lecturette 149: the maxwellian speed distribution

Physics with Calculusby

David V. Anderson & Peter Tarsi

assisted by students

Carolyn McCrosson & Andrew Trott

Lecturette 149: The Maxwellian Speed DistributionIn the previous lesson we discussed the concept of a particle distribution function . If tells you, according to the speed or velocity of particles how many particles have any particular speed. Next we explore a special distribution function, which in many cases is both a good physical

approximation and is easily manipulated algebraically.

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Model of James Clerk Maxwell

Lecturette 149 presents information about the Maxwellian Speed Distribution.

Well known 19th century physicist James Clerk

Maxwell, using concepts from the field of

statistical mechanics, derived a formula for the

speed distribution, , of particles (molecules or atoms) likely to be encountered when the particles randomly collide with

one another. is the number of particles per unit interval

of . While the derivation of this distribution is beyond this course’s scope, the resulting Maxwellian Distribution is given in

LCN 584 as shown below:


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f (v)

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f (v)

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What is Statistical Mechanics?

Ann asks, “What on earth is statistical mechanics- I’ve never heard of it?”

Surely you remember our lessons on the kinematics and dynamics of one and two particle systems? The basic model for statistical mechanics is simply the combination, or overlay, or superposition of a vast number of such single particle systems all made to occupy the same volume. It also assumes two-body collisions or interactions.

Now statistical mechanics, as a formal physics course, in generally taught in graduate school or in upper-level undergraduate courses. I might add that the ideal gas law and the zeroth law of thermodynamics, which we recently covered, can also be derived from statistical mechanics. Does this answer your question?

“ So I infer from what you are saying that I must take this Maxwellian formula on faith and perhaps wait years until I take a statistical mechanics course where I’ll finally understand where this distribution originated?”

I’m afraid that’s right. Nevertheless, the Maxwellian distribution is also seen experimentally- giving us further assurance that it is correct and giving us more reason to study it now.




Variables and Parameters of Maxwellian


Returning now to the Maxwellian formula in LCN 584, we remind ourselves that

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f(v) is the numb e r of pa rticles pe r unit inte rval of v . Since w e a nticipat e usin g

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f(v) a s an integr a nd , we the n interpr e t

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f(v)dv a s the nu mber of part icles with s peed s betw e en v an d v + dv. v can als o be interpr e ted a s the veloci ty vect or magnitud e , which , by defin ition, is the speed . The Maxwellian formula’s thre e pa ramet e rs , m, k, an d T, are , re spe ctively, the part icle or mole cular mas s , Boltzmann’ s co nsta nt, an d the absol ute temp e rature measur e d in degr e es Kelvin . Finally N is eithe r the numbe r of pa rticles in the volum e unde r s tudy or it could be regarde d a s the numbe r densit y, in which cas e the volum e under s tudy is the unit volume. Now we turn to conside r ma them a tical metho ds fo r finding the zer oth and fir s t mome nts of this distributio n.



Partial Integration Methods for the Maxwellian


As we learned earlier in LCN 577 and LCN 581, the zeroth and first moments of a particle speed distribution function equal the particle number,

N , and the average speed, , respectively.

Calculating the moments of LCN 584 requires a high level of competence in integral calculus. Our interest here is the zeroth moment and the first moment of the Maxwellian speed distribution.

Given the foregoing definition of as a particle number distribution function, it should be evident that the zeroth moment of this distribution will generate the number of particles within the volume of study, which in this case is the norm (which by definition is always the zeroth moment). Its first moment produces the average speed of those particles.

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Bob Asks How Zeroth Moment Produces N


Bob asks, “I don’t get it. How is this zeroth moment- this integral over the Maxwellian going to produce the number of particles?”

I can answer it two ways. First, the particle number N is part of the coefficient in the Maxwellian formula and thus once the integration is carried out the result will be proportional to N- this is something that is clear from the mathematics of it.

The other way to understand it is by remembering what the integral means. It is telling you to add up all the different particles according to their speeds. Graphically, remember that

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f(v)dv is a histogr a m of width dv and of he ight

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f(v). Sin ce the la tte r is the nu mb er of pa rticles per unit v, it follows tha t

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f(v)dv is th e numbe r of pa rticles in tha t his togra m. The integra l simpl y s a ys tha t you add all thes e histo grams togeth e r. Since the integra l range s ove r all possible pa rticle speed s , it captur e s e very pa rticle in the volume of intere s t. Thus adding the hi s togra ms produc e s the tota l numbe r of pa rticles in the volume of intere s t.



Partial Integration Result for Zeroth Moment of Maxwellian

Getting back to the zeroth moment, it is clear from the form of the Maxwellian formula in LCN 584 that evaluating the zeroth moment will require evaluating an integral of the form shown below in LCN 585.


Students are not expected to demonstrate their ability to derive the result shown in LCN 585 using integration by parts and the “squares” reduction. For those who are interested in its derivation we’ll discuss the methods employed in a moment. Even for students who know how to derive LCN 585, you are nevertheless advised to memorize it given the significant time required to go through the derivation each time an evaluation is needed.



Integration by “Tricks”


We say “integration by tricks” not to imply the use of an improper technique, but we say it to suggest the method is quite unusual and not at all obvious.

Through partial integration one can show

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x2e−px2dx= e−px2

2pο∞∫ο

∞∫ dx which is the “eas y” s tep. The

latte r integ ral cann ot be e valuate d in close d form exce pt a s a de finite integra l, which is accomplis hed , by evalu a ting the related integral tha t runs over

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−∞,∞[ ]. The “rela ted” integra l is

then multiplied by itsel f re s ulting in an integral ove r the entir e (x, y) plane. A fter conv e rsion to polar

coordinat e s , the resu lting integra nd is trivially integr a ted t o produc e

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x2e−px2dx=14ο

∞∫ πp3 as

desired.



Partial Integration for First Moment

The other integral, required to evaluate the average speed of the Maxwellian distribution, is based on the partial integration relationship shown below in LCN 586.


In this case, evaluating the integral in LCN 586 is not difficult but proceeds from one step of partial integration applied to the substituted form

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x3e−px2ο∞∫ dx= 1

2p2ue−u

ο∞∫ du where

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u= px2

where the pa rtial integr a tion of the reduce d integral is

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ue−uο∞∫ du= −ue−u[ ]

ο

∞− (−e−u

ο∞∫ du)=0−0−0+1=1.

Unlike th e pre ceding LCN 585, you are expe cted to be abl e to deri ve LCN 586.



Norm or Zeroth Moment of Maxwellian


Using LCN 585 to evaluate the zeroth moment of LCN 584 one obtains the relation shown below in LCN 587.

By definition this integral over any particle distribution function results in the number of particles, N. A good homework exercise is to use LCN’s 584 and 585 to prove this relation.

Next we consider the first moment of a particle speed distribution function. It is

of course, the average speed of those particles.



What about Velocity Distributions?Ann asks, “You keep referring to speed. Is this also true for velocity distributions?”

Generally, a velocity distribution is different as it is a vector quantity- unlike speed, which is a scalar. Yet, in the specific case of the Maxwellian the speed always enters as a square meaning that the velocity squared, which equals the speed squared, can be used as well.

We recommend that you not confuse yourselves by thinking of these as velocity distributions as there are other particle distributions functions, that you may encounter later, for which the vector nature of the distribution will be important and where the concept of a speed distribution may not be well defined.

I hope this answers your question.




Evaluating the Maxwellian First Moment

Let’s see, where were we? We were about to use LCN586 to evaluate the Maxwellian distribution’s first moment. Doing that results in the following formula shown below in LCN588.


Students often look at this integral and don’t at first see how it produces an average speed. If you remember that an integral is a special sum and that this particular sum is adding speeds that are multiplied by the percentage of particles at each speed ( divided by N) you will realize that it is an average.

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f (v)dv



Most Probable Speed of Maxwellian


Another characteristic speed of the Maxwellian distribution is its most probable speed, which is simply that speed where

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f(v) is max imum. Sinc e ma xima occu r wher e derivativ e s vanish , on e ca n appl y this principle to the Ma xwellian dist ribution fun ction, a s given in LCN 58 4, to de te rmine the mo s t probable spee d a s show n belo w in LCN 589.

In the next Lecturette (# 150) we will compare the three characteristic speeds, the root- mean-square, the average, and the most probable, of the Maxwellian distribution of speeds.



Lecturette Summary: The Maxwellian Speed DistributionIn studying the motions of a large number of point particles (or monatomic molecules), usually in instances where the particles collide among themselves, their speeds or velocities are found to be distributed according to the Maxwellian formula. The numbers of particles and their average speed are given by the zeroth and first moments of the Maxwellian distribution, which we have shown can be evaluated in closed form. Another characteristic of the Maxwellian distribution is the most probable speed, which is also given in a closed formula. Many experiments also validate this distribution of speeds. Thus the Maxwellian is not only a good physical approximation, it is relatively easy to use in evaluating macroscopic properties associated with it- such as particle numbers and their average speeds.




Preview of Lecturette 150: Characteristic Speeds of Maxwellian & PV Diagrams


We finish the discussion on Maxwellian speed distributions by evaluating the three characteristic speed formulas. Then, in terms of the PV diagram we discuss the regions and limits in which the ideal gas law is valid and develop a descriptive nomenclature including the concept of the critical point.



Physics with Calculus: Credits

• David Anderson Instructor• Peter Tarsi Instructor• Carolyn McCrosson Student• Andrew Trott Student

lecturette 149: the maxwellian speed distribution

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