1 / 11

Chapter 24a: Molecules in Motion Basic Kinetic Theory

Chapter 24a: Molecules in Motion Basic Kinetic Theory. Homework: Exercises (a only) : 4, 5, 6, 8, 10. Kinetic Model of Gases. Motion in gases can be modeled assuming Ideal gas Only contribution to energy of gas is the kinetic energy of the molecules (kinetic model)

ostinmannual
Télécharger la présentation

Chapter 24a: Molecules in Motion Basic Kinetic Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 24a: Molecules in MotionBasic Kinetic Theory Homework: Exercises (a only) : 4, 5, 6, 8, 10

  2. Kinetic Model of Gases • Motion in gases can be modeled assuming • Ideal gas • Only contribution to energy of gas is the kinetic energy of the molecules (kinetic model) • Basic Assumptions of Kinetic Model: 1. Gas consists of molecules of mass, m, in random motion 2. Diameter of molecule is much smaller than average distance between collisions • Size negligible 3. Interaction between molecules is only through elastic collisions • Elastic collision - total kinetic energy conserved

  3. Pressure and Molecular Speed • Bernoulli’s Derivation* - • Assumptions • Collection of N molecule, behaving as rigid spheres • Random velocities in volume, V • Consider movement in x direction (normal to a wall) • Heading toward wall is positive x • Fraction with velocity u + du specified by density function, f(u), where • dN(u)/N = f(u) du • f(u) probability velocity between u ® u +du • No need to define this function further at this point *slightly different approach than book

  4. Pressure and Molecular Speed • Bernoulli’s Derivation - • Kinematics • In time dt, only molecules striking wall are those with positive velocity (u>0) between u and u+du • Assumes initial distance ≤ udt • # of Molecules striking wall during dt are those within volume V=Audt at t= 0 with right velocity N/V(f(u)du))isN/Vf(u) Audt du • Momentum change with each collision is Dp= mu - (-mu) = 2mu. Incremental momentum change is:

  5. Pressure and Molecular Speed • Bernoulli’s Derivation - • Pressure (P) = force/area, so • Integrating over all + velocities since only they contribute to pressure on wall

  6. Consequences of Kinetic Model of Equation of State (PV = 2/3 Ek) • Kinetic model looks a lot like Boyle’s law (PV = constant @ constant T) • Since the mean square speed of molecules should only depend on temperature, the kinetic energy should be consatnt at constant T • In fact if NRT = 2/3 Ek, Kinetic Model is the ideal gas equation of state • Since Ek = 0.5Nmc2,it follows that <c2>1/2a T1/2 and <c2>1/2a (1/m)1/2 {<c2>1/2 is the root mean square speed} • The higher the temperature, the faster the mean square speed • The heavier the molecule, the slower the mean square speed

  7. Distribution of Speeds, f(u)du • Earlier didn’t define f(u) exactly • James Clerk Maxwell (1860) derived distribution • Key assumption: probability of particle having a velocity component in one range (u +du) independent of having another component in another range • If this is true, the distribution function, F(u,v,w), is the product of components in each direction F(u,v,w) = f(u)f(v)f(w) • Not fully justified until advent of quantum mechanics • Law proved by Boltzmann in 1896 • Verified experimentally • If v is the speed of a molecule, • Features of distribution • Narrow at low temperatures, broadens as T increases • Narrow for high molecular masses, broadens as mass decreases

  8. Using Maxwell Distribution • Evaluating Maxwell distribution • Over narrow range, Dv,evaluate f(v) Dv • Over wide range, numerically integrate, • Mean speed of particle • Mean speed (not mean root speed) is calculated by integrating the product of each speed and the probability of a particle having that speed over all possible speeds

  9. Molecular Speed Relationships • Root mean speed, c, c = (3RT/M)0.5 • Mean speed, c , c = (8RT/πM)0.5 • Most probable speed, c* • Maximum of Maxwell distribution, i.e, df(v)/d(v) = 0 c* = (2RT/M)0.5 • Most probable speed = (π/4)0.5 x mean speed = 0.886 c • Relative mean speed, crel,speed one molecule approaches another • Range of approaches, but typical approach is from side

  10. Collisions • Collision diameter, d - distance > which no collision occurs, i.e., no change in p of either molecule • Not necessarily molecular diameter, except for hard spheres • N2, d = 0.43 nm • Collision frequency, z - number of collisions made by molecule per unit time z = s x crel x N s = collision cross section = πd N = # molecules per unit volume = N/V crel = relative mean speed • For ideal gas, in terms of pressure , z = (s crel p)/kT • As T increases, z increases because the rel. mean speed a T1/2 • As p increases, z increases because the # of collisions a density of gas a N/V • N2 at 1 atm and 25°C, z = 5 x 109 s-1

  11. Collisions • Mean free path, l - average distance between collisions l = mean speed ¸ collision frequency • Doubling p halves l • At constant V, for ideal gas, T/p is constant so l independent of T • l(N2) = 70 nm • At 1 atm & RT, most of the time, molecules far from each other • Collision Flux, zw - # collisions per unit time and area zw = p/(2πmkT)0.5

More Related