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# Chapter 24a: Molecules in Motion Basic Kinetic Theory - PowerPoint PPT Presentation

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 MotionBasic Kinetic Theory

Homework:

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

• 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

• 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

• 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:

• Bernoulli’s Derivation -

• Pressure (P) = force/area, so

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

• 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

• 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

• 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

• 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

• 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

• 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