20 B Week II Chapters 9 -10)• Macroscopic Pressure

•Microscopic pressure( the kinetic theory of gases: no potential energy)

• Real Gases: van der Waals Equation of State

London Dispersion Forces: Lennard-Jones V(R )

and physical bonds

Chapter 10

• 3 Phases of Matter: Solid, Liquid and Gas of a

single component system( just one type of molecule, no solutions)

Phase Transitions:

A(s) A(g) Sublimation/Deposition

A(s) A(l) Melting/Freezing

A(l) A(g) Evaporation/Condensation

For a fixed mass( of gas, e.g., Air) how does the Volume occupied by the gas change when the gas is cooled or heated?

Lets do an experiment!

• Define a Thermodynamic Temperature scale to make a thermometer( to measure T)

Assumes we know nothing about the Boltzmann distributionso T is a parameter

• Charles’ Law V/T = const

Quantifying the temperature scale requires some materialProperty that changes if the material is heated or cooled.

The thermal expansion coefficient = (1/V)(V/T)

assume fractional rate of change of V with respect to T is ideal.

The temperature scale should be material independent(Universal)

appears to be Universal for low pressure gases

Over the T range where water freezes and boils, Charles observed that = (1/V)(V/T) ~const for low pressure gases.

= (1/V)(V/T)=(1/V0){V-V0}/{T-T0}

V0 and T0, are the initial volume and Temperature, thermometer material and V and T, , are the final volume and Temperature, respectively!

The Experiment: we need 3 T’s to be accurateThe material for the experiment is air

20±?Room Temp!

Assigned values

= (1/V0){V-V0}/{T-T0} Thermal expansion

Let V0 be the gas Volume at T0 = 0

The Freezing point of water!

T = (1/{(V/V0) – 1}

Re-arrange to V=V0T}

If T has units of °C (Celsius), the boiling point of water must be set to 100 °C!

All measurements are for air at low pressure

V=V0T} solve for T at V=0 gives T=C

Fitting the V vs T the equation to the data, we can extrapolate to absolute zero!

V0= 1.5 L

V0= 1.0 L

V=V0T} so as V goes to zero T goes to The absolute T=0 KC

However, the mass of gas is fixed!So V can approach zero but cannot be zero!

V0= 1.5 L

V0= 1.0 L

In terms of an absolute temperature T=-273.15 C= 0 K (Kelvin)

V~T since as V approaches 0, T 0 V = TConst.

This is most useful form of Charles’ Law

V~ N (the number of gas atoms/Molecules) (Since the more molecules the greater the volume)

V~1/P from Boyle’s Law

Therefore V~ NT/P or V=kNT/P: PV=kNT

The is Ideal or Perfect Gas Law where k = Boltzmann Constant

: PV=kNT

PV=kNT: The Idea Gas Law! Applied two different gases, e.g., N2 and O2

(Or even better! The Law describing the behavior of aTheoretically Perfect/Ideal Gas)! Can be used for all gases at Low Pressure and high enough temperature

~ ultrahigh vacuumultralow pressures!

Effusion Cell

~ very small hole or area A.If P=PO2 + PN2(partial pressures)P=NO2kT/V + NN2kT/VP = (NO2 + NN2 )kT/V=NkT/VN=NO2 + NN2 which is true

Fig. 9-9, p. 377

+ =

Partial Pressures add. P= P1 + P2 + P3 + etc

When dealing with the behavior of large numbers N~NA

NA=6.022 x 1023 mole-1 then we must average over the behavior of the individual atom/molecule in the system

For example.

If i is the total energy of the ith atom/molecule, Kinetic + Potential, then the average energy per atom/molecule

is <>= (1 + 2 + 3 ………. + N )/N=∑i/NThis is the behavior we observe and is the domain ofStatistical Mechanics: the science that describes the behavior of a system with large numbers of atoms/molecules based on the behavior of individual atoms/molecules, e.g.,Gases!

The Kinetic Theory of Gases uses this idea to describe the

Behavior of an ideal gas!

Think about this for a min!

The average behavior can be determined by watching one

particle for a very long time(infinitely long in principle),

or a large number of particles (infinitely large in principle)

for a short time, i.e., snapshot or an instant

<(vx)2

The Kinetic Theory

Nanoscopic theory of gas pressure

watch the average behavior of one particle and describe the behavior of a system with a large number of

particles

L

The Kinetic Theory

Nanoscopic theory of gas pressure

watch the average behavior of one particle and describe the behavior of a system with a large number of

particles

- FA =F = ma=d(mv)/dt

mv = linear momentum(mv)=(mvx)f - (mvx)i

(mv)=-mvx - mvx=-2mvx

t=2L/vx

L

-FA =F = ma=d(mv)/dt ~ -2mvx/(2L/vx)= - m(vx)2/L

FA=2m(vx)2/L; <FA>=m<(vx)2>/L average force on the wall

Px=P= N<FA>/A=Nm<(vx)2>/AL=(N/V) m<(vx)2>

The Kinetic Theory

Microscopic pressure

one particle at a time

-FA = F = ma=d(mv)/dt force atom/molecule

mv = linear momentum(mv)=(mvx)f - (mvx)i

(mv)=-mvx - mvx=-2mvx

t=2L/vxL

V=AL

Fig. 9-11, p. 379

v2 = (vx)2 + (vy)2 +(vz)2 =u2 then

< v2 >=<u2>=<(vx)2 + (vy)2 +(vz)2>=3<(vx)2> or <(vx)2> = <u 2 >/3

The average of speed in all directions must be the same <(vx)2>=<(vy)2>=<(vz)2> for the random motion of an atom/molecule

P= (N/V) m<(vx)2>= (N/3V) m< u2 > but <KE>= (m/2)< u2 >

So P=(N/V)<KE>(2/3) recall that PV=NkT so <KE>=(3/2)kT

Since <>=<PE> + <KE> = <KE> for a prefect gas,

that is a gas described by PV=NkT

Motion in all directions x,y, and z are equally likely: <vx>=<vy>=<vz>

Fig. 9-9, p. 377

+ =

Partial Pressures add. P= P1 + P2 + P3 + etc

20 B Week II Chapters 9 -10)• Macroscopic Pressure

•Microscopic pressure( the kinetic theory of gases: no potential energy)

• Real Gases: van der Waals Equation of State

London Dispersion Forces: Lennard-Jones V(R )

and physical bonds

Chapter 10

• 3 Phases of Matter: Solid, Liquid and Gas of a

single component system( just one type of molecule, no solutions)

Phase Transitions:

A(s) A(g) Sublimation/Deposition

A(s) A(l) Melting/Freezing

A(l) A(g) Evaporation/Condensation

Fig. 9-18, p. 392

A AR

<V(R )> = 0

For RVery LargeDensity N/V is lowThereforeP=(N/V)kT is low

Real Gases and Intermolecular Forces

well depth ~ Ze or Mass but it’s the # of e

Lennard-Jones Potential

V(R ) = 4{(R/)12 -(R/)6}

Ar+ ArkT >>

Real Gases and Intermolecular Forces

well depthBond DissociationD0= h

~ hard sphere diameter

Lennard-Jones Potential

V(R ) = 4{(R/)12 -(R/)6}

The London Dispersion

or Induced Dipole Induced Dipole forces

Weakest of the Physical Bonds but is always present!