Question

The density of nitrogen gas (N2) at standard temperature and pressure (STP, 0°C and about 1atm) is 1.25 kg/m3, while the density of liquid nitrogen at atmospheric pressure is 0.81 x 103kg/m3. Let D denote a typical distance between nitrogen molecules in its gas (i) Assuming that the nitrogen atoms in the liquid are basically touching each other, esti- ii) Using the ideal gas equation of state determine the volume of 1 mole of gas at STP. phase and d denote the approximate average size of the nitrogen atom mate the ratio D/d, the size of the atom, d, and the distance, D (iii) Air is about 80% nitrogen. Explain what your answer to part (i) tells you about treating air as an ideal gas at STP

Answer 1

i)For the liquid nitrogen, Density= Mass/Volume=2*Atomic weight of nitrogen/Volume occupied by 2 nitrogen atoms just touching each other=$At. Wt of nitrogen/(4/3\pi d^{3}) kg/m^{3}.$ where d is the size of a nitrogen atom.

For the nitrogen gas,Density=Mass/Volume=2*At.wt of nitrogen/volume occupied by 2 atoms at a distance D from each other=$At. Wt of nitrogen/(4/3\pi (D+d)^{3}) kg/m^{3}.$

Therefore Density of liquid nitrogen/Density of nitrogen gas=$4/3\pi (D+d)^{3}/(4/3\pi d^{3})=((D+d)/d)^{3}=0.81*10^{3}/1.25=698$. This reduces to

. i.e$(D/d+1)=698^{1/3}=8.9$.

Thus $D/d=7.9$.

The value of d can be calculated from the first equation i.e for liquid nitrogen,

Density=

$2*At.wt.of nitrogen/(4/3\pi d^{3})=2*14.007*1.67*10^{-27}kg/(4/3\pi d^{3})=0.81*10^{3}kg/m^{3}$.

on substituting the values of atomic weight of nitrogen and density of liquid nitrogen.

On simplification we obtain d=2.3*10^-10 m.

Now D/d=9.9. Therefore D=7.9*2.3*10^-10 m=1.8*10^-9 m.

ii)From the equation of state for an ideal gas:

$PV=nRT$ where n=1 mole we obtain

V=RT/P=8.31 J/molK*273K/10⁵ N/m²=2.27*10^-2 m³=0.023m³=23 litres.

This is the volume of 1 mole of gas at STP.

iii)It can be concluded that the density of air is approximately the same as that of nitrogen gas and that the typical distance between air molecules is about the same as D calculated for N₂ in part i).