The mean free path of molecules of a 5 g oxygen gas in a container is 760 Angstroms. Assume the diameter of oxygen is 3.55 Angstroms.
a/ Determine the volume of the container.
b/ If the total translational kinetic energy of the oxygen gas is 1.95 kJ, calculate the mean time between collisions.
a/ To determine the volume of the container, we need to use the ideal gas law equation, which is:
PV = nRT
Where:
P = pressure of the gas
V = volume of the container
n = number of moles of gas
R = ideal gas constant
T = temperature of the gas in Kelvin
Since we are given the mean free path of the molecules, we can calculate the number of molecules per unit volume using the formula:
N/V = 1/λ
Where:
N = number of molecules
V = volume of the container
λ = mean free path
First, let's convert the mass of oxygen gas to moles using the molar mass of oxygen, which is approximately 32 g/mol:
Number of moles (n) = mass / molar mass
= 5 g / 32 g/mol
= 0.15625 mol
Now, let's rearrange the ideal gas law equation to solve for volume (V):
V = (nRT) / P
Assuming standard temperature and pressure (STP), which is 273 K and 1 atm respectively, and using the ideal gas constant value R = 0.0821 L·atm/(mol·K), we can substitute these values into the equation:
V = (0.15625 mol)(0.0821 L·atm/(mol·K))(273 K) / (1 atm)
= 3.41 L
Therefore, the volume of the container is approximately 3.41 liters.
b/ To calculate the mean time between collisions, we can use the formula:
τ = 1 / (𝑁σ𝑣)
Where:
τ = mean time between collisions
N = Avogadro's number (6.022 x 10^23 mol^-1)
σ = collision cross-section
𝑣 = mean velocity of the gas particles
The collision cross-section can be estimated as the area of a circular cross-section with the same diameter as the oxygen molecule:
σ = πr^2
= π(1.775 Å)^2
= 9.906 Å^2
Now, let's calculate the mean velocity of the gas particles using the formula:
v = √(8kT / πm)
Where:
v = mean velocity
k = Boltzmann constant (1.381 x 10^-23 J/K)
T = temperature of the gas in Kelvin
m = mass of one oxygen molecule
The mass of one oxygen molecule can be calculated by dividing the molar mass by Avogadro's number:
Mass of one oxygen molecule (m) = molar mass / Avogadro's number
= 32 g/mol / (6.022 x 10^23 mol^-1)
= 5.31 x 10^-23 g
Now, let's convert the given total translational kinetic energy to Joules:
Total translational kinetic energy (E) = 1.95 kJ = 1.95 x 10^3 J
Finally, we can substitute all the values into the formula for mean time between collisions:
τ = 1 / (𝑁σ𝑣)
= 1 / ((6.022 x 10^23 mol^-1)(9.906 Å^2)(√(8(1.381 x 10^-23 J/K)(273 K) / π(5.31 x 10^-23 g))))
≈ 2.01 x 10^-12 seconds
Therefore, the mean time between collisions is approximately 2.01 picoseconds.