Find Height of Air Sample w/ 6.21 moles of Particles @ 15C

[Jilly]

A. Given that the average diameter of all particles in a sample is 16 nm, te density of all particles is 1.25 x 10^-4 g/cm^3, and the density of the sample of air is 100 microgram/cm^3, how many particles are in 1 L of this air sample?

V(particle) =2.14x10^-18 cm^3
x density = 2.675x10^-22 g

1 L = 1000cm^3
1000cm^3x100microgram/cm^3 = 100000 microgram= 0.1g

0.1g/2.675x10^-22 = 3.74x10^20 particles

B. In what layer of the atmosphere was this sample of air likely taken from (you can assume a sea level temperature of 15C)?

3.74x10^20 particles/ 6.02x10^23 particlees/mol = 6.21 moles

P = [(6.21 mol)(0.08206L•atm/mol•K)(288K)]/1L = 0.015 atm ... Aaaaand that's where I get stuck,

I'm pretty sure I'm supposed to use the equation:
P(h) = P(0) x e^-Mgh/RT

Where:
P(h) = pressure at any given height
P(0) =pressure at sea level
M = 28.97 g/mol
g = 9.81 m/s^2
R = 0.08206 L•atm/K•mol

And solve for h to determine the height my particle is at and then reference atmospheric layers but... I have too many extra variables...

 

[SteamKing]

A. Given that the average diameter of all particles in a sample is 16 nm, te density of all particles is 1.25 x 10^-4 g/cm^3, and the density of the sample of air is 100 microgram/cm^3, how many particles are in 1 L of this air sample?

V(particle) =2.14x10^-18 cm^3
x density = 2.675x10^-22 g

1 L = 1000cm^3
1000cm^3x100microgram/cm^3 = 100000 microgram= 0.1g

0.1g/2.675x10^-22 = 3.74x10^20 particles

B. In what layer of the atmosphere was this sample of air likely taken from (you can assume a sea level temperature of 15C)?

3.74x10^20 particles/ 6.02x10^23 particlees/mol = 6.21 moles

How do your get 6.21 moles from 3.74*10²⁰ particles / 6.02*10²³ particles/mol?

An order of magnitude calculation suggests n = 10²⁰ / 10²³, which is nowhere close to 6.

 

[Jilly]

Apologies, misread on the calc.

The Calc for pressure is still 0.015.
Number of moles should equal 0.000621

 

[SteamKing]

P = [(6.21 mol)(0.08206L•atm/mol•K)(288K)]/1L = 0.015 atm ... Aaaaand that's where I get stuck,

I'm pretty sure I'm supposed to use the equation:
P(h) = P(0) x e^-Mgh/RT

Where:
P(h) = pressure at any given height
P(0) =pressure at sea level
M = 28.97 g/mol
g = 9.81 m/s^2
R = 0.08206 L•atm/K•mol

And solve for h to determine the height my particle is at and then reference atmospheric layers but... I have too many extra variables...

Which extra variables do you have which are preventing you from finding the altitude?

 

[Jilly]

Well, in order to calculate for 'h' I'd need P(h), and Temperature at P(h) ...

 

[Jilly]

I just don't understand what I'm not seeing to solve this... What am I missing? =\

 

[SteamKing]

What does the 0.015 atm from your calculations represent?

As far as figuring out from which layer of the atmosphere the sample came, you could always use the Barometric Formula:

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

 

[Jilly]

What does the 0.015 atm from your calculations represent?

As far as figuring out from which layer of the atmosphere the sample came, you could always use the Barometric Formula:

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

The equation I posted is equation 2 on that page.

 

[SteamKing]

The equation I posted is equation 2 on that page.

Then, if you read that page carefully and check the definition of the variables for equation 2, you'll find that you know all that you need in order to find the altitude where the sample could have been taken.

Remember, the pressure of 0.015 atm. is the pressure at altitude. If you carefully list all of the variables and their values, you'll find that altitude is the only one which is unknown.

 

[Jilly]

Hmmmm
0.015 is pressure at sea level...

so 0.015 should be P(0)

Is it that I'm simply misunderstanding the question and I should assume that temperature of 15C is constant? But how would I find pressure at the height the particle is at? And if I'm not assuming the temperature is constant (since in real life it obviously wouldn't be) how do I calculate that? I'm pretty sure I'm just not understanding the information given...

 

[SteamKing]

Hmmmm
0.015 is pressure at sea level...

so 0.015 should be P(0)

Really? I thought the pressure at sea level on Earth was, you know, 1 atmosphere. It's kinda like a definition, or something.

Read Post #9 more carefully, especially the last two sentences.

Is it that I'm simply misunderstanding the question and I should assume that temperature of 15C is constant? But how would I find pressure at the height the particle is at? And if I'm not assuming the temperature is constant (since in real life it obviously wouldn't be) how do I calculate that? I'm pretty sure I'm just not understanding the information given...

From the Wiki article on the Barometric Formula:

There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). The first equation is used when the value of standard temperature lapse rate is not equal to zero; the second equation is used when standard temperature lapse rate equals zero.

The lapse rate is defined as the rate at which atmospheric temperature decreases with increase in altitude.

Equation 1:

Equation 2:

Read the quoted section above carefully. You should have enough information now to solve your problem.