# CML100 Physical Lectures I II

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### Transcript of CML100 Physical Lectures I II

28-Jul-15 5

Thermodynamics Variables: , P, T, V, M, E, S, Avogadro number of particles, NA=6.023 X 10

23

Thermodynamic Limit: N , V

In Statistical Thermodynamics, we do care about the molecular details, establish connection between molecular details (microscopic states) and macroscopic properties.

In Thermodynamics, we talk only about Macroscopic properties, dont pay attention to molecular details of the system

Thermodynamic Equilibrium State: A state at which macroscopic properties stop changing

System: A system is the part of the world in which we are interested.

System

Surroundings

Boundary or wall

System

Boundary or wall

Surroundings Surroundings

Surroundings

Surroundings: the rest is surrounding.

SYSTEM AND SURROUNDING

Boundary could be real, imaginary; thermally conducting, insulating, rigid, flexible, permeating, non-permeating etc.

Surroundings

OpenSurroundings

Surroundings

Closed System

IsolatedSystem

Diathermal /Adiabatic Walls

Adiabatic Container

Endothermic

Exothermic

Diathermal Container

Open ~IsolatedClosed

System is Coffee

28-Jul-15 10

Suppose an object A (which we can think of as a block of iron) is in thermal equilibrium with an object B (a block of copper), and that B is also in thermal equilibrium with another object C (a ask of water). Then it has been found experimentally that A and C will also be in thermal equilibrium when they are put in contact. This observation is summarized by the Zeroth Law of thermodynamics as:

If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A. The Zeroth Law justies the concept of temperature and the use of a thermometer, a device for measuring the temperature.

Zeroth Law of Thermodynamics and Thermal Equilibrium

If TA=TB and TB=TC, then TC=TA

To achieve thermal equilibrium the objects have to be in contact through a diathermic boundary

First Law of Thermodynamics, Internal Energy, Enthalpy, and Heat Capacities

First Law of Thermodynamics

The internal energy of an isolated system is constant (or conserved).

U=Constant

SSYSTEM

U

SURROUNDINGS

ADIABATIC/INSULATING WALL

SUR

RO

UN

DIN

GS

SUR

RO

UN

DIN

GS

SURROUNDINGS

Isolated system

ADIABATIC/INSULATING WALL

System: GAS, LIQUID, SOLID,

for Isolated system

wqUSys ddd

Supply heat

Do someWork

To change the energy of the system by amount dU:

SSYSTEM

U

SURROUNDINGS

ADIABATIC WALL AD

IAB

ATIC

WA

LL

AD

IAB

ATI

C W

ALL

ADIABATIC WALL

SUR

RO

UN

DIN

GS

SUR

RO

UN

DIN

GS

SURROUNDINGS

wd qd and/orS

SYSTEMU+dU

SURROUNDINGS

ADIABATIC WALL AD

IAB

ATIC

WA

LL

ADIABATIC WALL

SUR

RO

UN

DIN

GS

SURROUNDINGS

It has been found experimentally that the internal energy of a system may be changed either by doing work on the system or by heating it.

Heat and work are equivalent ways of changing a systems internal energy.

SSYSTEMU+dUdU >0

SURROUNDINGS

ADIABATIC WALL AD

IAB

ATIC

WA

LL

ADIABATIC WALL

SUR

RO

UN

DIN

GS

SURROUNDINGSSSYSTEM

U

SURROUNDINGS

ADIABATIC WALL AD

IAB

ATIC

WA

LL

AD

IAB

ATI

C W

ALL

ADIABATIC WALL

SUR

RO

UN

DIN

GS

SUR

RO

UN

DIN

GS

SURROUNDINGSS

SYSTEMU+dUdU >0

ADIABATIC WALL AD

IAB

ATIC

WA

LL

ADIABATIC WALL

SUR

RO

UN

DIN

GS

SURROUNDINGS

Dia

the

rmal

Wal

l

wd

dq

If energy of the system increases, then U is +ve

If energy of the system decreases, then U is -ve

If work is done on the system, then w is +ve

If work is done by the system, then w is ve

If heat is supplied to the system, then q is +ve

If heat is liberated by the system, then q is -ve

WORK

HEAT

WORK

HEAT

ENERGY, U

w < 0

w > 0

q < 0

q > 0

The Sign Convention

i.e. internal energy plus the surrounding energy is also conserved.

wq dddUSys

Supply heat Do some Work

Work done or heat supplied is by the surroundings, so

)dd(dUSurr wq

0dUdU SurrSys

ConstantUU SurrSys

Different forms of FirstLaw of Thermodynamics

ConstantUUniverse Universe is an isolated system

wqU ddd Sys Differential form of First Law

Function)(Path d

Function)(Path - d

function) state is (U dU

12

2

1

12

2

1

12

2

1

Sys

-wwww

qqqq

UUU

wqU Sys

Integrated form of First Law

Exact Inexact Inexact

2

1

2

1

2

1

Sys ddd wqU

Total change in internal energy when the system goes from state 1 to 2 is

Thermodynamic Variables

State Function Path Function

PATH 1

State A State B

System System

321 www (UB-UA)1= (UB-UA)2= (UB-UA)3

PA, VA, TA, nAPB, VB, TB, nB

PATH 2

PATH 3

For fixed number of moles: nA=nB

Depends on how the process has been carried out from one

state to another. Example: Work

Depends on the states (initial and final), does matter

how those states have been attained. Example:

Internal Energy

Between two states the change in a state variable is always the same regardless of the

path the system travels.

Differential of a state function is called exact differential, df in this case.

fffdf AB

AB

A

B

AB fffd Differential of a path function is called inexact differential, df in this case.

Exact Differential and State Function:

Inexact Differential and Path Function:

df=Infinitesimal change in f

f=Macroscopic change in f

A

D

C

B

B

A

fdfdfdfdf 0... Cyclic Integral is zero for state function

A

D

C

B

B

A

fdfdfdfd ... will depend on the path

xa xb

yb

ya a

bExamples: (i)dz=ydx

(ii)dz=ydx+xdy

AREA I

AREA II

b

a

b

a

ydxdz I Area

II AreaI Area)( b

a

b

a

xdyydxdz

AREA I and AREA II are path dependentAREA I+AREA II is not path dependent i.e. the sum of these areas is independent of the shape of the curve (path).

Test to know about exact and inexact differentials

dyyxNdxyxMdz ),(),(

dyy

zdx

x

zdz

xy

yx

zyxM

),(

xy

zyxN

),(

yxxyy

z

xx

z

y

yxx

N

y

M

Exact or not ??Examples: (i)dz=ydx

(ii)dz=ydx+xdy

Consider, as if dz was exact then

As mixed partial derivatives are equal

(1)

(2)

Comparing (1) and (2),

Then has to be satisfied if dz in Eq. 1 is exact.

Eulers Criteria for exactness

and

o Inexact differentials can be made exact by multiplication of integrating factor.dq is inexact but dq/T is exact!

o Sum of two inexact differentials can be an exact differential.dz=ydx+xdy is exact but ydx and xdy are inexact differentials.

ENERGY, WORK, AND HEAT

Work and Heat are two forms of energy transfer

Mechanical

Electrical

Thermal

Energy (U) is State Function Heat (q) and Work(w) Both are Path Functions, not state functions

UUUdU AB

AB

A

B

AB wwdw

A

B

AB qqdq

coordinatereaction

theoft independen ;)( FF

Work done GeneralizedForce

GeneralizedReaction Coordinate

)(2

1

2

1

FdFwdw

Generalized Work

Examples:

dQfdLddVPqdU ext d

When volume expansion, surface expansion, elongation or electrical work are involved, first law can be written as

2

1

2

1

2

1

2

1

2

1

dQfdLddVPqdUU ext

Reversible and Irreversible Process

fA fA +df fA +2df fA +3df fB

fA fA +df fA +2df fA +3df fB

Reversible Process: Changes in the system are brought in infinitesimal amount step by step, so that the system can adjust the changes.

Isothermal Reversible P-V Work for Compression

A reversible change in thermodynamics is a change that can be reversed by an infinitesimal modification of a

variable.

PPP Gasex

External pressure is changed by infinitesimal amount that at every step the pressure exerted by the gas is equal to external pressure i.e. at each step one ensures mechanical equilibrium

S

DIATHERMAL

SUR

RO

UN

DIN

GS

M

L

S

DIATHERMAL

M

L

m S

DIATHERMAL

SUR

RO

UN

DIN

GS

M

L

m

2/ LMgPex 2/)( LgmMPex

2/)2( LgmMPex

m

GAS GASGAS

Reversible and Irreversible Process

PPP Gasex

S

DIATHERMAL

SUR

RO

UN

DIN

GS

M

L

S

DIA

THER

MA

LDIATHERMAL

SUR

RO

UN

DIN

GS

M

L

2/ LMgPP exi 2/)( LgMMPP exf

GAS GAS

M

Isothermal Irreversible Compression against Constant External Pressure

VPVVPdVPw exifex

V

V

ex

f

i

)(

)( if VVV

Vf ,T

Vi ,T

Vf Vi

P

V

Pf

Pi

Pex

irrevifexirrev AreaVVPw )(

Areairrev

Reversible and Irreversible Process

When a piston of area A moves out through a distance dz, it sweeps out a volume

The work required to move an object a distance dz against an opposing force of magnitude F is

dw=|F|dz

|F|=Pex A

dV=Adz

The external pressure Pex is equivalent to a weight pressing on the piston, and the force opposing expansion is

When the system expands through a distance dz against anexternal pressure Pex, it follows that the work done is

dw=PexAdz=-PexdV Total work done when the volume changes from Vi to Vf

f

i

V

V

exdVPw

Pex

P

of the Piston

Force (F) exertedby the piston

|F| is the magnitude of force exerted by the pistonbecause of its mass, lets say.

wqU ddd Sys

2

1

d

dVPqU

dVPqdU

ext

ext

First law for isothermal expansion

First law for adiabatic process

2

1

0d

dVPU

dVPdU

q

ext

ext

Change in Temperature of theSystem is expected in an adiabaticprocess

Supporting Slides

Work is a Path Function

SS

Free Expansion or Expansion in Vacuum

0 f

i

V

V

exdVPw0exP

Expansion/Compression against Constant External Pressure

VPVVPdVPw exifex

V

V

ex

f

i

)( )( if VVV

veiswssionfor CompreVVV

veiswionfor ExpansVVV

if

if

0)(

0)(

Vf Vi

P

V

Pf

Pi

Pex

Isothermal Reversible P-V Work for Compression

rev

V

V

exrev AreadVPwf

i

Arearev < Areairrev

ArearevWork done on the gas in reversiblecompression is less than that in irreversiblecompression.

Adiabatic Process (Compression/Expansion)

System

Surroundings

ADIABATIC

AD

IAB

ATI

C ADIA

BA

TIC

ADIABATIC Adiabatic Wall (Isolated system)

No heat is allowed to be transferred between System and Surroundings

dq=0

Change in Temperature of the System is expected. Pex=Pmechanical

dVPwddUdU exSys

Adiabat and Isotherm

T1

T1

T2

System

ADIABATIC

AD

IAB

ATI

C AD

IAB

ATIC

ADIABATIC

System

ADIABATIC

AD

IAB

ATIC A

DIA

BATIC

ADIABATIC

(P1, V1, T1) (P2, V2, T2)

The P-V curve for reversible adiabatic process are steeper than those of isothermal process

Adiabatic Cooling Adiabatic Heating