THERMODYNAMICS

A GENERALLY GLOOMY SUBJECTTHAT TELLS US THAT THE UNIVERSEIS RUNNING DOWN, EVERYTHING ISGETTING MORE DISORDERED ANDGENERALLY GOING TO HELL IN AHANDBASKET

James Trefil - NY Times

THE LAWS OF THERMODYNAMICSTHE THIRD OF THEM,THE SECOND LAW,WAS RECOGNIZED FIRST

THE FIRST, THE ZEROTH LAW,WAS FORMULATED LAST

THE FIRST LAW WAS SECOND

THE THIRD LAW IS NOT REALLY A LAW

P. Atkins ~ The Second Law

Zero’th and third laws TemperatureFirst law Conservation of energy

THE SECOND LAW ENTROPY

QUESTIONS

Define T, S.

What is thermodynamics?

What are the laws of thermodynamics?

∆∆Gm – Wha t i s i t?

THERMODYNAMICS

Expresses relationships between the macroscopic properties of a system without regard to the underlying physical (i.e., molecular) structure.

E.g., EQUATION OF STATE OF AN IDEAL GAS

PV = nRT

•Where does this come from?

•What is the molecular machinery? ie. can we obtain this equation from a molecular theory ?

•Can we also obtain equations of state for liquids and solutions?

•How do we describe interactions?

DEFINITIONS

1. VOLUME AND PRESSURE

Cross section area = A

BOYLE’S LAW

Volume V

Force F P = FA

P V = constant

P1

~ V

(at constant T,n)

E X P E R I M E N T

T H E O R Y

1/V

P

DEFINITIONS2. TEMPERATURE - -What is it?

Thermodynamic definition: - something that determines the direction heat will flow

Normal scales — °C, °F arbitrary

If there is no heat flow, the two bodies are at the sametemperature (zero’th law of thermodynamics)

HEAT

CHARLES LAWV ~ T (at constant n, P)

Note: third law. Can’t get to absolute zero in a finite number of steps

Is there anabsolutescale oftemperature ?

-273 CO

V

T

0 CO

DEFINITIONS3. The amount of stuff; n.

• Don’t confuse mass and weight

• Numbers of molecules - - large ! • Use moles (mol) as a unit (Latin - - massive heap)

1 mol of particles =# of atoms in 12 gms of 12C1 mol = 6.022 x 1023

AVOGADRO’S PRINCIPLE V ~ n (constant T,P)

FINAL DEFINITIONS

ENERGY AND ENTROPY

Energy

• Started with the concepts of HEAT and WORK

• ~200 years ago these were considered to be different things and had different units

HEAT Caloric A weightless form of matter that flowed in and out of materials

WORK NEWTON Force x distance moved

FINAL DEFINITIONS

UNITED BY CONCEPT OF ENERGY, defined as the capacity to do work

1Joule = 1 kg m2 s-2

Joule - a wierd Manchester Brewer interested in the mechanical equivalent of heat

Thermodynamics grew up around the question of the transformation of energy, particularly

HEAT MECHANICAL WORK

NOTE: It’s easy to turn mechanical work into heat

Turning heat into mechanical work is much harder. Cannot turn 100% of the heat into work

THERE IS A MISSING QUANTITY

THE FIRST LAW

Conservation of EnergydE = dQ - PdV

ENTHALPY: HEAT SUPPLIED AT CONSTANT PRESSURE

Heat Q F dl = P dV(F dl = P Adl)

dl

p dQ = d H

H = E + P V

GETTING WORK OUT OF HEAT WORK HEAT - easy

HEAT WORK - harder

CARNOT - Impossible to take heat at a certain temperature and convert it to work with no other changes in the system or the surroundings

HOT

COLD

NO USEABLE WORK

USEABLE WORK

THE CONCEPT OF ENTROPY AROSE FROM THIS ANALYSIS

∆∆S =∆∆Q

T__ rev

ENTROPY - THEMODYNAMICDEFINITION

∆S = ∆Qrev

T and

∆S > ∆Q

irrev

T

From Carnot’s analysis

AnalogyW = – PdVQ = – T∆S

(note sign change - direction of Q)

i.e., S is the quantity that defines the relationship between heat and temperature

ENTROPY

. . ..

. ..

.. .. ..

16TONS

16TONS

16TONS

THUD !!

PE = mgh

heat

doesn'tlevitate !

.

.

.

. .

. .

...

. .

. . ..

. .

... .. ..

doesn't happenspontaneously

doesn't happenspontaneously

OTHER MANIFESTATIONS OF ENTROPY

Also happens spontaneously

ENERGY AND MATTER TEND TO DISPERSE CHAOTICALLY - THE SECOND LAW

HOT

Happens spontaneously

COLD

THE SECOND LAW OF THERMODYNAMICSAND FREE ENERGY

For a process to occur spontaneously,S must increase; eg mixing

∆∆Stot = ∆∆S sys + ∆∆Ssurr

But,if heat is released by the system

∆∆ G = -T∆∆S tot = ∆∆ H -T∆∆ S

Define free energy

∆∆Ssurr = - ∆∆Q sysT

= - ∆∆H sysT

FREE ENERGY

∆G → advantage of using the free energy is that it contrives a way to relate overall changes to changes in the system alone

WHY IS IT CALLED THE FREE ENERGY?

– Because it is the maximum amount ofnon-expansion work that can be obtainedfrom a system (T,P const)

∆G – ∆H → heat tax!

SUMMARY

PV = nRT∆Gm = ∆Hm - T∆Sm

NOW WHAT?

WE HAVE DISCUSSED THE ORIGIN OF THETHERMODYNAMIC EQUATIONS

NEXT STEP INTRODUCE THE MOLECULES

1. From F = ma derive the gas laws and provide a molecular machinery

2. Show that by looking at averages and distributionsof “mechanical” properties we can obtain S, Getc.

FEYNMAN - LECTURES ON PHYSICSFEYNMAN - “If in a cataclysm all human knowledge was destroyedexcept one sentence that could be passed on to future generations – – – what would contain the most information in the fewest words ?”

“ALL THINGS ARE MADE OF ATOMS – LITTLEPARTICLES THAT MOVE AROUND INPERPETUAL MOTION, ATTRACTING EACHOTHER WHEN THEY ARE A LITTLE DISTANCEAPART, BUT REPELLING UPON BEINGSQUEEZED INTO ONE ANOTHER .”

BASIC LAWS OF PARTICLES

Mechanics (various forms) Electricity + magnetism

BASIC LAWS OF STUFF

Thermodynamics

THE PROBLEM IS TO LINK THEM

PRESSURE

If there is no opposing force onthe piston, each collision movesit a little bit.

How much force, F, do we need to put on the piston to prevent movement?

First: Remember P = F/A

Second: How much force is imparted to the piston by the collisions ?

F = mx. .

= ddt (mx

. )i.e., WE NEED TO CALCULATE HOW MUCH MOMENTUM PER SEC IS DELIVERED BY THE COLLISIONS

PRESSURETHIS IS EASY!

a) Calculate momentum delivered by one collisionb) Count # of collisions/sec

Assume perfectly elastic conditions

Then the particles have + mvx momentum before - mvx momentum after

Change = mvx – (-mvx) = 2mvx

v x

PRESSURE

Now calculate the # of collisions in time t

FIRST: Define the number density

(# of particles/unit vol)

In a time t, only those particles that are close Enough and have sufficient velocity will hit the piston

Vol occupied by the molecules that will make it is

The number hitting the piston = N vx t A# per sec = N vx A

N =

nV

vx = xt

v x t A

this doesn’t get there in t

v x

v x

PRESSUREHENCE

BUT a) All molecules don’t have the same vxb) Some are moving away from the piston

F = NvxA.2mvx

P = FA

= 2 Nm v x2

S o re p l a c e v x2 w i th

12 v 2

x

P = N m

N o w

= v 2

3

v 2x

v 2yv 2

x v 2z= =

= 13 v x

2 + v y2 + v z

2 v 2x

THE IDEAL GAS LAW- KINETIC THEORY

HENCE

Recall the thermodynamic definition

TWO BODIES ARE AT THE SAME T IF THERE IS NO HEAT FLOW BETWEEN THEM.

2

3 2P =

2 N m v

2 = 23

nV

m v2

OR P V = 23 n KE

P V = n'R T COMPARED TO

IS n’ THE # OF MOLES ?IS KE ~ T ?

TEMPERATURE - KINETIC THEORY

WHAT HAPPENS AT EQUILIBRIUM?

•Forces balance•Atoms gain or lose energy depending on wether the piston is moving towards them or away from them during the collisions

FOR COLLISIONS OF PAIRS OF MOLECULES

THEN – IF TWO GASES ARE AT THE SAME T, THE MEAN KE OF THE CENTER OF MASS MOTIONS ARE EQUAL.

COLD H O T

12 m 1 v 1

2

= 12 m 2 v 2

2

High densityLow v

Low densityHigh v

PISTON

KINETIC THEORY- THE CONSTANTS k and R

HENCE

At same T, P, V, n is a constant!

Absolute scale of temp

K E ~ T

12 m v 2 = 3

2 k T

P V = n'R T

zero is w hen

= 0!

V

T

PV = 23 n = n k T1

2 m v 2

12 m v 2

IMPORTANT POINTS

•A simple consideration of the motion of particles gives a fundamental understanding of P, T, the ideal gas law, absolute T and absolute zero, etc.

•Can we stretch this approach and ultimately get a molecular interpretation of S , ∆∆G , etc.?

THE DISTRIBUTION OFENERGY AND MATTER

a) The distribution of velocities?

b) The distribution of molecules in space?

i.e., THIS WILL LEAD TO A DESCRIPTION OF ENTROPY, ∆∆G etc. IN TERMS OF STATISTICAL ARGUMENTS

All of the thermodynamic quantities we have dealtwith so far deal with how much material is presentand,on average,how fast the molecules are moving

WHAT ABOUT

EXAMPLEThe distribution of molecules in the atmosphere

ASSUMPTIONS Constant T !No wind !

i.e., if we know P, we know n, if P is a constantBUT, in the atmosphere it varies

P V = n k T o r P = Nk T ( N = nV )

hh + dh

Area A

P at (h + dh) must be less than P at h by an amount that is proportional to the weight of gas in Adh

# of moles in Adh = NAdh

ForceArea

Nadh A

=

Ph+dh - Ph = dP = - mgNdh

P = NkT, or dP = kTdN

EXAMPLE (cont.)

hh + dh

Area A

THIS IS A GENERAL RESULTBOLTZMANN’S LAW

SIMILARLY

HENCE

N = Noe -m gh/kT = Noe -P E /kT

N ~ e -P E /kT

n >u ~ e -K E /kT

ForceArea

Nadh A

=

dNdh = -

mgN kT

STATISTICAL MECHANICS

PROPERTIES OF MOLECULES

Classical mechanicsQuantum mechanics

PROPERTIES OFSTUFF (Bulk Materials)

Thermodynamics(P, T, V, G, S etc+ relationships between them)

Can bridge the “gap”by considering the average properties of all theparticles of the system and the distribution of matter and energy.

ENTROPY A measure of the distribution of energy and matter in a system

How do weget from the

molecules tobulk properties ?

o b ta i n e d P V = 23 n

m v 2

2

BOLTZMANN’S TOMBWe have seen that Entropy is associated with the distribution of energy and matter in a system.Thiscan be expressed formally in terms of the equationcarved on Boltzmann’s tomb

S = k lnW

Which,today,is normally written

S = k ln Ω

Where Ω is the number of arrangementsAvailable to the system[At a given V, E, N]

ENTROPY

S = kln ΩΩ Why does the relationship have this form ?

Consider an ordered pack of cards- 1 arrangement

Now shuffle; what is the number of Possible arrangements ?

ΩΩ = 52! (~4.45 X 10 )66

Now consider two packs shuff led separately

ΩΩ = ΩΩ1ΩΩ2

BUT S IS A THERMODYNAMIC PROPERTY THAT MUST BE ADDITIVE = S 1 + S 2

S = kln ΩΩ = kln ΩΩ1ΩΩ2= klnΩΩ1 + klnΩΩ2

FREE ENERGY REVISITED

BUT,IF HEAT IS RELEASED BY THE SYSTEM

For this to occur S must increase ie. ∆∆S > 0

∆∆Stot = ∆∆S sys + ∆∆Ssurr

∆∆ G = -T∆∆ S tot = ∆∆ H -T∆∆ S

∆∆Ssurr = - ∆∆Q sysT

= - ∆∆H sysT

WHAT YOU SHOULD KNOW

THERMODYNAMICS•Describes relationships between macroscopic properties

•2nd law: In a spontaneous process S always increases

•Free energy: What is it?

STATISTICAL MECHANICSRelates the “mechanical” properties ofatoms and molecules to macroscopic orThermodynamic quantities

EXAMPLES:PressureTemperature(Heat is motion!)

WHAT IS ENTROPY?