Calculate the number of collisions per second of one hydrogen molecule at 24°C and 2.00 bar. The diameter of a hydrogen molecule is 270 pm?

To calculate the number of collisions per second of a hydrogen molecule at a given temperature and pressure, we can use the kinetic theory of gases. This theory provides a framework for understanding the behavior of gas molecules in terms of their motion and interactions. Let's break down the calculation step by step.

Key Variables and Constants

First, we need to identify the key variables and constants involved in our calculation:

• Temperature (T): 24°C, which we convert to Kelvin by adding 273.15, giving us 297.15 K.
• Pressure (P): 2.00 bar, which we convert to pascals (Pa) since 1 bar = 100,000 Pa. Thus, 2.00 bar = 200,000 Pa.
• Diameter of a hydrogen molecule (d): 270 pm (picometers), which is 270 x 10^-12 meters.
• Boltzmann's constant (k): 1.38 x 10^-23 J/K.

Calculating the Mean Speed of Hydrogen Molecules

Using the ideal gas law, we can find the mean speed (v) of the hydrogen molecules. The formula for the mean speed of gas molecules is:

v = sqrt((3kT)/m)

Here, m is the mass of a hydrogen molecule. The mass of a hydrogen molecule (H₂) is approximately 2.016 g/mol. To convert this to kilograms, we use:

m = (2.016 g/mol) / (1000 g/kg) * (1 mol / 6.022 x 10²³ molecules) ≈ 3.34 x 10^-27 kg

Now, substituting the values into the mean speed formula:

v = sqrt((3 * (1.38 x 10^-23 J/K) * 297.15 K) / (3.34 x 10^-27 kg))

Calculating this gives us:

v ≈ 1.84 x 10³ m/s

Calculating the Collision Rate

The collision rate (Z) can be calculated using the following formula:

Z = (π * d² * v * n)

Where n is the number density of the gas, which can be found using the ideal gas law:

n = P / (kT)

Substituting the values:

n = (200,000 Pa) / ((1.38 x 10^-23 J/K) * (297.15 K)) ≈ 4.83 x 10²⁵ molecules/m³

Now we can calculate the collision rate:

Z = π * (270 x 10^-12 m)² * (1.84 x 10³ m/s) * (4.83 x 10²⁵ molecules/m³)

Calculating this gives:

Z ≈ 2.73 x 10¹⁰ collisions/second

Final Result

Therefore, the number of collisions per second of one hydrogen molecule at 24°C and 2.00 bar is approximately 2.73 x 10¹⁰ collisions/second. This high number reflects the rapid motion and frequent interactions of gas molecules under these conditions.