Answer
(a) $2.69\times 10^{25}~molecules/m^3$
(b) The average distance between molecules is $3.34\times 10^{-9}~m$
(c) The total mass is $0.028~kg$
The mass density is $1.25~kg/m^3$
Work Step by Step
(a) We can find the number density:
$\frac{6.022\times 10^{23}}{0.0224~m^3} = 2.69\times 10^{25}~molecules/m^3$
(b) To find the average distance between molecules, we can assume the molecules are spread evenly in a cube with a volume of $0.0224~m^3$
We can find the length of each side of the cube:
$L = (0.0224~m^3)^{1/3} = 0.282~m$
We can find the number of molecules along one side of the cube:
$(6.022\times 10^{23})^{1/3} = 8.445\times 10^7~molecules$
We can find the average distance between molecules:
$\frac{0.282~m}{8.445\times 10^7} = 3.34\times 10^{-9}~m$
The average distance between molecules is $3.34\times 10^{-9}~m$
(c) A nitrogen atom consists of 7 protons and 7 neutrons.
The mass of a nitrogen molecule $N_2$ is $28~u$
Since the mass of one nitrogen molecule is $28~u$, we know the mass of one mole of nitrogen molecules is $28~grams$ which is $0.028~kg$
We can find the mass density:
$\frac{0.028~kg}{0.0224~m^3} = 1.25~kg/m^3$
