6.022 \times 10^{23} \approx 6.022 \times 10^{23} \, molecules \]

### Step 3: Calculate the Average Collision Rate
The average collision rate \( Z \) can be calculated using the formula:
\[ Z = \frac{1}{4} n \sigma V_{rms} \]
Where:
- \( n \) is the number density of molecules, given by \( n = \frac{N}{V} \)

Calculating \( n \):
\[ n = \frac{N}{V} = \frac{6.022 \times 10^{23}}{25 \times 10^{-3}} = \frac{6.022 \times 10^{23}}{0.025} \approx 2.4088 \times 10^{25} \, molecules/m^3 \]

Now substituting \( n \), \( \sigma \), and \( V_{rms} \) into the collision rate formula:
\[ Z = \frac{1}{4} \times (2.4088 \times 10^{25}) \times (0.3 \times 10^{-9}) \times (200) \]

Calculating \( Z \):
\[ Z = \frac{1}{4} \times (2.4088 \times 10^{25}) \times (0.3 \times 10^{-9}) \times (200) \]
\[ Z = \frac{1}{4} \times (2.4088 \times 10^{25}) \times (6 \times 10^{-8}) \]
\[ Z = \frac{1}{4} \times (1.44528 \times 10^{18}) \]
\[ Z \approx 3.6132 \times 10^{17} \, collisions/s \]

### Step 4: Final Result
Thus, the average collision rate for an \( O_2 \) molecule is approximately:
\[ Z \approx 3.61 \times 10^{17} \, collisions/s \]