The Maxwell-Boltzmann Distribution shows electrons having a continuum of Electron Levels; but don’t electrons have very specific energy levels?
I'm looking at the Wikipedia for the Maxwell-Boltzmann distribution, specifically this part:
What I've circled in red is the Maxwell-Boltzmann distribution, as I understand it. It is a probability density function of the speeds in a gas. But according to Wikipedia, what I've circled in blue is also the probablity density function of the speeds in a gas.
So clearly I'm misunderstanding something, what gives?
I am taking a kinetics class and one of the very first assignments tasks us with deriving various quantities from the Maxwell-Boltzmann distribution, like the average velocity. I am not really sure to start with these types of problems because they don’t involve using figures as such, they are manipulations of formulas and I’m not really sure how to “think through” these types of questions. We are given a table of useful integrals to use, and pretty much nothing else. Professor also said we should be able to do this on an exam, so it’s not a one-off thing. Any help in getting started would be helpful.
heya, i’m confused about a question in my tutorial.
for part (c), why does it seem like the area under the graph has increased? does an increase in pressure lead to an increased number of molecules? or is it just an issue with my perception?
Hello Grant, love your work!
Here's an idea for a video (or even an app/demo with a user-operable interface) to demonstrate the relationship between the Bernoulli effect, the venturi effect, and the time dilation effect in special relativity - something like an enhanced version of Mark Mitchell's java demo of the Bernoulli effect (which was here: http://home.earthlink.net/%7Emmc1919/venturi.html , but I can no longer find).
On an unbounded, diagonally striped/patterned background, there is a square, rigid, impermeable enclosure (a kind of "corral"). Inside this corral, a number of small identical "balls" move around randomly (at around 2cm/sec, say), colliding with each other and bouncing off the walls of the corral. The momentum/kinetic energy of each ball is initially the same and all collisions are perfectly elastic, so after a time, the balls settle into a Maxwell-Boltzmann speed distribution. (A graph could display the time-evolution of the speed distribution of the balls). (An additional enhancement may include the ability of balls to spin in response to glancing collisions.)
Attached to each of the four walls of the corral there is a pressure gauge, showing the pressure exerted by the balls bouncing off them (vector collisions per second per unit length). Another graph could display the time-evolution of the pressure on each wall. Of course, at equilibrium, the pressure on the walls will be the same.
Now for the interesting part: the corral can move - horizontally or vertically - against the patterned background (hence the pattern, which reveals the motion of the corral). When it does so (and now moves with a constant horizontal velocity less than the RMS velocity of the balls, at 1 cm/sec, say), the balls initially bunch up at the back, but with elastic collisions off the walls, they soon re-establish a uniform Maxwell-Boltzmann speed distribution relative to the corral. At this new equilibrium, the balls appear to move uniformly slower, and the pressure on the four walls is now lower. This demonstrates the Bernoulli effect. (An enhancement might explore what happens when the corral rotates.)
Now for an even more interesting bit: You, the observer, are attached to the corral - "sitting on the fence", so to speak - and you're holding a "clock" - a tiny box containing just one ball (in effect a small corral) - which "ticks" whenever the ball bounces off the walls of the box. Of course, th
... keep reading on reddit ➡I'm trying to generate a graph for the maxwell boltzmann (energy) distribution that ACTUALLY has numbers on the x-axis for once - no internet resources show this.
I initially went to the wiki page to find a formula for the distribution and found this - labelled as equation (9), which gives a familiar bell-ish curve to the distribution. It looks kinda like a surge function or a chi squared distribution with this formula. This is also the shape you see in most chemistry text books when they use it to discuss reaction rates and such.
HOWEVER when I try and couple this with my other knowledge of how this distribution works - ie. that the proportion of particles (area under the curve) with energy greater than E should be exp(-E/kT) it simply doesn't check out. A distribution with that property has to cross y = 1 at the x axis, not begin at the origin like all the texts show. Looking further I found this hyperphysics page, and again we see a distribution that's clearly just an exponential decay - no bump in the middle, No beginning at the origin.
This is maddening, because the exp(-E/kT) relation crops up in reaction rates (solving the rate constant using the Arrhenius equation), yet does not check out with any of the graphics of the distribution itself at all.
WHAT is going on here?!? and how could I ever plot a graph of this distribution with numbers on the x-axis like I want? Any clarification would be GREATLY appreciated here.
note: looking to plot (kinetic) energy, not velocity. The reason for this is because in chemistry (where I want to use this), texts use this distribution to contrast with the idea of activation energy. It's easier to compare energy to energy than to switch between energy and velocity in a student's head.
My book gave me the hint of doing <F(E)>=1/n *Integral from 0 to inf of [ F(E)*g(E)*f(E)]. In which, F(E) is the Maxwell-Boltzmann distribution (If I'm not mistaken, F(E) could be any function I want to calculate the average value of), g(E) the function of density of states for a gas, but what is f(E)? Is it just 1/(1+e^[(E-Ef)/kt]) ? (Ef=Fermi energy).
If that's it, is there any tips for solving this integral? Because I have an example but I'm not a huge fan of how they solved it. And also, could I make the assumption of f(E)=1/e^[(E-Ef)/kt] ?
Thanks in advance!
P.S: Let me know if I wrote something wrong, english is not my first language nor the language I'm studying this subject on, so translations might be off.
As the distribution becomes asymptotic to x-axis at higher velocities, is there a possibility(however small) that certain air molecule will be travelling at tremendous speed inside that box?
If yes, what is the momentum imparted by that molecule to the walls of the box as it collides with them?
Since high velocity means high temperature, how much heat will that molecule transfer to the wall?
Basically if we could somehow look at each individual molecule we will surely find some of the molecules(like 10 out of 1 mole) having temperature as high as surface of sun?
Or am I understanding it all wrong?
I am studying the kinetic theory of gasses at university at the moment and we have to learn the derivation of the maxwell probability distribution function. However, the information that they have given me at the moment isn't very helpful. Does any one know of a book or some other material which has this derivation laid out in a easier to follow way. Any suggestions will be helpful :)
I have been reading Max Tegmark's book "Our Mathematical Universe" and in later chapters he briefly discusses that the Maxwell-Boltzmann Distribution shows that the probability of a particular arrangement of molecules, such as the exact molecular structure of your brain, complete with all your thoughts and memories is more likely to occur in other regions of space and/or other universes (Many-Worlds interpretation of Quantum Mechanics) than through evolution.
This is known as the Boltzmann Brain and is a hypothesized self aware entity which arises due to random fluctuations out of a state of chaos.
What I mean is that when the light hit the surface, it reflects a certain colour. I'd like to know first if every light (I say light because I don't wanna make a mistake and maybe say that photons reflect back to the eyes when it comes to seeing)that hit a surface has the same wavelength/energy or an average which can be shown by some kind of distribution. If the latter, are there any article I can read or information I can have on this distribution? I know this is a weird and badly explained question, so i am not surprised if any readers can't understand what I ask. For the others, you can try your best to answer this, thanks in advance.
This one is quite good in that it [mostly] doesn't have any 'hand wavy' leaps of logic in it. That said, the reasoning presented with equation 11 has stumped me, the equation makes sense, but it also appears as if it constrains each velocity vector component (the x,y and z parts) to have the same value, hence it seems like we are now restricting analysis to a subset of velocities where v*x* == v*y* == v*z*
Please be kind, I'm not a mathematician or physicist by trade, I'm an enthusiastic amateur!
Thanks.
Say there were a large number of flies, or similarly sized insects, in a container. If it is a closed system would the flies obey the Maxwell-Boltzmann distribution or would they move around in another way? Also, if they did obey the distribution would they also obey things like mean free path or diffuse like gas molecules would if there was a hole?
Hello,
I am trying to understand the Maxwell Boltzmann distribution for ideal diatomic gases. I do not have a physics background and have been studying this on my own through a textbook. I've found the probability density function for the speed of a monatomic gas from the Boltzmann distribution by assuming the total energy of each atom is from the translational kinetic energy of m/2*(vx^2 + vy^2 + vz^2 ).
However, to my understanding diatomic gases have additional energy in the form of rotational and vibrational modes. If the temperature of the diatomic gas is low enough such that rotational modes are not activated, does this distribution for monatomic gases also apply for the diatomic gas? How will it differ when rotational modes become activated?
I am thinking about this question which seems quite stupid today, in Boltzmann distribution the ground state is always the most occupied state, while in Maxwell-Boltzmann distribution there is a non-zero most probable speed which most particles have. Why is this?
My book gave me the hint of doing <F(E)>=1/n *Integral from 0 to inf of [ F(E)*g(E)*f(E)]. In which, F(E) is the Maxwell-Boltzmann distribution (If I'm not mistaken, F(E) could be any function I want to calculate the average value of), g(E) the function of density of states for a gas, but what is f(E)? Is it just 1/(1+e^[(E-Ef)/kt]) ? (Ef=Fermi energy).
If that's it, is there any tips for solving this integral? Because I have an example but I'm not a huge fan of how they solved it. And also, could I make the assumption of f(E)=1/e^[(E-Ef)/kt] ?
Thanks in advance!
P.S: Let me know if I wrote something wrong, english is not my first language nor the language I'm studying this subject on, so translations might be off.
