Thermodynamics versus Statistical Mechanics

Thermodynamics versus Statistical Mechanics

1. Both disciplines are very general, and look for description of macroscopic (many-body) systems in equilibrium

2. There are extensions (not rigorously founded yet) to non-equilibrium processes in both

3. But thermodynamics does not give definite quantitative answers about properties of materials, only relations between properties

4. Statistical Mechanics gives predictions for material properties

5. Thermodynamics provides a framework and a language to discuss macroscopic bodies without resorting to microscopic behaviour

6. Thermodynamics is not strictly necessary, as it can be inferred from Statistical Mechanics

1. Review of Thermodynamic and Statistical Mechanics

This is a short review

1.1. Thermodynamic variables

We will discuss a simple system:

• one component (pure) system

• no electric charge or electric or magnetic polarisation

• bulk (i.e. far from any surface)

The system will be characterised macroscopicallyby 3 variables:

• N, number of particles (Nm number of moles)

• V, volume

• E, internal energy

only sometimes

in this case system is isolated

Types of thermodynamic variables:

• Extensive: proportional to system size

• Intensive: independent of system size

Not all variables are independent. The equations of state relate the variables:

f (p,N,V,T) = 0

For example, for an ideal gas

NkTpV or

N, V, E (simple system)

p, T, (simple system)

RTNpV m

Boltzmann constant1.3805x10-23 J K-1

No. of particlesGas constant

8.3143 J K-1 mol-1

No. of moles

Any 3 variables can do. Some may be more convenient than others. For example, experimentally it is more useful to consider T, instead of E (which cannot be measured easily)

Thermodynamic limit:

V

NVN , ,

in this case system is isolated

in this case system interchanges energy with surroundings

1.2 Laws of Thermodynamics

Thermodynamics is based on three laws

1. First law of thermodynamics

SYSTEM

Energy, E, is a conserved and extensive quantity

QWdE

(hidden)(explicit)

change in energy involved in

infinitesimal processmechanical work

done on the system

amount of heat transferred to the

system

proportional to system size

in an isolated system

1.2 Laws of Thermodynamics

Thermodynamics is based on three laws

1. First law of thermodynamics

SYSTEM

Energy, E, is a conserved and extensive quantity

QWdE

(hidden)(explicit)

change in energy involved in

infinitesimal processmechanical work

done on the system

amount of heat transferred to the

system

inexact differentialsW & Q do not exist (not state functions)

exact differentialE does exist (it is a

state function)

Thermodynamic (or macroscopic) work

...2211 dXxdXxdXxWi

ii

ii Xx , are conjugate variables (intensive, extensive)

xi

intensive

variable

-p -H ...

Xi

extensive

variable

N V M ...

xidXi dN -pdV -HdM ...

independent of system size

(explicit)(hidden)SYSTEM

surroundings

In fact dE = dWtot= W + Q

Only the part of dWtot related to macroscopic variables can be

computed (since we can identify a displacement). The part related

to microscopic variables cannot be computed macroscopically

and is separated out from dWtot as Q

01 EEdE 0

00

0

In mechanics:

where

and F is a conservative force

dEdW

01

1

0

EErdFW

(explicit)(hidden)SYSTEM

surroundings

01 EEdE 0

00

0

In fact dE = dWtot= W + Q

Only the part of dWtot related to macroscopic variables can be

computed (since we can identify a displacement). The part related

to microscopic variables cannot be computed macroscopically

and is separated out from dWtot as Q

A

system’s pressure = F / AF = external force

volume change in slow compression

• mechanical work (through macroscopic variable V):

• heat transfer (through microscopic variables):

molecules in base of container get kinetic energy from fire, and transfer energy to gas through conduction (molecular collisions)

gas0 pdVW if 0dV

0Q

the system performs work

the system adsorbs heat

from reservoir 1

the system transfers heat to

reservoir 2

HEAT ENGINE

Equilibrium state

A state where there is no change in the variables of the system(only statistical mechanics gives a meaningful, statistical definition)

A change in the state of the system from one equilibrium state to another

Thermodynamic process

0,, kTpvTvpf

/1/ NVvspecific volume

It can viewed as a trajectory in a thermodynamic surface defined by the equation of state

For example, for an ideal gasinitial state

final statereversible

path

Tvpf ,,

• quasistatic processa process that takes place so slowly that equilibrium can be assumed at all times. No perfect quasistatic processes exist in the real world

• irreversible processunidirectional process: once it happens, it cannot be reversed spontaneously

• reversible processa process such that variables can be reversed and the system would follow the same path back, with no change in system or surroundings. The system is always very close to equilibrium

A quasistatic process is not necessarily reversible

the wall separating the two parts is slightly non-adiabatic (slow flow of heat from left to right)

T1 > T2

B

A

V

V

AB pdVW

Calculation of work in a process

The work done on the system on going from state A to state B is

One has to know the equation of state p = p (v,T) of the substance

In a cycle E = 0 but 0W

pdVWQ

-

work done by the system

work done by the system along the cycle

Therefore:

the heat adsorbed by the system is equal to the work done by the system

on the environment

Types of processes• Isochoric: there is no volume change

00 WdV

• Isobaric: no change in pressure

dQQdE

dHpVEddEpVddEpdVQ

pVEH is the enthalpy.

B

A

V

V

AB VpVVppdVW

HdHQ Also: important in chemistry and

biophysics where most processes are at constant

pressure (1 atm)

isobaric

isoc

hori

c

QdQE

• Isothermal: no change in temperature, i.e. dT = 0 For an ideal gas

WQdENkTE 02

3

WQ (ideal gas)

WEdWdE

Adiabatic coolingIf the system expands adiabatically W<0 and E decreases

(for an ideal gas and many systems this means T decreases: the gas gets cooler)

Adiabatic heatingIf the system contracts adiabatically W>0 and E increases

(for an ideal gas this means T increases: the gas gets hotter)

isothermal

• Adiabatic: no heat transfer, i.e. Q = 0

adiabatic

isotherm

isotherm

Adiabatic cooling

work done by the system

p

E<0for an ideal gas and many other

systems this means T<0

VA VB

2. Second law of thermodynamics

There is an extensive quantity, S, called entropy, which is a state function and with the property that

In an isolated system (E=const.), an adiabatic process from state A to B is such that

BA SS

In an infinitesimal process 0dS

The equality holds for reversible processes; if process is

irreversible, the inequality holds

S can be easily calculated using statistical mechanics

the internal wall is

removedideal gas

expanded gas

Example of irreversible process

0initialfinal SSS

entropy of ideal gas in volume V entropy of ideal gas in volume V/2

V/2 VV/2 Arrow of time

isolated system

The entropy of an ideal gas is N

VvvTNkSS ,log 2/3

0

• entropy before:

2/3

0initial 2log T

vNkSS

• entropy after:

2/30final log vTNkSS

2loginitialfinal NkSSS • entropy change:

The inverse process involves S<0 and is in principle prohibited

• at equilibrium it is a function

• it is a monotonic function of E

),,( EVNSS iEVNSS ;,,

The existence of S is the price to pay for not following the hidden degrees of freedom.

It is a genuine thermodynamic (non-mechanical) quantity

An adiabatic process involves changes in hidden microscopic variables at fixed (N,V,E). In such a process

maximum

(N,V,E)time evolution from

non-equilibrium state

EVENVN N

S

TV

S

T

p

E

S

T ,,,

,,1

S is a thermodynamic potential: all thermodynamic quantities can be derived from it (much in the same way as in mechanics, where the force is derived from the energy):

Since S increases monotonically with E, it can be inverted to give E = E(N,V,S)

SVSNVN N

E

V

Ep

S

ET

,,,

,,

),,( EVNSS

),,( SVNEE

entropy representation of thermodynamics

energy representation of thermodynamics

equations of state

equations of state

Equivalent (more utilitarian) statements of 2nd law

Kelvin: There exists no thermodynamic process whose sole effect is to extract heat from a system and to convert it entirely into work (the system releases some heat)

As a corollary: the most efficient heat engine operating between two reservoirs at temperatures T1 and T2 is the Carnot engine

Clausius: No process exists in which the sole effect is that heat flows from a reservoir at a given temperature to a reservoir at a higher temperature(work has to be done on the system)Clausius

Lord Kelvin

Carnot

Historically they reflect the early understanding of the problem

S is connected to the energy transfer through hidden degrees of freedom, i.e. to Q. In a process the entropy change of the system is

T

QdS

where

reversible process

irreversible process

If Q > 0 (heat from environment to system) dS > 0

In a finite process from A to B: B

A T

QS

For reversible processes T-1 is an integrating factor, since S only depends on A and B, not on the trajectory

alternative statement of

2nd law

The name entropy was given by Clausius in 1865 to a state function whose variation is given by dQ/T along a reversible process

i

ii ppkS log

where pi is the probability of the system being in a

microstate iIf all microstates are equally probable (as is the case if E = const.) then pi =1/, where is the number of

microstate of the same energy E, and

loglog11

log1

kkkSi

It can be shown that this S corresponds to the thermodynamic S

by Boltzmann in terms of probability arguments in 1877 and then by Gibbs a few years later:

CONNECTION WITH ORDER

More order means less states available

Wahrscheindlichkeit(probability)

Gibbs

A clearer explanation of entropy was given Clausius

Boltzmann

Does S always increase? Yes. But beware of environment...In general, for an open system:

0env dSdSdSdS ei

0idSedS

entropy change due to internal processes

entropy change due to interaction with

environment

>< 0 entropy change of

environment

entropy change of systemmay be positive or negative e.g. living beings...

Processes can be discussed profitably using the entropy concept.For a reversible process:

B

A

AB T

QSSS

• If the reversible process is isothermal:

STQT

QQ

TSSS

B

A

AB 1

S increases if the system absorbs heat, otherwise S decreases

00 ST

QSSS

B

A

AB

Reversible isothermal processes are isentropicBut in irreversible ones the entropy may change

• If the reversible process is adiabatic:

B

A

TdSQIn a finite process: (depends on the trajectory)

In a cycle:

WTdSQ work done by the system in the cycle

Q

heat absorbed by system

Q

S = 0

CARNOT CYCLE

Q=-W

Isothermal process. Heat Q1 is absorbed

Adiabatic process. No heat

0 BCS

11 /TQSAB

Isothermal process. Heat Q2 is released

22 /TQSCD

Change of entropy:

02

2

1

1 T

Q

T

QS

2

2

1

1

T

Q

T

Q 01 21

2

1

2

1 QQT

T

Q

Q

T1

T2

Adiabatic process. No heat

0 DAS

it is impossible to perform a cycle with

W 0 and Q2 = 0

Efficiency of a Carnot heat engine:

1systemby absorbedheat

systemby donework

Q

W

1

2

121

1221 1T

T

SST

SSTT

Carnot theorem:

The efficiency of a cyclic Carnot heat engine only depends on the operating temperatures (not on material)

By measuring the efficiency of a real engine, a temperature T2

can be determined with respect to a reference temperature T1