the molecule diameter of nitrogen is 3.5*10^-8 cm. calculate the mean free path at temprature 27 degree centigrate and p=1 atm

To calculate the mean free path of nitrogen molecules at a temperature of 27 degrees Celsius and a pressure of 1 atm, we can use the formula derived from kinetic theory. The mean free path (\( \lambda \)) is given by the equation:

Mean Free Path Formula

The formula for mean free path is:

λ = (k * T) / (√2 * π * d² * P)

Where:

• λ = mean free path
• k = Boltzmann's constant (1.38 × 10^-23 J/K)
• T = absolute temperature in Kelvin
• d = diameter of the molecule in meters
• P = pressure in Pascals

Step-by-Step Calculation

First, we need to convert the given values into the appropriate units:

1. Convert Temperature

The temperature in Celsius needs to be converted to Kelvin:

T = 27 + 273.15 = 300.15 K

2. Convert Diameter

The diameter of nitrogen is given as 3.5 × 10^-8 cm. We need to convert this to meters:

d = 3.5 × 10^-8 cm × (1 m / 100 cm) = 3.5 × 10^-10 m

3. Convert Pressure

The pressure is given as 1 atm. We convert this to Pascals (1 atm = 101325 Pa):

P = 1 atm = 101325 Pa

Plugging in the Values

Now that we have all the values in the correct units, we can substitute them into the mean free path formula:

λ = (1.38 × 10^-23 J/K * 300.15 K) / (√2 * π * (3.5 × 10^-10 m)² * 101325 Pa)

Calculating Each Component

Let's break this down:

• Numerator: 1.38 × 10^-23 J/K * 300.15 K ≈ 4.14 × 10^-21 J
• Diameter squared: (3.5 × 10^-10 m)² = 1.225 × 10^-19 m²
• Denominator: √2 * π * (1.225 × 10^-19 m²) * 101325 Pa

Calculating the denominator:

Denominator ≈ 1.414 * 3.14159 * 1.225 × 10^-19 m² * 101325 Pa ≈ 5.03 × 10^-14

Final Calculation

Now we can calculate the mean free path:

λ ≈ 4.14 × 10^-21 J / 5.03 × 10^-14 ≈ 8.23 × 10^-8 m

Convert to Centimeters

To express the mean free path in centimeters:

λ ≈ 8.23 × 10^-8 m × (100 cm / 1 m) ≈ 8.23 × 10^-6 cm

Conclusion

The mean free path of nitrogen molecules at 27 degrees Celsius and 1 atm pressure is approximately 8.23 × 10^-6 cm. This value indicates how far a nitrogen molecule travels on average before colliding with another molecule, which is an important concept in understanding gas behavior and properties.