What is the Ideal Gas Law in Space?

[kikko]

Homework Statement

A nebula-a region of the galaxy where new stars are forming-contains a very tenuous gas with 100 atoms/cm^3. This gas is heated to 7500 K by ultraviolet radiation from nearby stars. Put the answer in atmospheres.

Homework Equations

pV=nRT
R=8.31
n=N/N_A
N_A = 6.02*10^23
N/V = number density

The Attempt at a Solution

100 = N/V
100V=N
100V=nN_A
100V/N_A = n
p=nRT/V
P= 100V/N_ART
P=(((100(1*10^-6))/(6.02*10^23))(8.31)(7500))/(1*10^-6)

P=1*10^-17 Pascals = 1*10^-22 atm


Not where where I went wrong.

 

[cdotter]

Why are you dividing by 1*10^-6?

N/v = 100 atoms/cm^3 = 0.0001 atoms/m^3
n/v = (.0001 atoms/m^3)*(1/(6.022*10^23 atoms/mol))=1.66*10^-28 mol/m^3
p=(n/v)rt=(1.66*10^-28 mol/m^3)(8.314 J/mol*K)(7500 K)=1.04*10^-23 Pa = 1.03*10^-28 atm.

 

[kikko]

Actually 1.03*10^-28 atm was the first answer I used in mastering physics and got it wrong.

I was going along the lines pV=nRT so p=(nRT)/V = ((100V/N_A)RT)/V, and V=1*10^-6 m^3

 

[cdotter]

Oops. We're both using the wrong gas constant. The units don't work out.

 

[kikko]

Isn't there just Boltzmann's Constant which equals the gas constant R divided by Avogrado's number, which we divided the gas constant by avogrado's number in the problem when multiply n and R.

 

[cdotter]

I'm losing it.

The answer wants atmospheres so just use a gas constant with atmospheres. R=0.0821 L*atm*K^-1*mol^-1.

n=1.66*10^-22 mol/cm^3 = 1.66*10^-19 mol/L.
p=(1.66*10^-19 mol/L)(0.0821 L*atm*K^-1*mol^-1)(7500 K)

Moles cancel, liters cancel, Kelvin cancel, and you get atmospheres.

p = 1.02*10^-16 atm.

I hate these types of problems. More on getting caught up in unit analysis than actually solving a problem.

 

[kikko]

That works out pretty straightforward with the R=0.0821 L*atm*K^-1*mol^-1. Thanks a ton. I'm wondering how we get that number for R. Is it just a given, like 8.31 we're supposed to just know when we do problems? My physics text doesn't mention any other values for R, and I can't remember the professor mentioning it either.