Answer:
[tex]3.726\times 10^{-17}\ Pa[/tex]
[tex]6.02464\times 10^{17}\ m^3[/tex]
844.58565402 km
Explanation:
[tex]\frac{N}{V}[/tex] = Density of atoms = [tex]10^6\ atoms/m^3[/tex]
n = Amount of substance = 1 mol
V = Volume
R = Gas constant = 8.314 J/mol K
[tex]k_b[/tex] = Boltzmann constant = [tex]1.38\times 10^{-23}\ J/K[/tex]
T = Temprature = 2.7 K
L = Side of cube
From ideal gas law we have the relation
[tex]PV=Nk_bT\\\Rightarrow P=\frac{Nk_bT}{V}\\\Rightarrow P=\frac{N}{V}k_bT\\\Rightarrow P=10^6\times 1.38\times 10^{-23}\times 2.7\\\Rightarrow P=3.726\times 10^{-17}\ Pa[/tex]
The pressure is [tex]3.726\times 10^{-17}\ Pa[/tex]
From ideal gas law
[tex]PV=nRT\\\Rightarrow V=\frac{nRT}{P}\\\Rightarrow V=\frac{1\times 8.314\times 2.7}{3.726\times 10^{-17}}\\\Rightarrow V=6.02464\times 10^{17}\ m^3[/tex]
The volume is [tex]6.02464\times 10^{17}\ m^3[/tex]
Volume is given by
[tex]V=L^3\\\Rightarrow L=V^{\frac{1}{3}}\\\Rightarrow L=\left(6.02464\times 10^{17}\right)^{\frac{1}{3}}\\\Rightarrow L=844585.65402\ m=844.58565402\ km[/tex]
The length of the side of the cube is 844.58565402 km
