Calculate the number of collisions per second of one hydrogen molecule at 24 °C and 1.00 bar. The diameter of a hydrogen molecule is 270 pm.
To calculate the number of collisions per second of one hydrogen molecule, we can use the kinetic theory of gases. The formula for calculating the number of collisions per second is given by:
Z = (N/V) * √(8 * k * T / (π * m))
Where:
Z is the number of collisions per second,
N is the Avogadro constant (6.022 × 10^23),
V is the volume occupied by the gas (in m^3),
k is the Boltzmann constant (1.38 × 10^-23 J/K),
T is the temperature in Kelvin,
m is the mass of one hydrogen molecule (in kg).
First, we need to convert the given temperature from Celsius to Kelvin:
T(K) = T(°C) + 273.15
Given:
Temperature T = 24 °C = 24 + 273.15 = 297.15 K
Pressure P = 1.00 bar
Diameter d = 270 pm = 270 × 10^-12 m
Now, we need to calculate the volume occupied by the hydrogen molecule.
The radius of the hydrogen molecule, r = d/2 = (270 × 10^-12 m)/2 = 135 × 10^-12 m.
The volume of the hydrogen molecule, V = (4/3) * π * r^3
= (4/3) * π * (135 × 10^-12)^3
Next, we need to calculate the mass of one hydrogen molecule using the molecular mass of hydrogen (from the periodic table) and the Avogadro constant.
Mass of one hydrogen molecule, m = (Mw/Mn) * (1/N)
= (2.016 g/mol) * (1 kg / 1000 g) * (1 mol / 6.022 × 10^23)
= 2.016 * 1 / (1000 * 6.022 × 10^23) kg
Now, we can calculate the number of collisions per second:
Z = (N/V) * √(8 * k * T / (π * m))
Substitute the known values into the equation to get the answer.