Kinetic Theory of Gases. Chapter 33. Kinetic Theory of Gases

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1 Kinetic Theory of Gases Kinetic Theory of Gases Chapter 33 Kinetic theory of gases envisions gases as a collection of atoms or molecules. Atoms or molecules are considered as particles. This is based on the concept of the particulate nature of matter, regardless of the state of matter. A particle of a gas could be an atom or a group of atoms (molecule). For gases following the relationship PV = nrt (IGL); Observations KT Postulate Gas density is very low Particles are far apart Pressure is uniform in all directions IGL is independent of particle type Particle motion is random Gas particles do not interact The Kinetic Theory relates the 'micro world' to the 'macro world. Dalton s Law of Partial Pressures nrt P V Gas particles do not interact KT (IGL): Applicable when particle density is such that the inter-particle distance >> particle size (point masses). Low pressures and high temperatures e.g 1atm and room temp.

2 Postulates Gas particles are very far apart. Gas particles in constant random motion. Gas particles do not eert forces on each other due to their large intermolecular distances. Pressure is due to particle collisions (elastic) with the walls of the container from translational motion - the microscopic eplanation of pressure. Collisions with the wall are elastic, therefore, translational energy of the particle is conserved with these collisions. Each collision imparts a linear momentum to the wall, which results the gaseous pressure. In Newtonian mechanics force defined as the change of momentum, here, due to the collision; pressure is force per unit area. In KT, the pressure arising from the collision of a single molecule at the wall is derived and then scaled up to the collection of molecules in the container, to obtain the ideal gas law (IGL); PV = nrt Gas kinetic theory derives the relationship between root-mean-squared speed and temperature. The particle motions are random, therefore velocities along all directions are equivalent. Therefore the average velocity (vector) along any dimension/direction will be zero. Now, the root-mean-squared velocity = root-mean-squared speed ; it is nonzero. A distribution of translational energies; therefore, many velocities would eist for a collection of gaseous particles. Velocity is a vector quantity (v). Speed is a scalar (). v = v v = v + v y + v z v = = + y + z <v > = < > ; symbol <> average of. Most properties of gases depend on molecular speeds. What is the distribution of the particle velocities?

3 Root mean square velocity, translational energy: m Assume: (Chapter 31) v kbt m v v v v y z v v v because of randomness y z Therefore, < v > = < v > + < v > + < v > = < v > + < v > + < v > y z 3 v m v kbt Using m < v > m3 < v > 3k T m < v > 3 kbt; < v > B by definition re me mber m v ( l ast sli de ) 3kBT = 3kT m m 1/ k T B For a gas sample of n moles occupying a volume V (cube), with an area of each side A. Consider a single particle of mass m, velocity v. Particle collides with the wall. (elastic collisions) t Number density of particles = nn N A V where N A = Avagadro Number Half of the molecules moving on ais with a (velocity component in the direction) within the volume v t collides with one surface in the direction. v t Change of momentum p = mv () mv mv mv () mv mv N coll = number of collisions on the wall of area A in time t.

4 Change of momentum on a surface during t = Pressure = Force per unit area = Force on the surface = Rate of change of momentum = Pressure arises because of the molecular motion of gases; microscopic/molecular model of pressure. usi ng m k T kt v B The fact that component velocities of all molecules are not the same, necessitated the definition of an average in each direction. Thus <v j > arises because of a (probability) distribution of v j values f(v j ) in each direction (j =, y, z) PV = nrt Ideal gas law- IGL Pressure: macroscopic/empirical model. velocity distribution function Being a probability function; all ν j f () v dv 1 j j

5 Derivation of distribution functions f(v i ) v v(v v, v,) y z ;also from another relationship v = v + v + v for a molecule y z v [v + v + v ] 1/ y z Math supplement In general? Because derivatives of three independent variables are equal, the derivatives must be constant, say = -. Upon rearrangement and integration, Similarly where A = integration constant Note the distribution (probability) function!

6 Evaluating A: Distribution function (Assumption slide 9) Math supplement Mean/average averaging Now, & m kt Substituting for in f(v j ); use tables Math supplement

7 velocity distribution function Deriving the distribution function for v Changing the volume element (in cartesian) to variable v, spherical coordinates replace dv dv dv by 4 d y z and v v v by v y z

8 rise eponential decay Notice the shape, blue. ave of Xe, H, He? Earth and Jupiter (300) and fied varied t v t rise eponential decay Notice the shape, blue. ave of Xe, H, He? Earth and Jupiter (300) At lower angular velocities slower moving molecules go through the second slit. Detector signal proportional to the number of particles reaching the detector.

9 Most probable velocity v mp Most probable velocity v mp : differentiate F(v), set to zero. Notice the shape. Mean (average) velocity: Root mean square velocity: Using: (Assumption slide 4) and v v v v 3 v y z v 3 v v rms

10 Number of collisions per unit time on the wall of area A: average component of velocity <v > = # molecules in light blue volume of the cube colliding per unit time = A Rate of collisions on surface = <v > N c = number of particles colliding. Collisional Flu Z c : Number of collisions on the wall per unit time per unit area. & Substituting in Z c ;

11 Particle collision rates: (Hard sphere model) Particles interact when spheres attempt to occupy the same region of the phase. (consider one moving particle orange, 1; all other particles stationary are red label Because the collisional partners are moving in reality, an effective speed, <v 1 >, of orange particle will be considered in the model to emulate the collisions the orange particle encounters; = collisional cross-section. V v dt cyl ave Because the collisional partners are moving, an effective speed <v 1 > used to model the system; N Collisional partner (red) density = V Volume covered by orange in dt = V cyl V v dt cyl ave Collisions by it in time dt = N V cyl V Particle collisional frequency of it = z 1 m1m m m 1 ; reduced mass For a sample of one type of gas;

12 Total collisional frequency Z 1 : Total number of collisions in the gaseous sample. For a sample of one type of gas; Accounts for double counting Mean Free Path: Average distance a particle would travel between successive collisions two types of molecules, say 1 and. For one type of molecules, Effusion: Effusion is the process in which a gas escapes through a small aperture. This occurs if the diameter of the aperture is considerably smaller than the mean free path of the molecules (effusion rate = number of molecules that pass through the opening (aperture) per second). Once the particle passes through it generally wont come back because of the low partial pressure on the other side Pressure of the gas and size of the aperture is such the molecules do not undergo collisions near or when passing through the opening.

13 Collisional Flu Z C : High P Z C = number of collisions per unit time per unit area. (by one type of molecule) Low P Left effusion; right - diffusion. Effusion occurs through an aperture (size A) smaller than the mean path of the particles in motion whereas diffusion occurs through an aperturethrough which many particles can flow through simultaneously. Note: Upon substitution and simplification; ;using IGL Effusion rate decreases with time because of the reduction in gas pressure inside the container. And rate of loss of molecules Integration yields; Upon substitution

14 -v 0 v 0 -v 0 v 0 o f ()()() v v v f v dv f v dv mv / kt mv kt 0 0 / 0 f () v0 v v0 e d 0 o 0 mv 0 / kt

15 0 f () v0 v v0 e d 0 mv / v v kt / m 0 mv / kt z erf () z e d erf () z f () z v z 0 z erf () z e d 1 erf () 1 e d covers v kt / m 0 0 erf () z probability v kt / m?