Question

(a) In the deep space between galaxies, the density of atoms is as low as 106 atoms/m^3 , and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m^3) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

Answer 1

Answer:

$3.726\times 10^{-17}\ Pa$

$6.02464\times 10^{17}\ m^3$

844.58565402 km

Explanation:

$\frac{N}{V}$ = Density of atoms = $10^6\ atoms/m^3$

n = Amount of substance = 1 mol

V = Volume

R = Gas constant = 8.314 J/mol K

$k_b$ = Boltzmann constant = $1.38\times 10^{-23}\ J/K$

T = Temprature = 2.7 K

L = Side of cube

From ideal gas law we have the relation

$PV=Nk_bT\\\Rightarrow P=\frac{Nk_bT}{V}\\\Rightarrow P=\frac{N}{V}k_bT\\\Rightarrow P=10^6\times 1.38\times 10^{-23}\times 2.7\\\Rightarrow P=3.726\times 10^{-17}\ Pa$

The pressure is $3.726\times 10^{-17}\ Pa$

From ideal gas law

$PV=nRT\\\Rightarrow V=\frac{nRT}{P}\\\Rightarrow V=\frac{1\times 8.314\times 2.7}{3.726\times 10^{-17}}\\\Rightarrow V=6.02464\times 10^{17}\ m^3$

The volume is $6.02464\times 10^{17}\ m^3$

Volume is given by

$V=L^3\\\Rightarrow L=V^{\frac{1}{3}}\\\Rightarrow L=\left(6.02464\times 10^{17}\right)^{\frac{1}{3}}\\\Rightarrow L=844585.65402\ m=844.58565402\ km$

The length of the side  of the cube is 844.58565402 km