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Thermodynamics
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28-Jul-15 5
• Thermodynamics Variables: ρ, P, T, V, M, E, S,…• Avogadro number of particles, NA=6.023 X 1023
• 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, don’t 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 flask 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 justifies 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 system’s internal energy.
S
SYSTEMU+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 ΔU is +ve
If energy of the system decreases, then ΔU 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 A
B
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
Δf=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
),(
yxxy y
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.
Euler’s 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 A
B
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
f’A f’A +df’ fA +2df’ fA +3df’ … f’B
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
Arearev
Work done on the gas in reversiblecompression is less than that in irreversiblecompression.
Adiabatic Process (Compression/Expansion)
System
Surroundings
ADIABATIC
AD
IAB
ATI
C AD
IAB
ATIC
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
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