The Velocities of Gas Molecules


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1 he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes but coercial use is not allowed without the perission of the author A knowledge of the velocities of gaseous olecules helps us understand a nuber of cheical phenoena As an exaple, the rates of cheical reactions involving gaseous species depend on the kinetic energy of the olecules, which, in turn, depends on the velocity of the olecules When gas velocities are easured experientally, a nuber of interesting features are observed Further down this page you will exaine the effect of olecular ass on gas velocity, and later in the docuent you can look at other factors At the end of the first part of this docuent you will be asked to write a paragraph about gas velocities based on your observations here he questions that you are to answer are shown in boldface Note that in the following graph, the olar ass of the three gases is given in kg/ol, so that N has a olar ass of 8 etc In this first plot, change the asses of the three gases several ties, and change the teperature as you wish Observe how the plots change How will you deterine which curve corresponds to which gas? (You ay also need to change the range for u, the velocity, depending on your choice of asses and teperature Consult your MathCad guide if you do not reeber how to change the range of a variable) Page
2 fraction with given velocity trace trace trace velocity (/sec) 6 44 Molec Wt in kg/ol in K u, 5 5 Velocity in /sec Do all olecules in a gas saple at a particular teperature have the sae velocity? Why do cheists speak of the distribution of gas velocities? Page
3 Before proceeding to the next page, predict the behavior of gas velocity as the teperature changes In the next graph, you choose a olecular weight and then plot the distribution of velocities as the teperature changes (you can choose three different teperatures at any one tie) 5 fraction with given velocity trace trace trace velocity (/sec) Page
4 Now you can begin to explore the question of the kinetic energy of gas olecules at a particular teperature Reeber that the forula for kinetic energy is ke = (v /) Since there is a distribution of velocities in the gas saple, there is also a distribution of kinetic energies You will have noticed in the graphs above that one velocity is ore likely than any others for a given olecular weight at a given teperature We call that the ost probable velocity We can use that velocity to calculate a kinetic energy that is representative of the gas saple Go back to the first graph Using the crosshair feature, deterine the ost probable velocity for each of the three gases Now calculate the kinetic energy for each gas, based on the ost probable velocity that you easure What do you conclude about the kinetic energy of different gases at the sae teperature? Do your conclusions support the stateent that teperature is a easure of available energy? Write a paragraph that discusses the velocity of gaseous olecules and the factors which influence those velocities and kinetic energies Alternatively, prepare a 5 inute talk on the sae topic In both cases you will find it useful to have soe graphics to support your words Page 4
5 he Velocities of Gas Molecules  A Further Look In this section we will exaine the distribution of olecular velocities in ore detail Let's begin with a atheatical for of the velocity distribution function his equation was derived by the Scottish physicist Jaes Clerk Maxwell, and, independently, by the Austrian physicist Ludwig Boltzann, in the latter part of the 9th century It is not unheard of for two researchers to reach the sae result independently of one another, although it will probably becoe less likely in the future as scientific results are counicated ore and ore rapidly in print and electronic edia What is now known as the MaxwellBoltzann distribution law is given by the expression: p( u ) 4 π e π Here u is the velocity in /sec, R the gas constant in J/Kol, the teperature in K, and the olar ass in kg/ol u One way to interpret p(u) is that it is the fraction of gas olecules that has a velocity between a given value u and u+δu, where δu represents a sall change in u Go back to the graphs of the velocity distributions What do you notice about the area under each of the curves? You can use calculus to evaluate the area under a curve since that area is equal to the integral of the distribution function over the values of u Below is an integral sign with MathCad placeholders for the integrand (the function to be integrated), the variable of integration (u in this case) and the liits of integration his is a nuerical integral and the result will be a nuber, not another function ry changing the function (using p through p6 as possibilities) as evaluate the integral Does the nuerical value ake sense given the nature of the distribution? What do you observe if the upper liit for integration is too sall? u 5 p( u) du = Since there is a range of velocities at any teperature, how do we characterize the velocity? One way would be to look at the peak of the distribution curve he velocity at this peak is called the ost probable velocity (u p ) How would you use calculus to find the value of u p? he axia of a function are values of the arguent of the function (u in our case) for which the first derivative is zero (Yes, this is also true for the inia, and we will see how to distinguish one fro the other for this proble) You can ake use of MathCad's sybolic capabilities to differentiate p(u) with respect to u, and derive an expression for u p (You could also find dp(u)/du by hand but as you will see it is not a trivial proble) Load the MathCad sybolic processor (see handout) Under derivation forat, click on "show derivation coents" and "vertically, inserting lines" In the expression for p(u) below, place the vertical cursor to the right of one of the u's (but before the superscript) hen choose "differentiate on variable" in the sybolic enu Page 5
6 4 π π e u u Since this is a cubic equation in u (it is, isn't it), there are three roots, or three values of u for which the derivative is zero Place the cursor to the right of one of the u's in the derivative expression and choose "solve for variable" fro the sybolic enu Which of the three roots corresponds to the ost probable velocity? Why? Another easure of the velocity is the average velocity Look at the shape of the velocity distribution function Is the average velocity going to lie to the left (lower velocity) or the right (higher velocity) of the ost probable velocity For properties j, defined by distribution functions, F(j), the definition of the average value of that property is given by: j av j F( j) dj F( j) dj Using this expression, evaluate the average velocity Hint  substitute u for j and p(u) for F(j) and then perfor the appropriate integration You already know the value of the denoinator in the expression for the average velocity (what is it) he third coonly used easure of gas velocities is the root ean square velocity, u rs he text contains a derivation of the pressure exerted by a gas based on the kinetic olecular theory In this odel, the velocity appears as u, and the average force exerted on the walls of a container by a gas saple depends on the average of u he square root of the average of u is the root ean square velocity he expression for this velocity is: u rs Page 6
7 Using the expression that you obtained for the ost probable velocity and the average velocity, copare these three approaches to characterizing the velocity of a gas saple If the ost probable velocity is assigned a value of, what are the relative values of u av and u rs? In the first part of this docuent you easured the ost probable velocity fro the velocity distribution plot How do the values that you easured copare with those you would calculate using the forula above? You also calculated the ost probable kinetic energy for a gas based on your easured ost probably velocity Recalculate the kinetic energies using the average and root ean squared velocities Do you reach the sae conclusion that you did above about the kinetic energy of gas olecules as a function of teperature? Most of the tie when cheists speak of the average kinetic energy of a saple of olecules they are speaking of a kinetic energy based on the root ean squared velocity Using the expression above for the root ean squared velocity, derive a sybolic expression for the kinetic energy Does this expression ake sense in light of what you have observed above regarding the kinetic energy of different gases at the sae teperature? Page 7
8 As a final exaple, how would you find the fraction of nitrogen olecules that have a kinetic energy (based on the root ean squared velocity) greater than two ties the average kinetic energy? One approach to this proble is to find the velocity that gives a kinetic energy twice the average kinetic energy u his is a atheatical stateent of the proble  the kinetic energy we want is twice the average kinetic energy Solve this expression for u, the velocity which produces this kinetic energy he solution we want is 6 his akes sense when you think about it We want a kinetic energy that is twice the average Since the kinetic energy is proportional to the velocity squared, the velocity to produce a kinetic energy that is twice the average ust be the square root of two ties the velocity to produce the average kinetic energy he solution to our proble is then one inus the fraction which have a velocity less than the desired velocity (why is this the case): 6 4 π e u du = π u What percentage of gas olecules satisfy our criterion at any teperature? Additional exercises: A fast ball thrown by professional baseball pitcher ay approach a velocity of iles per Page 8
9 A fast ball thrown by professional baseball pitcher ay approach a velocity of iles per hour What fraction of oxygen olecules have a velocity greater than this at roo teperature? he velocity that a body ust have in order to copletely escape the earth's gravitational field is known as the escape velocity he value of the escape velocity is about 7 iles per second Using calculations siilar to those above, what do you conclude about the nuber of gas olecules which escape the earth's gravitational field ry a variety of reasonable olecular weights and choose reasonable teperatures Explain how your results are consistent or inconsistent with your understanding of the earth's atosphere wo additional concepts which are useful when describing the behavior of gas olecules are the collision nuber, Z, and the ean free path, λ he collision nuber is the nuber of collisions a particular gas olecule would experience with other olecules in a second hese collisions are very iportant in deterining the rates of cheical reactions he nuber of such collisions ust depend on the nuber of gas olecules in a given volue, N/V, the size of the olecules, d, and the average velocity of the olecules he expression that is derived for Z is: Z 4 N π V d You should recognize eleents of the average velocity in this expression Closely related to the collision nuber is the ean free path, the distance that a olecule travels, on average, between collisions Since Z is the nuber of collisions per second, /Z is the tie between collisions and /Z ties the average velocity is the distance between collisions hus the expression for the ean free path is: λ u av Z Substitute the appropriate expressions for u av and Z and obtain an expression for the ean free path Write a brief paragraph explaining why the forula you have obtained sees reasonable Page 9
10 he value of d for O olecules is 6* 8 Calculate the collision nuber and ean free path for O olecules at roo teperature and one atosphere pressure Suppose you were devising an experient in which it was iportant for olecules, on average, to go at least one icrosecond without experiencing a collision Using O as the saple, and roo teperature conditions, what pressure would be necessary to achieve this desired tie between collisions? (o begin with, what will the relationship between the nuber of collisions per second and the average tie between collisions be?) Page
11 A digression: One way toconsider the MB distribution, is that it represents the balance between the nuber of ways of achieving a particular velocity u, a nuber which goes as u, and the probability of having a particular velocity, and hence a particular kinetic energy, at any given teperature his latter is given by the ter exp(u/r) We will frequently encounter this coparison between the kinetic energy of the olecules and the theral energy R (or k if we are discussing single olecules) We can evaluate the constant ters by realizing that we wish the result to be a distribution function which integrates to unity Do this, and see if you get the sae for of the distribution seen earlier (he solution below would not appear in the docuent students see) N e has solution(s) u u du liit exp,, u u ( ) R R π erf u u left R siplifies to R π π p( u ) 4 π e π 4 π ( π) siplifies to π u u Page
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