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Chapter 24a: Molecules in Motion Basic Kinetic TheoryPowerPoint Presentation

Chapter 24a: Molecules in Motion Basic Kinetic Theory

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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

- Size negligible

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

- Assumptions

*slightly different approach than book

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:

- In time dt, only molecules striking wall are those with positive velocity (u>0) between u and u+du

- Kinematics

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 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

- 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}

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 this is true, the distribution function, F(u,v,w), is the product of components in each direction
- 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

- Key assumption: probability of particle having a velocity component in one range (u +du) independent of having another component in another range

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 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

- Maximum of Maxwell distribution, i.e, df(v)/d(v) = 0
- 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

- Not necessarily molecular diameter, except
- 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

- For ideal gas, in terms of pressure ,
- 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 Collision Flux, zw - # collisions per unit time and area

- 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

zw = p/(2πmkT)0.5

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