Transcript thermodynamic states

Interfacial Physics and Thin-Film Processing
A7. Basics of Thermodynamics
Fall, 2013
Instructor: J.-W. John Cheng
Mech. Engr. Dept., Nat’l Chung Cheng Univ.
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Outline

1. Thermometer and Zeroth Law

2. Heat (Enthalpy) and 1st Law

3. Entropy and 2nd law

4. Gibbs free energy and equilibrium

5. Chemical potential

References


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[BS99] Bowley, R. and Sanchez, M., Introductory Statistical Mechanics,
2nd ed., 1999, Ch 1 & Ch 2
[MS95] Moran, M. J. and Shapiro, H. N., Fundamentals of Engineering
Thermodynamics, 3rd ed., 1995, Ch 6 & Ch 14
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1. Thermometer and
Zeroth Law
3
Temperature

First, we postulate existence of temperature
through experience

In fact, we have T = f(P, V)


“This is a very powerful mathematical statement: it says that
the temperature has a unique value for any choice of pressure
and volume; it does not depend on any other quantity.” (p. 4,
BS99)
Thermal equilibrium



When 2 systems A and B are brought into contact and
there is no long a flow of energy between them,
A and B are said to be in thermal equilibrium.
The apparent observation for thermal equilibrium is
that A and B have same temperature.
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Zeroth Law and Thermometer

Zeroth law



If systems A and B are separately in thermodynamic
equilibrium with system C,
then systems A and B are in thermodynamic
equilibrium with each other.
Thermometer – an implication of Zeroth law



If we want to know if A and B are at the same
temperature, we do not need to bring two into contact
It can be answered by observing if they are individually
in thermal equilibrium with a third body.
This third body is usually a thermometer (溫度計)
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3 Aspects of Thermodynamic Equilibrium

Thermal equilibrium (defined previously)

Mechanical equilibrium


Chemical equilibrium


A condition of balance maintained by force balance
See next page for description
Thermodynamic equilibrium


A system is in thermodynamic equilibrium when thermal,
mechanical, and chemical equilibria have been reached.
At thermodynamic equilibrium, the system has welldefined temperature, pressure, and chemical potential.
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Chemical Equilibrium


Chemical equilibrium concerns systems where the #
of particles can change
E.g. a chemical reaction
C+D

If there are too much C and D, the reaction proceeds to form CD

If there are too much CD, the reaction proceeds to form C and D


CD
In chemical equilibrium, there is a balance between these two
rates of reaction, so the numbers of #’s of C, D, and CD remain
constant.
E.g. phase changes

Water and ice co-exist at a temperature around 0oC
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Thermodynamic Coordinates (States)

Thermodynamic Coordinates/States

When in thermodynamic equilibrium, properties of the
system only depend on thermodynamic ‘coordinates’,


E.g.,



such as the pressure and volume;
consider a pure gas with no chemical reactions between gas
particles and having constant number of particles
T = f(P, V)
Thermodynamic coordinates are more commonly
referred to as thermodynamic states
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Functions and Equations of States

Function of States and Equation of States

When a quantity only depends on the present value of
thermodynamic coordinates

such as the pressure and volume as that of the temperature
T = f(P, V) shown above

we say that the quantity is “a function of states” and

the governing equation “an equation of states.”

Generally, equations of states are very complicated
and do not give rise to a simple math formula.

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The ideal gas is an exception. PV = nRT  T = (PV)/(nR)
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Some Definitions

Theory of thermodynamics


Adiabatic wall


An ideal heat-insulating wall
Thermally isolated


Is concerned with systems of a large number of
particles which are contained in a vessel of some kind.
Referring to a system is surrounded by adiabatic walls
Diathermal

Referring to a system which allows energy to pass
through its walls
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Some Definitions

contd
Isothermal

Any two systems in thermal equilibrium with each
other are called isothermal to each other
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Exact Differential (A Math Tool)

Consider a function of states, G = g(x, y)

The total derivative is defined dG as
G
G
dG 
dx 
dy
x
y

Expressing dG = A(x,y)dx + B(x,y)dy, we have
A B

y x

Conversely, a change dG =A(x,y)dx + B(x,y)dy is
called an exact differential if we have
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A B

y x
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Inexact Differential

Inexact Differential

When a change dG =A(x,y)dx + B(x,y)dy with
A B

y x


The change is called an inexact differential
To differentiate from exact differential, we will
put a bar on top of it
dG  A( x, y)dx  B(x, y)dy
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Application of Exact Differential

Consider a 2-D force field



F  Fx ex  Fy ey

Q. When does there exist a potential function
u(x,y) s.t.

F  u
Hint:
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du   Fx dx    Fydy 
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2. Heat (Enthalpy) and 1st Law
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First Law

Internal energy U is a function of state.


It includes potential energy, kinetic energy, and others
First law

Energy is conserved if heat Q is taken into account.


U = W + Q
U: internal energy, W: external work, Q: external heat
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Enthalpy (焓)

Consider a system

Under a pressure P, a change dV implies that external
world does work to the system by -PdV

Resulting in an increase in internal energy of the system.

This change in internal energy is described as follows
U  Q  W  dU  dQ  PdV

Enthalpy (under constant pressure)

Note the bar on top
of Q reminds us
that dQ is not an
exact differential
Above deduction from 1st law of thermodynamics implies
a new useful variable, the enthalpy H = U + PV
P const
dU  dQ  PdV  dQ  dU  PdV  d( U  PV)
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 dQ
H  U  PV

dH
(for isobaric process)
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Heat Capacities

General concept of heat capacity

The amount of heat absorption dQ required for dT increase in temperature of
the system
C


dQ
dT
Note that dQ is condition dependent; different condition gives rise to
different value of C
Cv: heat capacity at constant volume
V  const
dU  dQ  W  dQ  PdV  dQ
 CV 

dQ
 U 


dT V  T  V
CP: heat capacity at constant pressure
dU  dQ  W  dQ  PdV  dQ  dH P
 CP 

dQ
 H 


dT P  T  P
Note CP  CV

The difference is small for liquid and solid in comparison to that for gas because
gas expands significantly with temperature increase
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3. Entropy and 2nd Law
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Motivation for Entropy

1st law states
dQ  dU  PdV


Note that external heat change is an inexact differential
It is interesting to know that


by multiplying an integrating factor an inexact
differential can sometimes become exact
E.g.
df  z 3dy  (2z  yz 2 )dz

e yz df  z 3dy  (2z  yz 2 )dz
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

* e yz df  d z 2e yz


* Performing a path integration of eyzdf along a particularly chosen path
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Motivation for Entropy

A quest


contd
“Can we find an integrating factor which multiplies dQ
and produces an exact differential?
If we can do this then we can construct a new function
of states and call it the entropy.” (p.25)
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Entropy of Ideal Gas

Ideal gas is

A collection of n moles of gas molecules whose internal
energy is the total kinetic energy of the gas
U

3nRT
2
and satisfy the following equation of states
PV  nRT

1st law says
3nR
dV
dQ  dU  PdV 
dT  nRT
2
V
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Entropy of Ideal Gas

contd
Consider an integrating factor, 1/T
1
3nR dT
dV
dQ 
 nR
T
2 T
V

f
i
Tf 3nR dT
Vf
1
dV
dQ  
  nR
Ti
Vi
T
2 T
V
 Vf
3nR  Tf 

ln    nR ln 
2
 Ti 
 Vi



 3nR
  3nR


ln Tf  nR ln Vf   
ln Ti  nR ln Vi 
 2
  2

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Independent of
process path!
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Entropy of Ideal Gas

contd
Above integration result implies existence of new
function of states, S
1
Sf  Si   dQ
i T
 3nR
  3nR


ln Tf  nR ln Vf   
ln Ti  nR ln Vi 
 2
  2

f

Entropy S of ideal gas at temperature T & volume V
w.r.t. a reference entropy S0
3nR
S
ln T  nR ln V  S0
2
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Formal Definition of Entropy

Definition of Entropy (unit: J/K)

For a reversible process, the entropy S of the system is
defined to be
S
0

dQ rev
 S0
T
Thus, the 1st law for reversible processes can be
expressed as
dQ rev
dS 
T
 dU  TdS  PdV
Note this equation is only
valid for reversible processes
* “A reversible process is defined as one which may be exactly reversed to bring the system
back to its initial state with no other change in the surroundings.” (p. 16)
* The subscript rev in dQrev is to remind that the underlying process is a reversible one.
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2nd Law of Thermodynamics

Clausius inequality as 2nd law of thermodynamics*


For a general process, reversible or irreversible, we have
And cycle
 dQ 
  T   cycle  0
= 0, when process is reversible;
cycle > 0, when process is irreversible



cycle “is a measure of the effect of the irreversibilities present
within the system executing the cycle.”
or as “the entropy introduced by internal irreversibilities during
the cycle.” (p. 203, [MS95])
In most textbook, the 2nd law refers to the principle of entropy increase and
derive Clausius inequality as a corollary. But in some books, the 2nd law starts with
Clausius inequality and derive principles of entropy increase as a corollary.
*
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Entropy Balance Equation

Entropy balance eq.

The first implication of the Clausius ineq.

Consider a scenario of a cycle consisting of


a reversible return path from state 2  state 1
From Clausius inequality, we have


a forward path, either irreversible or reversible, from state 1 
state 2 and
2
1
2  dQ 
 dQ  1  dQrev 

   
  cycle
 cycle  S2  S1   

1
 T  2 T 
 T 
entropy balance eq.
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Entropy Balance Equation

Entropy balance in differential form
S2  S1  
2
1


 dQ 
dQ

  cycle  dS 
 d
T
 T 
1st term of RHS

entropy transfer accompanying heat transfer;

positive value means transferring into system
2nd term of RHS, cycle  0 always


contd
entropy produced within the system by the action of
irreversibilities
Interpretation of entropy balance eq

(An entropy change) = (entropy transfer due to heat
transfer) + (entropy induced by action of irreversibilities)
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Principle of Increase of Entropy

Principle of Entropy Increase

Another implication of the Clausius inequality

System under consideration

Consider an enlarged system comprising a system of interest
and that portion of the surroundings affected by the system as it
undergoes a process.


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Since all energy and mass transfers taking place are
included within the enlarged system,
The enlarged system is considered thermally isolated, i.e.,
dQ = 0
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Principle of Increase of Entropy

contd
Entropy balance of enlarged system gives
S isolate  
2
1
dQ
  cycle,isolate
T
thermally
isolation
 0   cycle,isolate  0

which implies


Since   0 in all actual processes, the only processes that can
occur in nature are those with entropy increase of the isolated
system
The above is the so-called the principle of increase of
entropy for thermally isolated system
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Themodynamic Equilibrium of Isolated
Systems



Implied by the principle of entropy increase,
the entropy of an isolated system increases as it
approaches the state of equilibrium, and
the equilibrium state is attained when the entropy
reaches a maximum
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Microscopic Definition of Entropy


Above discussion is the phenomenological definition
of entropy, the classical way
Modern statistical thermodynamics gives the
following microscopic definition


Let  denote the total number of possible microscopic
states available to a system
the entropy of the system is defined as
S  k B ln 

Principle of entropy increase implies equilibrium is
characterized with max disorder, i.e., largest 
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4. Gibbs Free Energy and
Equilibrium
33
Why Is Gibbs Free Energy Necessary?

Principle of entropy increase tells that



An isolated system reaches its thermodynamic
equilibrium when its entropy is maximum.
Limitation of principle of entropy increase is that it
is applicable only to isolated systems
How to describe thermodynamic behavior of a
more general system?
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2 Balance Eq’s of Thermodynamic Sys

(1) Energy balance equation (1st law)
dU = dW + dQ


U: internal energy, W: external work, Q: external heat
(2) Entropy balance equation (2nd law)
dQ
dS 
 d
T

Gibbs free energy is a clever application of these
two balance equations
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Predicting Process Direction

1st law for isobaric (恆壓) process
dQ  dU  PdV

Entropy balance in differential form
dS 

dQ
 d
T
(2)
(1), (2)  the only process allowed must satisfy
dQ  TdS  dU  PdV  TdS  Td  0

(1)
(3)
(3) can be used to study direction of process change

i.e., the system will change with the direction which
would result in negative value of LHS of (3)
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Gibbs Free Energy


Above inequality suggests a new function of
states, the Gibbs free energy
Definition of Gibbs Free Energy, G
G  U  PV  TS  H  TS
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Gibb’s Criterion for Equilibrium

Inequality of slide 35 in terms of Gibbs free energy
dG  dU  PdV  VdP  TdS  SdT
 dG  VdP  SdT   TdS  dU  PdV 
dG  VdP  SdT  0

(4)
For isothermal and isobaric processes, we have
dG  VdP  SdT  T,P  dG T,P  0


Thus, the process proceeds to state with lower G, and
the equilibrium state occurs at min Gibbs free energy,
i.e., when
dG T ,P  0
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5. Chemical Potential
39
Dependence on Size of the System


Intuitively, when the number of moles of the
particle increases, G also increases.
Specifically, we have (without proof)
G  G ( T , P, n )


and
G ( T , P,  n )   G ( T , P, n )
Any function of states is linearly proportional to the size
of the system, like G, is called an extensive property
Extending to multi-component system, we have
G  G(T, P, n1 , n 2 ,, n J )
and
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G(T, P, n1 , n 2 ,, n J )  G(T, P, n1 , n 2 ,, n J )
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Chemical Potential, i
From last slide,

G(T, P, n1 , n 2 ,, n J )  G(T, P, n1 , n 2 ,, n J )
Taking partial derivation w.r.t.  gives
G
G
G
G
n1 
n 2   
nJ
n1
n 2
n J
 1


G
 j1 n
j
J
nj
T ,P ,n q
Define chemical potential as
G
j 
n j
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G   j1 n j j
J
T ,P ,n q
41
i Being Independent of Size

Note that our assumption of extensiveness on G
will give for single component system
G
1 
n1


 1 (T, P)
T , P , n1
i is independent of the size of the system
i.e., chemical potential is a so-called intensive property
of the system
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Chemical Potential of Ideal Gas Mixture



Consider a binary mixture as an illustration
Let n1 and n2 be the numbers of moles of gases 1 and 2,
respectively
From previous discussion related to the ideal gas, we know
T
V
3
3
Ui  n i RT; Pi V  n i RT; Si  n i R ln    n i R ln  
2
2
 T0 
 V0 

Gibbs free energy of binary mixture of ideal gases is
G  U1  U2   P1  P2 V  TS1  S2 

Note
G   j1 n j j
J
By comparison, we will obtain i

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Equilibrium Criterion in Terms of
Chemical Potential

As noted before, equilibrium criterion of an
isothermal and isobaric process is
dG T ,P  0

Thus, this equilibrium criterion of an isothermal
and isobaric process can be reformulated as
dG T ,P  d  jn j 
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T ,P
   jdn j  0
44