finding the concentration of a gas (thermodynamics)

**raul_l** · Jan16-07, 10:29 AM

Hi.

1. The problem statement, all variables and given/known data

The root mean square velocity of oxygen molecules is 480m/s while the pressure is 20kPa.

What is the concentration (particles/volume) of oxygen?

2. Relevant equations

PV=nRT
$LaTeX Code: E_{kin}=\\frac{m \\overline{v}^2}{2}=\\frac{3}{2}kT$

3. The attempt at a solution

$LaTeX Code: m \\overline{v}^2=3kT \\Rightarrow T= \\frac{m \\overline{v}^2}{3k}$

$LaTeX Code: PV=nR \\frac{m \\overline{v}^2}{3k} \\Rightarrow n=\\frac{3kPV}{Rm \\overline{v}^2}$ where $LaTeX Code: m=2 \\times 16 \\times 1.66 \\times 10^{-27} kg$

I set V=1m^3 and get n=8.137mol and therefore $LaTeX Code: \\frac{n \\times n_{a}}{V}=4.899 m^{-3}$ where $LaTeX Code: n_{a}=6.02 \\times 10^{23}$

Is this correct? I'm sure there's a simpler way to do this.

**GCT** · Jan16-07, 08:54 PM

Note that the mass refers to an individual oxygen molecule (from what I recall at the moment), (16 grams of Oxygen/mole of Oxygen)(1 mole/6.022 x 10^23 atoms)(1 kilogram/1000 grams)(2 atoms of Oxygen/1 diatomic molecule)=____.......the setup should be solved for n/V, this means that you need to incorporate a particular value of R, choose from the list on the page that can be linked to through the below text so that the final units for n/V is respect to moles/liter.

http://en.wikipedia.org/wiki/Gas_constant

**raul_l** · Jan17-07, 05:01 AM

Actually, since this is one of my physics class problems (not chemistry) concentration really does mean particles/volume in this case. Which means that I have chosen the right value for R, I think.

**Gokul43201** · Jan17-07, 05:12 AM

Originally Posted by raul_l

3. The attempt at a solution

$LaTeX Code: m \\overline{v}^2=3kT \\Rightarrow T= \\frac{m \\overline{v}^2}{3k}$

$LaTeX Code: PV=nR \\frac{m \\overline{v}^2}{3k} \\Rightarrow n=\\frac{3kPV}{Rm \\overline{v}^2}$ where $LaTeX Code: m=2 \\times 16 \\times 1.66 \\times 10^{-27} kg$

I set V=1m^3 and get n=8.137mol and therefore $LaTeX Code: \\frac{n \\times n_{a}}{V}=4.899 m^{-3}$ where $LaTeX Code: n_{a}=6.02 \\times 10^{23}$

You're missing a factor of 10^{24} in that final bit.

Is this correct? I'm sure there's a simpler way to do this.

The method is perfectly correct - I haven't checked the numbers, but I believe the final number looks close enough (I happen to know that the RMS speed of oxygen molecules at room temperature is about 500m/s, and at NTP, a mole of atoms occupies about 22.4 liters, so at a fifth of an atmosphere, the concentration would be roughly 6/(5*0.0224)*10^{23} per cubic meter, which is about 10% higher than your number, but this is very rough estimate.)

As for a simpler way, I think this is as simple as it gets. Only, notice that since R=k*Na, and M=m*Na, your final expression simplifies to n=3PV/Mv^2 (in moles).

**GCT** · Jan17-07, 08:01 PM

(n/v)=20 kPa(1000 Pa/1 kPa)(1.3806503 × 10-23 m2 kg s-2 K-1)3/[8.314472 m^3 · Pa · K-1 · mol-1(16 grams of Oxygen/mole of Oxygen)(1 mole/6.022 x 10^23 atoms)(1 kilogram/1000 grams)(2 atoms of Oxygen/1 diatomic molecule)(480 m/s)^2]= 0.00813782909722 moles/m^3

(0.00813782909722 moles Oxygen/L)(6.022x10^23 molecules Oxygen/mole)= 4.900600682 x 10^24 molecules/m^3

So there's everything done in a perfunctory fashion for ya, I wanted to see what answer would result with the "chemist" method.

I'm going to need to see if the units cancel out exactly.......