Average number of air molecules.

[eddzzz_2011]

Homework Statement

Hey guys. Hope you can help me out on this one. so here's the question:

At an altitude of 50km the average atmospheric temperature is 0^oC. What is the average number of air molecules per cubic centimeter at this altitude.

Homework Equations

M = 28.97 g mol^-1
g = 9.81 ms^-2
R = 8.314 J K^-1 mol^-1
Ph = pressure at any given height
P0 = pressure at sea level

Ph = P0e^-Mgh/RT

The Attempt at a Solution

Ph = P0e^-Mgh/RT

p @ 50km = 75.944Pa

75.944 x 1 x 10^-6
Ph = -------------------
8.314 x 273

= 3.346 * 10^-8. And then I need to divide this by 6.022 * 10^23

= 5.556 * 10^-32. Is this correct?

Cheers :)

 

[Borek]

75.944 x 1 x 10^-6
Ph = -------------------
8.314 x 273

What are you calculating here? Ph? That is - pressure?

 

[eddzzz_2011]

Ph = pressure at any given height.
M = 28.97 g mol^-1
g = 9.81 ms^-2
R = 8.314 J K^-1 mol^-1
P0 = pressure at sea level

for

... 75.944 x 1 x 10^-6
Ph = -------------------
...8.314 x 273

 

[Borek]

So what is 75.944?

 

[eddzzz_2011]

I did write down pressure at 50km = 75.944 Pa. But this is incorrect. I may have used the equation n = PV but n = number of moles and would be useless, wouldn't it?
---
RT

Ok. I've just found another equation in my handout I have been given. It states that "if air pressure at sea level is taken to be 1atm then the equation simplifies to:

log(Ph) = -[Mgh*10^5/2.303RT]

so using this equation

log(Ph) = - [(29.97*9.81*50000*10^5)/(2.303*8.314*273)]

log(Ph) = -[1.47*10^12)/(5227.169)]

log(Ph) = -[281222971]

Ph = 10^-281222971

Ph = 0

This is obviously not correct :/ did I miss out a conversion factor?

 

[Borek]

I did write down pressure at 50km = 75.944 Pa.

And then you used this pressure again to calculate pressure Ph? That's what you wrote.

Note - using your equation and numbers I didn't get the same 75.944 for the pressure at 50 km.

n = number of moles and would be useless, wouldn't it?

Would it? Knowing number of moles in cm³ you can easily calculate number of molecules. You almost did it using Avogadro's number, you just got it the wrong way.

This is obviously not correct :/ did I miss out a conversion factor?

As fast as I can tell you missed one and added one that is not necessary. 28.98 is in grams, and I don't understand what is 10⁵. Besides, now your pressure is in atm, previously it was in Pa. It is hardly a simplification, it is just a unit conversion (or a lousy wording in the handout).

 

[eddzzz_2011]

Surely I must have got that the wrong way round of P0 and Ph. Which equation do I use? Do I use "log(Ph) = -[Mgh*10^5/2.303RT]" or "Ph = P0e^-Mgh/RT" I have attached a copy of the handout for you to have a little look at for that last equation. I noticed the teacher took log instead what I think should be ln.

Using n=pV I don't have a value for V. So that wouldn't work.

Thanks for all of your help Borek. Your a real help!

 

[Borek]

Doesn't matter which equation you use - in both cases the final result will be the same. Use whichever equation you want to calculate the pressure at 50km, that's your step number one. This is simple plug and chug, nothing more complicated than keying several numbers into your calculator.

Using n=pV I don't have a value for V. So that wouldn't work.

No such equation as n=PV, more like n=PV/RT. Yes, you know the volume. You are asked about a number of molecules in 1 cm³.

 

[eddzzz_2011]

right.

using n=PV/RT

V=1cm^3 = 1*10^-6m^3
R = 8.314 JK-1mol-1
T = 273K
P @ 50km = ?

Have to find Ph

Plug in values into

Ph = P0e^-Mgh/RT

Ph = 101325e^-(28.97*9.81*50000/8.314*273)

I keep getting a value of 0 for this.

 

[Borek]

I already told you were the problem is - please reread my earlier posts.

 

[eddzzz_2011]

I know you said that 28.97 g mol-1 is not in its SI units? I don't know what to convert it to. Also the equation is Ph = P0e(-Mgh/RT) and not Ph = P0e^-Mgh/RT isn't it?

 

[Borek]

I know you said that 28.97 g mol-1 is not in its SI units?

Everything else is in kg, so molar mass must be in kg/mol as well.

Also the equation is Ph = P0e(-Mgh/RT) and not Ph = P0e^-Mgh/RT isn't it?

No idea what you mean - please elaborate on what e(-x) is.

Looks to me like you should brush up your math and units skills, otherwise you will be completely lost.

 

[eddzzz_2011]

right I understand the kg part. And the equation i mean is it e to the power or e-. ie e^(-x) or e(-x).

Edit: I will be back on later tonight.

 

[Borek]

And the equation i mean is it e to the power or e-. ie e^(-x) or e(-x).

It still doesn't make sense. e to power -x is e^-x. You are suggesting you see two different things, but I see only one.

 

[eddzzz_2011]

What I wrote seems fairly understandable to me. But I emailed the teacher and it is e^(-x) so ignore that one.

Ok so the answer I have is 193.544.

Then

n=PV/RT

n= (193.544*1)/(8.314*273)

n= 0.086 air molecules per cubic centimeter of air @ 50km?

 

[Borek]

No.

Watch your units - you put volume as 1, but 1 of what?

Then, what you are trying to calculate is number of moles, this is not yet number of molecules.

 

[eddzzz_2011]

Ah I want volume in m^3 yet it is in cm^3. so that's a factor of 10^-6.

so 1cm^3 = 1*10^-6 m^3

n= (193.544*1*10^-6)/(8.314*273)

n= 8.53 * 10^-8

So to work out number of molecules I have to times the number of moles by 6.022x10^23

number of molecules = 8.53 * 10^-8 * 6.022x10^23 = 5.135*10^8 number air molecules per cubic centimeter of air @ 50km

 

[Borek]

This is the right approach, but I am not convinced if your units are correct - and if they aren't, your answer is still orders of magnitude off. You ignore units all the time, so they are hard to follow and I am not going to browse the thread back to try to decipher them.

As a general rule: the final step of solving a question should be always check if the units of the answer are correct.

 

[eddzzz_2011]

Ok. Thanks for all of your help