Kinetic Theory of Gases: Elementary Ideas

Transcription

1 Kinetic Theory of Gases: Eleentary Ideas 17th February Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion of the derivation for the pressure of a rarefied collection of particles of ass. In the following, we provide a connection fro the one-diensional version to the full scalar pressure. The connection is not quite direct fro the discussion of Engel and Reid. In 3-D, for a collection of any particles on the order of Avogadro nuber), using average values of velocity and velocity coponents in Cartesian coordinates); these are not generalized coordinates as physicists would consider), the total kinetic energy is: KE total v v) 2 2 vx 2 + vy 2 + vz 2 ) 2 v2 x + 2 v2 y + 2 v2 z KE) x + KE) y + KE) z The last equality is really just a notational trick; there really does not exist a therodynaic or kinetic property KE) x or KE) y or KE) z! The lowercase is ass. The velocity is a vector so the v 2 we treat casually is really a dot product. The pressure coponents for the x, y, and z directions as we deterined in class are: 1

2 p x p y p z ) v 2 x 1 2 KE) x ) v 2 y ) v 2 z 1 2 KE) y 1 2 KE) z is the nuber of particles. is the volue of space we are considering. Fro statistical echanics which you will learn ore about in the future) we have the relation for the special case of a fluid or state of atter with extreely weak interactions or no interactions): KE) x KE) y KE) z 2 Avogadro 2 Avogadro 2 Avogadro Thus, 2 KE) x 2 KE) y 2 KE) z Avogadro Avogadro Avogadro Substituting the above relations for 2 KE x ), etc. equations yields: into the pressure p x p y p z ) ) ) 1 Avogadro 1 Avogadro 1 Avogadro Avogadro Recall that for pressure becoe: is the nuber of oles oles. Thus, the equations 2

3 p x p y p z oles oles oles Here, we stop and realize that we have a x-coponent of pressure, a y- coponent and a z-coponent. This is not an artificial result, as rigorously, pressure is a tensorial property it is a 3x3 atrix). The diagonal eleents which we have coputed) have special eaning in that they can be used to deterine the pressure as we know it. Rigorously, the scalar pressure that we norally easure and talk about) is deterined fro the trace of the pressure tensor or atrix). This is: ) ) ) p scalar total 1 3 p x + p y + p z ) 1 ) oles + + ) 3 ) oles R T ) oles pressure R T This should be a ore convincing arguent for the equality of kinetic theory description of fluids to that of an ideal gas. 2 Maxwell-Boltzann Distribution of elocities For a collection of particles that do not interact, we recall that the ideal gas description is sufficient to predict properties of fluids at certain liiting conditions. Within such a description, the particles, of certain asses, are considered to be oving with velocities that are distributed in a certain fashion. We now consider this distribution of velocities, starting with the distribution of velocity in one-diension. The generalization to three diensions follows. We are concered with finding a velocity distribution function. This atheatical description provides the probability of a particle having a velocity in the range v x + dv x, v y + dv y, and v z + dv z. We let the distribution function be: Ωv x, v y, v z ) fv x ) fv y ) fv z ) 3

4 An assuption regarding the nature of this distribution: the gas is isotropic such that the direction of particle oveent does not affect the properties of the fluid. In this sense, the velocity distribution is effectively dependent on the agnitude of the velocity. The agnitude of the velocity is: ν v 2 x + v 2 y + v 2 z We now consider the natural logarith of the distribution function: with ln Ωv x, v y, v z ) ln fv x ) + ln fv y ) + ln fv z ) ln Ων) ln fv x ) + ln fv y ) + ln fv z ) ν 2 v 2 x + v 2 y + v 2 z To deterine fv x ), we consider the derivative of the probability function Ων) with respect to the variable v x. Effectively, we are interested in finding that distribution of velocities along the x-direction) that distributes the total kinetic energy of the syste aong the available degrees of freedo in the collection of particles recall that the internal energy of an ideal gas is only dependent on the teperature and aount of fluid) under the constraint that the total nuber of particles is constant, the volue we consider reains constant, and the teperature is constant and thus known. One can show that ) ln Ων) v x ) ) d ln Ων) ν d ν v x ν v x )v y,v z vx ν. d ln fv x) d v x v y,v z d ln fv x) d v x v y,v z ) ν v x v y,v z ) v 2 v x + vy 2 + v2 z x v y,v z 1 2 2v x) v 2 x + v2 y + v2 z ) 1/2 v x ν 4

5 Using this last relation, we obtain: ) ) d lnων) ν d ln fv x) d ν v x v y,v z d v x ) d lnων) vx ) d ln fv x) d ν v d v x ) d lnων) d ln fv x) ν d ν v x d v x Since each direction is equivalent under our assuption of isotropy of the ediu), we can write for the v y and v z differentials and you can show yourself independently): ) d lnων) ν d ν ) d lnων) ν d ν d ln fv y) v y d v y d ln fv z) v z d v z Thus, the following equality is clear: d ln fv y ) v y d v y d ln fv x) v x d v x d ln fv z) v z d v z Since the individual functions of x, y, and z are equal to one another for all space, we can justifiably argue that each is a constant equal in all cases); in general we can write: d ln fv j ) v j d v j d fv j ) v j fv j ) dv j γ for j x, y, z We use the negative of a positive γ in order to obtain a probability function that is well-behaved as v j approaches. The solutions for the previous equation are: d fvj ) fv j ) γ v j dv j ln fv j ) 1 2 γ v2 j fv j ) A e γ v2 j /2 This last equation is the general expression of our solution; it applies to the individual probability distributions in each direction x,y,z) recall we 5

6 decoposed our total distribution into a product of the three individual distributions). We still need to consider 2 issing eleents: A γ To deterine an expression for A, we need to consider that the distribution is noralized over the doain for which it is defined. What does this ean operationally? Since this expression represents a probability, the su of the individual probabilities for each infinitesial region of space or in this case, each window in velocity space) ust su to 1. Thus we have: fv j ) dv j A 0 A e γ v2 j /2 dv j /2 dv j A e γ v2 j 2π 1 A γ γ A 2π We used the property of even integrands to change the lower liit of integration. Thus, the distribution becoes: fv j ) γ 2π e γ v2 j /2 To evaluate γ, we introduce fro statistical echanics which we just state here without proof, which is left for another course or to the individual student): v 2 x k B T where is the ass of the particles. Thus, 6

7 v 2 x k b T k B T γ 2π γ 1 2π γ 1 γ γ v 2 x fv x ) dv x vx 2 e γ v2 x /2 dv x ) 2 π γ We have thus arrived at the faous Maxwell-Boltzann velocity distribution in one diension: fv j ) v2 e j /2 k B T ) ) M 1/2 M v2 e j /2 R T ) 2 π R T where ass,, is in units of kg and olar ass, M, is in units of we use the relation R A k B for the conversion between the fors using and M. In three diensions Cartesian represenatation) the total distribution becoes: kg ol Ωv x, v y, v z ) Ωv x, v y, v z ) Ων) j1,3 j1,3 j1,3 fv j ) v2 e j /2 k B T ) v2 e j /2 k B T ) Exaple Proble 33.1 Engel and Reid) Copute average velocity, v x. 7

8 v x v x fv x ) dv x v x v2 e j /2 k B T ) dv x v x e v2 j /2 k B T ) dv x 0 The result reflects the vectorial nature of velocity. 3 Maxwell Distribution of Speeds We stipulated in the above discussion of the Maxwell-Boltzann distribution that the ediu defined by the particles is isotropic; that is, properties are not dependent on a direction on internal or external gradients). Furtherore, though we wrote Ων) as a function of speed, ν, there was no explicit dependence on speed, a non-vectorial quantity, in the final expression. With this in ind, we next consider the distribution of speeds that eerges fro the velocity distribution we derived. Again, bear in ind that speed is not vectorial, velocity is. Based on our earlier definition of the probability distribution based on speed, ν, we can write now: F ν) dν fv x ) fv y ) fv z ) dv x dv y dv z [ ) ] [ 1/2 ) ] v2 e x /2 k B T ) 1/2 v2 e y /2 k B T ) [ ) ] 1/2 v2 e z /2 k B T ) dv x dv y dv z ) 3/2 v2 e x+vy+v 2 z) 2 /2 k B T ) dv x dv y dv z We are alost to the point of having our expression for the speed distribution, since we can see clearly that v 2 x +v2 y +v2 z ν2. However, the differential volue eleent dv x dv y dv z is the troubling part. Here we consider the idea of change of variables fro Cartesian coponents of velocity to a spherical coordinate representation of speed. 8

9 Thus, the volue eleent becoes 4 π ν 2 dν after integrating over the angular diensions. The Maxwell speed distribution thus becoes: ) 3/2 F ν) dν 4 π ν 2 e ν2 /2 k B T ) dν ) M 3/2 F ν) dν 4 π ν 2 e M ν2 /2 R T ) dν 2 π R T Properties of this distribution include: At a particular teperature, there is a single axiu value of the speed Despite including a Gaussian factor, the distribution is not syetric due to the ν 2 ter which introduces contributions fro tail values at higher speeds At higher teperatures, the distribution becoes broader ore speeds, or translational energy levels accessible) At lower asses, the distribution becoes broader ore speeds are accessible) Coparison of distribution for gases of different ass At sae teperature, kinetic theory says all gases have soe energy depends on T); thus, heavier asses have distributions that are narrower and peaked at lower speeds. 4 Measures of the Maxwell Distribution of Speeds We can consider individual distributions of different gases or even a single gas) at different teperatures, for instance. But there are certain easures of the distribution of speeds for a particular gas that we can consider in order to ake quick coparisons. We consider these next. Most probable speed, ν p d F ν) d ν 4 π 4 π d 4 π d ν ) ) 3/2 ν 2 e nu2 /2 k B T ) 3/2 d d ν ν 2 e nu2 /2 k B T ) ) 3/2 e nu2 /2 k B T ] [2 ν ν3 k B T 9

10 The axiu probability occurs when the derivative is identically 0; we thus obtain: 2 ν p ν3 p k B T 0 2 kb T ν p Average Speed ν ave 0 ν F ν) dν 4 π ) 3/2 0 ν 3 e nu2 /2 k B T dν 8 kb T π 8 R T π M Root-Mean-Square Speed ν rs [ ν 2 ] 1/2 3k B T )1/2 1) ν rs > ν ave > ν p All easures are proportional to T 1/2 and M 1/2. 5 Collisions and Collision Frequency 5.1 Gas of Dissiilar Particles, 1 and 2 The rudientary concept of olecular collisions will be relevant to future discussions of reaction kinetics and echaniss. Here we consider what insights kinetic theory separate fro the field of reaction kinetics) can provide. First, consider a gas of dissiilar particles, 1 and 2. We want to estiate the nuber of collisions a single particle of 1 will ake with all other 2 particles. Then, we ll consider the total nuber of collisions per unit of tie. Since these properties are per unit of tie, they are intuitively frequencies 10

11 One of the assuptions that we will invoke is that the density is low or the distance between is particles is sufficiently large; or the particles have very weak effective interactions). All of these assuptions can justifiably argued and leave roo for further exploration. For now, we proceed with a siplistic odel. Consider Figure in Engel and Reid. Here we have a particle oving with soe effective speed which we can deterine based on our earlier discussion of easures of probability distributions of speeds; we ll see how to do this shortly). Consider the cylindrical volue swept out by a particle oving with effective speed ν eff. length of the cylinder generated by particle oving for a tie t: length ν eff t surface area of base of cylinder A σ π r 1 + r 2 ) 2 r 1 + r 2 ) is the effective radius of the cylinder The volue is then cylinder π r 1 + r 2 ) 2 ν eff t σ ν eff t. The effective speed, ν eff is deterined fro consideration of the apparent speed between two particles in a collision when both have a certain speed. We can approxiate this effective speed as: ν eff < ν 1 > 2 + < ν 2 > 2) [ 1/2 8 kb T π 1 [ 8 kb T )] 1/2 π 1 2 ) 8 kb T 1/2 π µ where the reduced ass µ is defined as: ) )] 8 kb T 1/2 + π 2 1 µ With the above definitions, we can now define the nuber of collisions that a single particle will have with other particles in the cylindrical volue in a unit of tie as follows: z ) cyl 2 t ) 8 σ kb T 1/2 π µ ) σ νeff t t 11

12 Are the units of this property consistent? For the collision frequency of a particle 1 with another particle 1, z 11, we have: z 11 1 σ 2 The total collisional frequency is thus: 8 kb T π 1 Z 12 1 z 12 Z σ 8 kb T π µ Z Mean Free Path 1 z 11 Z )2 σ 8 kb T π 1 Knowing the distribution of velocities, we can deterine the collision frequency as shown above). Knowing the previous two kinetic properties of a gas, we can now consider the Mean free path, the average distance a particle travels in between collisions. The question we are asking is: In a unit of tie, a particle travels a certain distance, along the way participating in a nuber of collisions. Thus, the ean free path the distance travelled between collision events) is: Mean F ree P ath v ave Z 12 12