A challenging statistical thermodynamics problem.

[Remy34]

Homework Statement

Consider the case of a gas in the atmosphere. Assume that the temperature is a constant. Based
on the Maxwell Boltzmann distribution, at sea level the atmosphere contains 78.1% N2, 21%
O2, 0.9% Argon, and 0.036 CO2. What are the ratios at the the top of Everest? (Molecular mass
of N2 = 28, Molecular mass of O2 = 32, Atomic mass of Argon = 40, Atomic mass of CO2 = 44.
Height of Everest = 8,848 m)Please walk me through this

Homework Equations

Maxwell Boltzmann statistics.

The Attempt at a Solution

To tell you the truth, I don't even know where to begin. I posted this on a different thread, but it wasn't properly addressed (it was ignored for the most part). I was just wondering if you guys can guide me through this problem. It would be greatly appreciated.

 

[fzero]

Homework Equations

Maxwell Boltzmann statistics.

The Maxwell-Boltzmann distribution gives an expression for the ratio of the number of particles in a given microstate, $N_i$, in terms of the total number of particles $N$, the energy of the microstate $E_i$, and the temperature. The sea-level values that you're given are equivalent to the values of $N_i/N$ for each component gas. Can you try to relate these to the Maxwell-Boltzmann formula? What is the appropriate expression for the energy? What changes when we consider a different altitude?

 

[Remy34]

The Maxwell-Boltzmann distribution gives an expression for the ratio of the number of particles in a given microstate, $N_i$, in terms of the total number of particles $N$, the energy of the microstate $E_i$, and the temperature. The sea-level values that you're given are equivalent to the values of $N_i/N$ for each component gas. Can you try to relate these to the Maxwell-Boltzmann formula? What is the appropriate expression for the energy? What changes when we consider a different altitude?

The trimester is over, and we didn't cover the statistical part of thermo in great detail (as a mater of fact we only began a week or so ago). This is supposed to be a challenge question. Any I thought that maybe as the pressure decreases so would the amount found at that height. But how can you relate it? Do you add up all the masses N and use it to divide Ni (individual masses)? As for relating the energy I am clueless.

 

[fzero]

The trimester is over, and we didn't cover the statistical part of thermo in great detail (as a mater of fact we only began a week or so ago). This is supposed to be a challenge question. Any I thought that maybe as the pressure decreases so would the amount found at that height. But how can you relate it? Do you add up all the masses N and use it to divide Ni (individual masses)? As for relating the energy I am clueless.

You really need to write down a Maxwell-Boltzmann formula at some point to solve this. Then ask what each quantity in the formula means and how you can relate it to the information in this problem.

$N$ and $N_i$ are numbers of particles (in a unit volume), not masses.

If I have an object with mass $m$ at an altitude $y$, what is its energy?

 

[Remy34]

You really need to write down a Maxwell-Boltzmann formula at some point to solve this. Then ask what each quantity in the formula means and how you can relate it to the information in this problem.

$N$ and $N_i$ are numbers of particles (in a unit volume), not masses.

If I have an object with mass $m$ at an altitude $y$, what is its energy?

I think I see what you mean. You find the number of atoms (for example .036/44 times avagadros number), and then add them all up. That would be the N in the Boltzmann-Maxwell equation. And the equation is as follows;

N/Z(e^-εj/kt)

I have to do that for each one, am I correct? First calculate N with the appropriate ε (lets say for CO2), and then the Nj (for CO2).

The energy would be m*y*g.

Am I on the right track?

 

[Remy34]

And another question. Why do the percentages add up to more than 100%?

 

[fzero]

I think I see what you mean. You find the number of atoms (for example .036/44 times avagadros number), and then add them all up. That would be the N in the Boltzmann-Maxwell equation. And the equation is as follows;

N/Z(e^-εj/kt)

I have to do that for each one, am I correct? First calculate N with the appropriate ε (lets say for CO2), and then the Nj (for CO2).

The energy would be m*y*g.

Am I on the right track?

Yes, you are on the right track, but you don't need Avagadro's number. The percentages are $N_i/N$. You want to fix the coefficient of the exponential by using the values at sea level, then the percentages at altitude can be computed by using the energy difference in the exponential.

And another question. Why do the percentages add up to more than 100%?

It's a round-off error. Not all of the quoted values are given to the same decimal place.