Chapter 24a molecules in motion basic kinetic theory
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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)

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Chapter 24a molecules in motion basic kinetic theory

Chapter 24a: Molecules in MotionBasic Kinetic Theory


Exercises (a only) : 4, 5, 6, 8, 10

Kinetic model of gases
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

Pressure and molecular speed
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

Pressure and molecular speed1
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:

Pressure and molecular speed2
Pressure and Molecular Speed

  • Bernoulli’s Derivation -

    • Pressure (P) = force/area, so

    • Integrating over all + velocities since only they contribute to pressure on wall

Consequences of kinetic model of equation of state pv 2 3 e k
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

Distribution of speeds f u du
Distribution of Speeds, f(u)du E

  • 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

Using maxwell distribution
Using Maxwell Distribution E

  • 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

Molecular speed relationships
Molecular Speed Relationships E

  • 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

Collisions E

  • 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

Collisions E

  • 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

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