Calculate the pressures, P, in atmospheres at which the mean free path, l, of a hydrogen molecule will be 1.50 μm, 1.50 mm, and 1.50 m at 20.0 °C. The diameter of a H2 molecule is 270 pm. Which formula would I use?
To calculate the pressures (P) at which the mean free path (l) of a hydrogen molecule (H2) will be a certain length, we can use the mean free path formula given by:
l = k * T / (sqrt(2) * d^2 * P)
where:
- l is the mean free path
- k is the Boltzmann constant (1.38 x 10^-23 J/K)
- T is the temperature in Kelvin (°C + 273.15)
- d is the diameter of the hydrogen molecule
- P is the pressure in atmospheres
The formula can be rearranged to solve for P:
P = (k * T) / (sqrt(2) * d^2 * l)
Therefore, to calculate the pressures at which the mean free path of a hydrogen molecule will be a certain length, we need to use the formula:
P = (k * T) / (sqrt(2) * d^2 * l)
To calculate the pressure at which the mean free path of a hydrogen molecule is a certain length, you can use the Kinetic Theory of Gases and the ideal gas law.
The formula you need to use is:
P = (d^2 * N) / (4 * V * l)
Where:
P is the pressure in atmospheres,
d is the diameter of the hydrogen molecule in meters,
N is Avogadro's number (6.0221 x 10^23 molecules per mole),
V is the molar volume of the gas at the given temperature in m^3/mol, and
l is the mean free path length in meters.
To find the molar volume, you can use the ideal gas law equation:
V = (R * T) / P
Where:
R is the ideal gas constant (0.0821 L * atm / mol * K),
T is the temperature in Kelvin, and
P is the pressure in atmospheres.
Now, let's substitute the values and calculate the pressures at different mean free path lengths:
Given:
l = 1.50 μm = 1.50 × 10^-6 m (mean free path)
d = 270 pm = 270 × 10^-12 m (diameter)
T = 20.0 °C = 20.0 + 273.15 K (temperature)
First, calculate the molar volume (V) using the ideal gas law equation:
V = (R * T) / P
Substituting the values:
V = (0.0821 * (20.0 + 273.15)) / P
V ≈ 22.414 L / mol / P
Next, substitute the values into the formula for pressure:
P = (d^2 * N) / (4 * V * l)
P = ((270 × 10^-12)^2 * 6.0221 x 10^23) / (4 * 22.414 * 10^-3 * 1.50 × 10^-6)
P = (2.43 × 10^-20 * 6.0221 x 10^23) / (1.694 x 10^-8)
P ≈ 8.67 × 10^14 atm
So, the pressure required for a mean free path of 1.50 μm is approximately 8.67 × 10^14 atmospheres.
Similarly, you can use this formula to calculate the pressure at which the mean free path is 1.50 mm or 1.50 m. Just substitute the respective values of l into the formula and calculate the pressure.