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?

1.17 kg/m^3 is

[1.17 kg/m^3*6.02*10^23 molecules/mole/28*10^-3 kg/mole]*
= 2.5*10^25 molecules/m^3

The mean separation is the -1/3 power or that, or about 3.4*10^-9 m.

The liquid separation distance is less by a factor (8.10/1.17)^1/3

To find the mean volume per nitrogen molecule, we can use the equation:

Mean volume per molecule = Total volume / Number of molecules

For gaseous nitrogen:
To find the total volume, we need to divide the mass of the gas by its density:
Total volume = mass of nitrogen / density of nitrogen (gaseous)

Since we have the density of gaseous nitrogen as 1.17 kg/m^3, we can use the known volume of one mole of gas to find the total volume. The volume of one mole of any gas at standard temperature and pressure (STP) is approximately 22.4 L.

Number of moles of nitrogen = mass of nitrogen / molar mass of nitrogen
Number of moles of nitrogen = 1 kg / 28.0 g/mol = 1000 g / 28.0 g/mol = 35.71 mol

Total volume of gaseous nitrogen = Number of moles of nitrogen x Volume of one mole of gas
Total volume of gaseous nitrogen = 35.71 mol x 22.4 L/mol = 799.5 L

Now we can calculate the mean volume per nitrogen molecule for gaseous nitrogen:
Mean volume per molecule (gaseous) = Total volume / Number of molecules
Number of molecules = Avogadro's number x Number of moles of nitrogen
Number of molecules = 6.02 x 10^23 molecules/mol x 35.71 mol ≈ 2.14 x 10^25 molecules

Mean volume per molecule (gaseous) = 799.5 L / 2.14 x 10^25 molecules

For liquid nitrogen:
The density of liquid nitrogen is given as 810 kg/m^3. To find the total volume, we divide the mass of the liquid by its density:
Total volume = mass of nitrogen / density of nitrogen (liquid)

Number of moles of nitrogen = mass of nitrogen / molar mass of nitrogen
Number of moles of nitrogen = 1 kg / 28.0 g/mol = 1000 g / 28.0 g/mol = 35.71 mol

Total volume of liquid nitrogen = Number of moles of nitrogen x Volume of one mole of gas
Total volume of liquid nitrogen = 35.71 mol x 22.4 L/mol = 799.5 L

Now we can calculate the mean volume per nitrogen molecule for liquid nitrogen:
Mean volume per molecule (liquid) = Total volume / Number of molecules
Number of molecules = Avogadro's number x Number of moles of nitrogen
Number of molecules = 6.02 x 10^23 molecules/mol x 35.71 mol ≈ 2.14 x 10^25 molecules

Mean volume per molecule (liquid) = 800 L / 2.14 x 10^25 molecules

To find the mean separation between nitrogen molecules in each case, we can use the equation:

Mean separation between molecules = (Total volume)1/3 / (Number of molecules)1/3

Mean separation between molecules (gaseous) = (799.5 L)1/3 / (2.14 x 10^25 molecules)1/3
Mean separation between molecules (liquid) = (800 L)1/3 / (2.14 x 10^25 molecules)1/3

Now you can use these equations and values to find the mean volume per nitrogen molecule and the mean separation between nitrogen molecules for both gaseous and liquid nitrogen.