Determining pressure in Interstellar space

[pdonovan]

Homework Statement

Interstellar space, far from any stars, is filled with a very low density of hydrogen atoms (H, not H2). The number density is about 1atom / cm³ and the temperature is about 3 K. Estimate the pressure in interstellar space. Give your answer in Pa and in atm.

Homework Equations

Average translational kinetic energy per molecule: Eavg = 1.5KbT = .5mv²
p = F/A = (1/3)(N/V)mVrms²
Eth = 1.5nRT
pV = nRT

The Attempt at a Solution

V = 10^-6 m³
T = 3K
m = 1u = 1.66x10^-27
n = 1.66x10^-24 (not sure if this is right?)

pV = nRT --> p = 4.14x10^-17
Where am I going wrong, or is this correct?

Thank you for any help/advice!

 

[D H]

Different approach, I get the same number:

$$\frac{10^6 \, \text{atom}} {\text{m}^3} \cdot \frac{1 \, \text{mole}} {6.0221415 \times 10^{23} \, \text{atom}} \cdot 8.3144621 \, \text{J}/\text{K}/\text{mole} \cdot 3 \, \mbox{K} \approx 4.142 \times 10^{-17} \, \text{pascal}$$

 

[Spinnor]

The only formula you need is PV = nRT

 

[D H]

That is exactly one of the formulae that pdonovan, Spinnor.

 

[pdonovan]

Thank you very much, that was the correct answer.

Now, how would I got about finding Vrms?

I know p = (1/3)(N/V)mVrms²
So, p = 4.14x10^-17
N = 1
V = 10^-6
m = 1u = 1.66x10^-27
And found Vrms = .86m/s, but this is incorrect. I think my m or v might be incorrect.

 

[D H]

Your value for m is correct (assuming units of kilograms; always carry units around). Your math is wrong somewhere. Show your work.

 

[pdonovan]

Now I have...

4.14x10^-17pa = (1/3)(1/10^-6)mVrms²

1.242x10^-22 = mVrms²

74819.28 = Vrms²
Vrms = 275.64m/s

So something is definitely wrong, because the atom should be moving very slowly.

 

[D H]

Yeah, your answer is too large. About 2 m/s too large. sqrt(74819.28) is about 273.53.

As a sanity check, you can always compute sqrt((3 * boltzmann's constant * 3 kelvin) / (1 amu)). You will get the same answer.

 

[pdonovan]

Then which values are wrong in the Vrms = sqrt(3KbT / m) formula? I have T = 3 and if m = 1, then Vrms = 1.11 x 10 ^ -11 which is wrong. And if m =1660x10^-27g it is too big (around 8.6).