Upload
berenice-murley
View
222
Download
3
Embed Size (px)
Citation preview
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
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