calculate no. Of binary collisions per cc of N2 gas at 1atm pressure at 25 C diameter of N2 molecule =3.74A (i no thers a dir formla but it is not easy 2 do in exams)

To calculate the number of binary collisions per cubic centimeter (cc) of nitrogen gas (N2) at 1 atm pressure and 25°C, we can use a simplified approach based on the kinetic theory of gases. This will involve a few steps, including determining the number density of the gas and applying the appropriate formula for collision frequency.

Understanding the Variables

Before diving into the calculations, let's clarify some key variables:

• Pressure (P): 1 atm, which is equivalent to 101325 Pa.
• Temperature (T): 25°C, which converts to 298 K (by adding 273).
• Diameter of N2 molecule (d): 3.74 Å, or 3.74 x 10^-10 m.

Step 1: Calculate the Number Density (n)

The number density of a gas can be calculated using the ideal gas law, which is given by:

PV = nRT

Where:

• P = pressure in pascals (Pa)
• V = volume in cubic meters (m³)
• n = number of moles
• R = ideal gas constant (8.314 J/(mol·K))
• T = temperature in kelvins (K)

Rearranging this gives us:

n = PV / RT

To find the number density (N), we need to convert moles to molecules using Avogadro's number (approximately 6.022 x 10²³ molecules/mol).

Calculating Number Density

First, we convert the pressure to pascals:

P = 1 atm = 101325 Pa

Now, substituting the values into the equation:

n = (101325 Pa) * (1 m³) / (8.314 J/(mol·K) * 298 K)

Calculating this gives:

n ≈ 40.79 moles/m³

Now, converting moles to molecules:

N = n * Avogadro's number = 40.79 moles/m³ * 6.022 x 10²³ molecules/mol

This results in:

N ≈ 2.45 x 10²⁵ molecules/m³

To find the number density in cc, we convert cubic meters to cubic centimeters (1 m³ = 10⁶ cc):

N ≈ 2.45 x 10²⁵ / 10⁶ ≈ 2.45 x 10¹⁹ molecules/cc

Step 2: Calculate the Collision Frequency (Z)

The collision frequency can be estimated using the formula:

Z = (π * d² * N * v)

Where:

• π: approximately 3.14
• d: diameter of the molecule in meters (3.74 Å = 3.74 x 10^-10 m)
• N: number density in molecules/cc
• v: average speed of the gas molecules, which can be calculated using the formula:

v = sqrt((3RT)/M)

Where M is the molar mass of nitrogen (approximately 28 g/mol or 0.028 kg/mol).

Calculating Average Speed

Substituting the values into the equation for v:

v = sqrt((3 * 8.314 J/(mol·K) * 298 K) / 0.028 kg/mol)

This results in:

v ≈ 517 m/s

Step 3: Putting It All Together

Now we can calculate the collision frequency:

Z = π * (3.74 x 10^-10)² * (2.45 x 10¹⁹) * (517)

Calculating this gives:

Z ≈ 1.48 x 10¹⁰ collisions/cc/s

Final Thoughts

In summary, the number of binary collisions per cubic centimeter of nitrogen gas at 1 atm and 25°C is approximately 1.48 x 10¹⁰ collisions per second. This calculation illustrates the dynamic nature of gas molecules and how frequently they interact with one another, which is fundamental to understanding gas behavior in various applications.