Energy of particle in electric field

I'm taking a physics class and the professor teaches us really basic things in lecture and then gives homework way beyond what he taught in lecture. Obviously I need to find some resource other than Stack Exchange, but in the mean time, perhaps you could explain some of the terms in this problem and give me a starting point for how to solve it.

Regarding the problem, given below, I'm not sure what the significance of "exactly one mean free path through the air before hitting a molecule" is. I know eV is an electron volt, which is a unit of energy (from high school) but I'm not sure how this is related to the strength of a field. I would assume this energy is related to the force on the particle, which I can relate to the strength of the field and the force on the particle, as well as the distance the particle travels, but that doesn't seem right because no figures are given for distance or charge. I haven't got a clue what cross-sectional area might have to do with this. I do, however, know what a Maxwell distribution is from chemistry class.

(a) Assume that an ion moves, on average, exactly one mean free path through the air before hitting a molecule. If the ion needs to acquire approximately $1.11\ \mathrm{eV}$ of kinetic energy in order to ionize a molecule, estimate the minimum strength of the electric field required at standard room pressure and temperature ($300\ \mathrm{K}$). Assume that the cross-sectional area of an air molecule is about $0.095\ \mathrm{nm^2}$. (Assume the velocities of the particles have a Maxwell distribution of velocities.)

(b) How does the strength of the electric field in Part (a) depend on temperature? (Use the following variable as necessary: $T$.)

(c) How does the strength of the electric field in Part (a) depend on pressure? (Use the following variable as necessary: $P$.)

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Wow, that is quite a question for a first course in electro-magnetism. Especially because it draws on ideas from statistical physics. –  Kevin Driscoll Apr 10 '13 at 1:14
Tell me about it. I appreciate the complexity of the problem, and realize it likely has to do with gas laws or something. If I only had to deal with one or two pieces in the puzzle, I could probably pull a solution together from reading on Wikipedia, but it looks like there's lots of pieces here. –  Big Endian Apr 10 '13 at 1:16
The problem is that I haven't got a clue even how field strength is related to the energy of a particle. I can't sort out the useful from the useless because I really don't have the first clue where to start. An answer to my question could easily come in the form of a link to an article on some basic physics related to particles in fields. I'm usually quite good at solving convoluted problems like these but I am embarrassingly ignorant on the subject at hand. Our lectures don't really cover this stuff. Like I said, I need to find a resource. –  Big Endian Apr 10 '13 at 1:37
I'm not sitting here just waiting for an answer. At the moment, I'm going through some materials provided with the course, looking for more information. –  Big Endian Apr 10 '13 at 1:39
Aha, I just looked up mean free path. So that is the distance a particle will travel before it makes contact with another particle, and it looks to be related to temperature and pressure, as one would expect. So is this the distance I use to relate field strength to energy of a particle? Let me do some more reading... –  Big Endian Apr 10 '13 at 1:49
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Since you are actively working on the problem, here are some references that can get you started. Maybe you can put all of the pieces together.

If you haven't taken statistical mechanics/thermodynamics, there is no way you are going to know how to calculate the mean free path of an ideal gas. Check out this hyperphysics article that has a full derivation.

As for the energy, we can use the work-energy theorem and some basic electrostatics to find the change in energy. $$\text{Work} = \int_{r_0} ^{r_f} \vec{F} \cdot d\vec{r}$$ We know the force on a charged particle due to a constant electric field $$\vec{F} = q \vec{E}$$ If we start with a charge at rest, then it will accelerate in a straight line in the direction of the electric field $$\int \vec{F} \cdot d\vec{r} = \int q |\vec{E}| \ dx = q |\vec{E}| \Delta x$$ This is the energy added to a positive charge, after it has moved a distance $\Delta x$ in a constant electric field. Hopefully, you can put the pieces together and come up with an answer.

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Can you explain why your formula relating charge, field strength, and distance delta x (which makes sense), is different from the formula I find on this page (which confuses me somewhat)? hyperphysics.phy-astr.gsu.edu/hbase/electric/ev.html#c2 –  Big Endian Apr 10 '13 at 2:01
The formulas are exactly the same, but using different quantities. Instead of the electric field, $\vec{E}$, hyperphysics is writing the energy in terms of the potential difference, $\Delta V$. Fundamentally, they are no different. That is, the potential difference is DEFINED as $-\int \vec{E} \cdot d\vec{r}$, which in this case simplifies to my statement because the field is constant. –  Kevin Driscoll Apr 10 '13 at 2:02
There's one last thing I don't understand though... And that is what charge I use for q. Would I use just a single unit charge, the charge of a single electron or proton? –  Big Endian Apr 10 '13 at 2:06
That's a good question. It isn't completely clear from the problem statement. I suppose that in a gas, a singly ionized atom/molecule would be the most reasonable thing. However, I'm not really sure. –  Kevin Driscoll Apr 10 '13 at 2:08
That makes sense, especially given that a weaker field could accelerate particles with more charge to the correct energy, but a stronger field will be needed to accelerate all charged particles (even those with a single charge) to the correct energy. Your answer and comments have been super helpful. Thanks! –  Big Endian Apr 10 '13 at 2:12

Perhaps this will help: the "mean free path" is defined as the average distance a particle (here the ion) moves before hitting an atom or molecule of the stuff it's traveling through. Hopefully you would have found that definition in your research, but the first sentence of the question is phrased oddly (in a way that makes you think it might be saying something nontrivial), so I can see how that would be a little confusing.

It should make sense that the mean free path depends on various characteristics of the ion - how fast it's going, how big it is, etc. - and also on the properties of the air, in particular density and temperature. There is a formula which you can look up (it appears in a number of questions on this site, on Wikipedia, and in any decent reference on statistical mechanics) which will let you calculate the mean free path of a particle from the properties I've listed. You'll need that to solve the problem.

The other thing you should know is that an electronvolt is defined as the amount of energy a particle of one elementary charge (an electron or proton) acquires when passing through a potential difference of one volt. There is a formula, which I expect has been covered in your class (or another class you've taken), which lets you calculate energy from charge and electrical potential. Using electronvolts as the unit of energy there just makes the calculation numerically easy. I think you can take it from there.

P.S. I strongly suggest doing the problem symbolically at first, i.e. don't plug in any numbers and just get a formula for the answer. That keeps you from worrying about variables you may not have values for. And as a bonus, it makes parts (b) and (c) trivial.

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Thanks for your answer. I gave Kevin Driscoll the accept because it gives some much needed clarification on the change in energy of a particle in a field, but I also found your answer very useful. I'm going to solve the problem now, like you say, symbolically. If I could have marked two answers I would have, but I can't. Thanks for your effort :) –  Big Endian Apr 10 '13 at 2:14