Gaseous nitrogen has a density of 1.17 kg/m3 and liquid nitrogen has a density of 810 kg/m3.
[The relative molecular mass of nitrogen is 28.0]
What is the mean volume per nitrogen molecule in each case?
What is the mean separation between nitrogen molecules in each case?
To find the mean volume per nitrogen molecule, we can use the formula:
Mean volume per molecule = 1 / (density * Avogadro's number)
For gaseous nitrogen:
Density = 1.17 kg/m^3
Avogadro's number = 6.022 x 10^23 particles/mol
Mean volume per molecule = 1 / (1.17 kg/m^3 * (6.022 x 10^23 particles/mol))
Calculating this, we find:
Mean volume per molecule of gaseous nitrogen = 1.392 x 10^-24 m^3/particle
Similarly, for liquid nitrogen:
Density = 810 kg/m^3
Avogadro's number = 6.022 x 10^23 particles/mol
Mean volume per molecule = 1 / (810 kg/m^3 * (6.022 x 10^23 particles/mol))
Calculating this, we find:
Mean volume per molecule of liquid nitrogen = 1.849 x 10^-27 m^3/particle
To find the mean separation between nitrogen molecules, we can use the formula:
Mean separation = (mean volume per molecule)^(1/3)
For gaseous nitrogen:
Mean volume per molecule = 1.392 x 10^-24 m^3/particle
Calculating the mean separation, we get:
Mean separation of gaseous nitrogen = (1.392 x 10^-24 m^3/particle)^(1/3) = 3.469 x 10^-8 m/particle
Similarly, for liquid nitrogen:
Mean volume per molecule = 1.849 x 10^-27 m^3/particle
Calculating the mean separation, we get:
Mean separation of liquid nitrogen = (1.849 x 10^-27 m^3/particle)^(1/3) = 7.348 x 10^-9 m/particle
To calculate the mean volume per nitrogen molecule, we need to use the formula:
Mean Volume = Total Volume / Number of Molecules
For gaseous nitrogen:
Given density = 1.17 kg/m^3
Relative molecular mass of nitrogen = 28.0 g/mol
First, convert the relative molecular mass to kg/mol:
Relative molecular mass = 28.0 g/mol * (1 kg / 1000 g) = 0.028 kg/mol
To find the total volume, we can use the relationship between mass, volume, and density:
Volume = Mass / Density
Using the density of gaseous nitrogen, which is 1.17 kg/m^3, we can find the total volume by assuming a mass of 1 kg:
Total Volume = 1 kg / 1.17 kg/m^3 = 0.8547 m^3
However, we need to convert the mass from kg to moles using the molar mass of nitrogen:
Number of Moles = Mass / Molar Mass = 1 kg / 0.028 kg/mol = 35.71 mol
Now we can calculate the mean volume per nitrogen molecule:
Mean Volume = Total Volume / Number of Molecules = 0.8547 m^3 / 35.71 mol ≈ 0.024 m^3/mol
For liquid nitrogen:
Given density = 810 kg/m^3
Using the same approach, assuming a mass of 1 kg, we can calculate the total volume:
Total Volume = 1 kg / 810 kg/m^3 = 0.00123 m^3
Converting the mass from kg to moles:
Number of Moles = Mass / Molar Mass = 1 kg / 0.028 kg/mol = 35.71 mol
Calculating the mean volume per nitrogen molecule:
Mean Volume = Total Volume / Number of Molecules = 0.00123 m^3 / 35.71 mol ≈ 0.0000345 m^3/mol
So the mean volume per nitrogen molecule in gaseous nitrogen is approximately 0.024 m^3/mol, and in liquid nitrogen, it is approximately 0.0000345 m^3/mol.
To calculate the mean separation between nitrogen molecules, we can use the formula:
Mean Separation = Volume of Sample / Number of Molecules^1/3
Using the same values as before:
For gaseous nitrogen:
Mean Separation = (0.8547 m^3 / 35.71 mol)^(1/3) ≈ 0.051 m
For liquid nitrogen:
Mean Separation = (0.00123 m^3 / 35.71 mol)^(1/3) ≈ 0.005 m
So the mean separation between nitrogen molecules in gaseous nitrogen is approximately 0.051 m, and in liquid nitrogen, it is approximately 0.005 m.