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

`3.726* 10^(-17)\ Pa`

`6.02464* 10^(17)\ m^3`

844.58565402 km

Step-by-step explanation:

`(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* 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=(Nk_bT)/(V)\\\Rightarrow P=(N)/(V)k_bT\\\Rightarrow P=10^6* 1.38* 10^(-23)* 2.7\\\Rightarrow P=3.726* 10^(-17)\ Pa`

The pressure is
`3.726* 10^(-17)\ Pa`

From ideal gas law

`PV=nRT\\\Rightarrow V=(nRT)/(P)\\\Rightarrow V=(1* 8.314* 2.7)/(3.726* 10^(-17))\\\Rightarrow V=6.02464* 10^(17)\ m^3`

The volume is
`6.02464* 10^(17)\ m^3`

Volume is given by

`V=L^3\\\Rightarrow L=V^{(1)/(3)}\\\Rightarrow L=\left(6.02464* 10^(17)\right)^{(1)/(3)}\\\Rightarrow L=844585.65402\ m=844.58565402\ km`

The length of the side of the cube is 844.58565402 km