1. Thermodynamics versus Statistical Mechanics
Both disciplines are very general, and look for description of macroscopic (many-body) systems in equilibrium
There are extensions (not rigorously founded yet) to non-equilibrium processes in both
But thermodynamics does not give definite quantitative answers about properties of materials, only relations between properties
Statistical Mechanics gives predictions for material properties
Thermodynamics provides a framework and a language to discuss macroscopic bodies without resorting to microscopic behaviour
Thermodynamics is not strictly necessary, as it can be inferred from Statistical Mechanics

2. in this case system is isolated
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 macroscopically
by 3 variables:
N, number of particles (Nm number of moles)
V, volume
E, internal energy
only sometimes
in this case system is isolated

3. N, V, E (simple system) p, T, m (simple system)
Types of thermodynamic variables:
Extensive: proportional to system size
N, V, E (simple system)
Intensive: independent of system size
p, T, m (simple system)
Not all variables are independent. The equations of state relate the variables:
f (p,N,V,T) = 0
For example, for an ideal gas or
No. of particles
No. of moles
Boltzmann constant
1.3805x10-23 J K-1
Gas constant
J K-1 mol-1

4. Any 3 variables can do. Some may be more convenient than others
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)
in this case system interchanges energy with surroundings
in this case system is isolated
Thermodynamic limit:

5. 1.2 Laws of Thermodynamics
in an isolated system
Thermodynamics is based on three laws
proportional to system size
1. First law of thermodynamics
Energy, E, is a conserved and extensive quantity
change in energy involved in infinitesimal process
amount of heat transferred to the system
mechanical work done on the system
(hidden)
SYSTEM
(explicit)

6. 1.2 Laws of Thermodynamics
Thermodynamics is based on three laws
inexact differentials
W & Q do not exist (not state functions)
exact differential
E does exist (it is a state function)
1. First law of thermodynamics
Energy, E, is a conserved and extensive quantity
change in energy involved in infinitesimal process
amount of heat transferred to the system
mechanical work done on the system
(hidden)
SYSTEM
(explicit)

7. independent of system size
Thermodynamic (or macroscopic) work
independent of system size
are conjugate variables (intensive, extensive) xi
intensive variable m -p -H
... Xi
extensive variable N V M
xidXi
mdN
-pdV
-HdM

8. surroundings SYSTEM In mechanics: where and F is a conservative force
(hidden)
(explicit)
In fact dE = dWtot= dW + dQ
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 dQ

9. surroundings SYSTEM (hidden) (explicit) In fact dE = dWtot= dW + dQ
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 dQ

10. F = external force A gas system’s pressure = F / A
volume change in slow compression
mechanical work (through macroscopic variable V):
gas if
heat transfer (through microscopic variables):
molecules in base of container get kinetic energy from fire, and transfer energy to gas through conduction (molecular collisions)

11. HEAT ENGINE the system adsorbs heat from reservoir 1
the system transfers heat to reservoir 2
the system performs work
HEAT ENGINE

12. Thermodynamic process
Equilibrium state
A state where there is no change in the variables of the system
(only statistical mechanics gives a meaningful, statistical definition)
Thermodynamic process
A change in the state of the system from one equilibrium state to another
It can viewed as a trajectory in a thermodynamic surface defined by the equation of state
For example, for an ideal gas
final state
reversible path
initial state
specific volume

13. quasistatic process reversible process irreversible process
a process that takes place so slowly that equilibrium can be assumed at all times. No perfect quasistatic processes exist in the real world
reversible process
a 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
the wall separating the two parts is slightly non-adiabatic (slow flow of heat from left to right)
T1 > T2
A quasistatic process is not necessarily reversible
irreversible process
unidirectional process: once it happens, it cannot be reversed spontaneously

14. work done by the system along the cycle
Calculation of work in a process
The work done on the system on going from state A to state B is
work done by the system -
One has to know the equation of state p = p (v,T) of the substance
work done by the system along the cycle
In a cycle DE = 0 but
Therefore:
the heat adsorbed by the system is equal to the work done by the system on the environment

15. Types of processes Isochoric: there is no volume change
isobaric
isochoric
Isobaric: no change in pressure
is the enthalpy.
Also:
important in chemistry and biophysics where most processes are at constant pressure (1 atm)

16. Isothermal: no change in temperature, i.e. dT = 0 For an ideal gas
adiabatic
isothermal
(ideal gas)
Adiabatic: no heat transfer, i.e. dQ = 0
Adiabatic heating
If the system contracts adiabatically W>0 and E increases
(for an ideal gas this means T increases: the gas gets hotter)
Adiabatic cooling
If the system expands adiabatically W<0 and E decreases
(for an ideal gas and many systems this means T decreases: the gas gets cooler)

17. for an ideal gas and many other systems this means DT<0
Adiabatic cooling
Dp<0
DE<0
for an ideal gas and many other systems this means DT<0
isotherm
isotherm
work done by the system VA VB

18. 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
In an infinitesimal process
The equality holds for reversible processes; if process is irreversible, the inequality holds

19. the internal wall is removed
Example of irreversible process
isolated system
the internal wall is removed
ideal gas
expanded gas
V/2
V/2 V
Arrow of time
DS can be easily calculated using statistical mechanics
entropy of ideal gas in volume V
entropy of ideal gas in volume V/2

20. The entropy of an ideal gas is
entropy before:
entropy after:
entropy change:
The inverse process involves DS<0 and is in principle prohibited

21. time evolution from non-equilibrium state
(N,V,E)
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
time evolution from non-equilibrium state
at equilibrium it is a function
it is a monotonic function of E
An adiabatic process involves changes in hidden microscopic variables at fixed (N,V,E). In such a process
maximum

22. 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):
equations of state
Since S increases monotonically with E, it can be inverted to give E = E(N,V,S)
equations of state
entropy representation of thermodynamics
energy representation of thermodynamics

23. Equivalent (more utilitarian) statements of 2nd law
Historically they reflect the early understanding of the problem
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)
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)
Lord Kelvin
Clausius
As a corollary: the most efficient heat engine operating between two reservoirs at temperatures T1 and T2 is the Carnot engine
Carnot

24. alternative statement of 2nd law
S is connected to the energy transfer through hidden degrees of freedom, i.e. to dQ. In a process the entropy change of the system is
alternative statement of 2nd law
reversible process
where
irreversible process
If dQ > 0 (heat from environment to system) dS > 0
In a finite process from A to B:
For reversible processes T-1 is an integrating factor, since DS only depends on A and B, not on the trajectory

25. A clearer explanation of entropy was given
The name entropy was given by Clausius in 1865 to a state function whose variation is given by dQ/T along a reversible process
A clearer explanation of entropy was given
Clausius
by Boltzmann in terms of probability arguments in 1877 and then by Gibbs a few years later:
It can be shown that this S corresponds to the thermodynamic S
Gibbs
Wahrscheindlichkeit
(probability)
where pi is the probability of the system being in a microstate i
CONNECTION WITH ORDER
More order means less states available
If all microstates are equally probable (as is the case if E = const.) then pi =1/W, where W is the number of microstate of the same energy E, and
Boltzmann

26. entropy change of system may be positive or negative
Does S always increase? Yes. But beware of environment... In general, for an open system:
<
entropy change of environment
>
entropy change due to internal processes
entropy change due to interaction with environment
entropy change of system may be positive or negative
e.g. living beings...

27. S increases if the system absorbs heat, otherwise S decreases
Processes can be discussed profitably using the entropy concept.
For a reversible process:
If the reversible process is isothermal:
S increases if the system absorbs heat, otherwise S decreases
If the reversible process is adiabatic:
Reversible isothermal processes are isentropic
But in irreversible ones the entropy may change

28. heat absorbed by system
In a finite process:
(depends on the trajectory)
In a cycle:
work done by the system in the cycle Q Q
DS = 0
heat absorbed by system

29. Q=-W CARNOT CYCLE T1 T2 Change of entropy:
Isothermal process. Heat Q1 is absorbed T1
Adiabatic process. No heat
Adiabatic process. No heat
Q=-W T2
Change of entropy:
Isothermal process. Heat Q2 is released
it is impossible to perform a cycle with W 0 and Q2 = 0

30. Efficiency of a Carnot heat engine:
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