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Chapter 0. Review of Thermodynamics
1.
Introduction to Thermodynamics and Statistical Mechanics
1.1 Thermal phenomena and thermodynamic systems
Both thermodynamics and statistical mechanics are concerned with thermal phenomena or thermal
properties of thermodynamic systems. Any phenomenon that depends on temperature is a thermal
phenomenon, and any temperature-dependent property of a macroscopic system is a thermal property.
Therefore, thermal phenomena and thermal properties of thermodynamic systems include almost any
phenomena and any properties of any systems one can see on earth (Fig. 0.1.1). In the last century,
these were extended to include objects in the sky and even our universe (Fig. 0.1.2).
Figure 0.1.1 Examples of
thermal phenomena
and thermodynamic
systems on earth. (a) A
typical car in 1920’s. (b)
A magnet floating
above a superconductor
(c) Bent rail-tracks in a
hot summer. (c) A
broken egg on a cup.
Figure 0.1.2 Examples of
thermal phenomena
and thermodynamic
systems in the sky. (a)
Sun. (b) A white dwarf
star. (c) A sketch of
evolution of our
universe.
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1.2 Scope of thermodynamics
Thermodynamics is a phenomenological theory of matter. As such, it draws its concept directly from
experiments. Thermodynamics was developed mainly in the first half of the 19th century by Carnot,
Joule, Clausius, Kelvin and others. It consists of the establishment of three laws of thermodynamics
and their applications. The laws of thermodynamics are the empirical laws which were drawn from a
large part of our experience and a large number of experimental observations. Thus, theoretical
conclusions from thermodynamics were found to be reliable and universal.
1.3 Roles of statistical mechanics
However, thermodynamics cannot make predictions of the properties for any specific systems.
Therefore, the quantities of a thermodynamic system such as heat capacity and equation of state can
only be obtained by experimental measurements. Furthermore, a thermodynamic (macroscopic) system
consists of a large number (typically 1023) of microscopic particles (e. g., molecules, atoms, electrons,
photons), but thermodynamics does not concern itself with the dynamical behavior of these
microscopic particles and hence provide not much insight into the law of thermodynamics. Therefore,
in the second half of the 19th century and the early part of the 20th century, after the establishment of
thermodynamics, kinetic theory of gas and statistical mechanics were developed by Joule, Clausius,
Maxwell, Boltzmann (kinetic theory of gas), Gibbs, Bose, Fermi, Einstein, Planck and others. Kinetic
theory of gases was rather successful in dealing with dilute gases but failed for condensed substances.
On the other hand, statistical mechanics starts with the fact that a macroscopic system is made up by an
extremely large number of microscopic particles. It is a formalism that aims at explaining the physical
properties of a macroscopic system on the basis of the dynamical behavior of its microscopic particles.
Thus, it is a first-principles theory.
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2.
State Variables (or Parameters) and Equation of State
K. Huang said “Thermodynamics has successfully described a large part of macroscopic experience,
which is the concern of statistical mechanics. If we familiarize ourselves with thermodynamics, the
task of statistical mechanics reduces to the explanation of thermodynamics.”
2.1 Thermodynamic (state) variables (parameters)
A macroscopic system has an extremely large number of freedoms, only a few of which are
measurable. Thermodynamics thus concerns itself with the relation between a small number of
variables (parameters) which are sufficient to describe the bulk behavior of the system in question.
Examples:
(1) Pressure P, volume V, and temperature T for a gas or liquid.
(2) Magnetic field B, magnetization M, and temperature T for a magnetic solid.
2.2 Steady state, equilibrium and state functions
If the thermodynamic variables are independent of time, the system is in a steady state. If, furthermore,
there are no macroscopic currents in the system (e.g., a flow of heat or particles), the system is in
equilibrium.
Any quantity which, in equilibrium, depends only on the thermodynamic variables, rather than on the
history of the same quantity, is called a state function.
The state variables can normally be taken to be either extensive (i.e., proportional to the size of the
system) (e.g., the internal energy U, and the entropy S) or intensive (i.e., independent of the system
size) (e.g., T, P, and the chemical potential ).
2.3 Equation of state
In equilibrium, the state variables (thermodynamic parameters) are not all independent and thus are
connected by equation of state. If P, V, T are the state variables of the system, the equation of state
takes the form
f ( P,V , T )  0
which reduces the number of independent variables of the system from three to two. The function f is
assumed to be given as part of the specification of the system in thermodynamics (e.g., determined
experimentally). A role of statistical mechanics is the derivation from the dynamics of the constituent
microscopic particles, of such equations of state.
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Examples of equation of state:
a) The Gay-Lussac’s law for the idea gas Pv  RT or PV  nRT  NkT
which in fact defines a temperature scale. Here N (n) is the number of the molecules (moles) in the gas,
R is the gas constant (8.314×103 J/kilomole-K ) and k = 1.38×10-23 J/K is Boltzmann’s constant.
Figure 0.2.1 The ideal-gas temperature scale based on Gay-Lussac’s law.
b) The van der Waals equation for a van der Waals gas
(P 
a
)(v  b)  RT  0 where a and b are constants. v is the mole volume of the gas.
v2
Figure 0.2.2 (a) Typical intermolecular potential and (b) idealized intermolecular potential.
c) The Curie law for a paramagnet M 
CB
 0 where C is the Curie constant.
T
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It is customary to represent the state of a system a point in the multi-dimensional state variable space
(e.g., P-V-T space).
Figure 0.2.3 Illustration of the equation of state.
3.
Laws of Thermodynamics
3.1 The first law of thermodynamics (the law of conservation of energy)
It states that in an arbitrary thermodynamic transformation (a change of state) the quantity
U  Q  W
is the same for all transformations leading from a given initial state to a given final state. Here Q is
the heat absorbed by the system and W is the configuration work done by the system. This defines a
state function U (the internal energy) since it is independent of the path. This is not shared by Q and
W .
The experimental foundation is Joule’s demonstration of the equivalence between heat and mechanical
energy – the feasibility of converting mechanical work completely into heat.
Figure 0.3.1 Joule’s experiment: Mechanical
energy can transfer to heat which was identified
as another form of energy.
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In an infinitesimal transformation dU  dQ  dW
where dW   y j dX j (yj is a generalized force and Xj is the generalized displacement).
j
Consider a system whose parameters are P, V, T. Then dW  PdV .
 U 
 U 
 U 
 U 
a) U  U ( P,V ). Then dQ  dU  dW  
 dP  
 dV  PdV  
 dP  [
  P]dV .
 P V
 V  P
 P V
 V  P
b) U  U ( P, T ).
 U 
 V 
 U 
 V 
dQ  [
  P
 ]dT  [
  P
 ]dP.
 T  P
 T  P
 P T
 P T
c)
U  U (V , T ).
 U 
 U 
dQ  
 dT  [
  P]dV .
 T V
 V T
These are called dQ equations.
Applications of the dQ equations – heat capacity (specific heat)
 Q   U 
CV  
 
 .
 T V  T V
 Q 
 H 
CP  
 
 where H  U  PV is the enthalpy.
 T  P  T  P
Applications of the first law
a) Analysis of Joule’s free-expansion experiment
Experimental finding T1 = T2.
Deductions: W  0 and Q  0 (since T1  T2 ) . Thus U  0 and U1  U 2 .
Figure 0.3.2 Joule’s
free-expansion
experiment.
Conclusions: For an ideal gas, U is a function of temperature alone.
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b) Internal energy of an ideal gas
dU
 dQ   U 
CV  
. Thus U  CV T  const  CV T .
 
 
 dT V  T V dT
3.2 The second law of thermodynamics
Experience tells that there are many processes that satisfy the law of conservation of energy yet never
occur. For example, a piece of stone resting on the floor is never seen cool itself spontaneously and
jump up to the ceiling, thereby converting the heat energy given off into potential energy.
The second law of thermodynamics is to incorporate such experimental facts (common sense) into
thermodynamics.
Equivalent statements of the second law:
Clausius statement: No process is possible whose sole effect is the transfer of heat from a colder to a
hotter body. The term “sole” effect is important, since it is possible to transfer heat from a colder
system to a hotter system using a refrigerator, but in this case external work must be done on the
working substance.
Kevin statement: No process is possible whose sole effect is the absorption of heat from a reservoir and
the conversion of all of this heat into work.
It is shown in textbooks that the truth of either the Clausius or the Kevin statements of the second law
is a necessary and sufficient condition for the truth of the other.
Applications of the second law of thermodynamics
The Carnot heat engine: An idealized engine in which all the steps are reversible. Using the second law,
one can derive the Carnot’s theorems:
a) No engine can be more efficient than a reversible engine working between the same limits of
W Q2  Q1
Q
temperatures.  

 1  1  1 since U  0 .
Q2
Q1
Q2
b) All reversible engines working between the same two limits of T have the same efficiency.
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Figure 0.3.3 The Carnot (reversible) cycle. ab: the gas absorbs heat Q2 and expands
isothermally and does work. bc: the gas expands adiabatically and further does work. cd: heat
(-Q1) is given off to the low-T reservoir and work is done on the gas. da: returns the working
substance to its original state adiabatically.
This allowed Kevin to define the absolute thermodynamic temperate scale as
Note:
3.3
Q2
Q1
T2 Q2
 .
T1 Q1
is independent of substance.
Entropy (a state function)
Clausius’ theorem: In any cyclic transformation throughout which the T is defined,

dQ
 0 holds
T
where the equality holds if the cyclic transformation is reversible.
Corollary: For a reversible transformation, the integral

dQ
is independent of the path and depends
T
only on the initial and final states of the transformation.
Therefore, we can define a state function, entropy, for any state A as
S ( A)  
A
O
dQ
where the path of integration is any reversible path joining O to A.
T
S ( B)  S ( A)  
B
A
B
dQ
dQ
  dS (dS 
).
A
T
T
Deductions:
a) For an arbitrary transformation,

B
A
dQ
 S ( B)  S ( A) .
T
The equality holds if the transformation is reversible.
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b) The entropy of a thermally isolated system never decreases. Furthermore, the equilibrium state of
an isolated system is the state of maximum entropy.
Proof of (a): For the path R, the assertion holds by definition. For the cyclic transformation,

I
dQ
dQ

 0 or
R T
T

I
dQ
dQ

 S ( B)  S ( A) .
R T
T
Figure 0.3.4 Reversible path R and
irreversible path I connecting A and B.
3.4 The third law of thermodynamics
From the second law, S  
dQ
 SO where SO is the T = 0 entropy of the system which cannot be
T
determined using the second law. The discussion on the value of SO led to the proposals of the third
law.
In 1906, based on the low temperature experiments, Nernst first proposed that, for pure condensed
substances (liquid and solid) at temperatures close to zero, the change in entropy associated with any
change in the external parameters tends to zero.
In 1911, based on statistical mechanics, Planck proposed that the entropy of every pure condensed
substance, in internal thermodynamic equilibrium, is zero (SO = 0) at the zero temperature.
In 1912, Nernst proposed another statement of the third law, i.e., a system cannot be cooled to absolute
zero by a finite change of the thermodynamic parameters. This statement of the unattainability of
absolute zero is often known as the third law of thermodynamics.
Among the three statements of the third law, the Planck statement is the strongest one. The two Nernst
statements are equivalent, and follow naturally if the Planck statement is true. However, the contrary is
not true.
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Implications:
a) Any heat capacity of a system must vanish at absolute zero. This is experimentally verified for all
A
substances so far. By the second law S ( A)   CR (T )
O
dT
.
T
By the third law, S ( A)  0 TA  0 . Thus, CR (T )  0 TA  0 , i.e., CR (T )  T  (  1).
Quantum statistical mechanics shows that the third law is a macroscopic manifestation of quantum
effects. For example, for a solid, the heat capacity CV is 3R in classical statistical mechanics but is
CV  T   T 3 ( T  0 ) in quantum statistical mechanics.
b) The coefficient of thermal expansion of any substance vanishes at absolute zero.
1  V 
1  S 
Using Maxwell relation,   
    .
V  T V
V  P T
Since S  SO (const) as T  0, S is independent of P. Thus,   0 T  0 .
4. Thermodynamic Potentials
For a mechanical system, the work done is related to the change in the mechanical potential energy.
Furthermore, the stable state is the state of the minimum potential energy.
The term thermodynamic potential derives from an analogy with mechanical potential.
4.1 The Helmholtz free energy
Consider a PVT system,
A  U  TS , which is a state function with dA  dU  TdS  SdT .
dQ
 dS . Thus, for an isothermal transformation (dT = 0),
T
dA  ( dQ  dW )  TdS or dA  dW  dQ  TdS  0 , i.e., dW  dA . The maximum amount of work
According to the second law,
that can be extracted at constant T, from a system is (–dA).
Theorems:
For a mechanical isolated system kept at constant T (dT = 0, dW = 0),
a) the Helmhotz free energy never increase ( dA  0 );
b) the state of equilibrium is the state of minimum Helmhotz free energy.
Now consider A  A(T ,V ) . For an infinitesimal reversible transformation
dA  (dQ  dW )  TdS  SdT   PdV  SdT since dQ  TdS and dW  PdV .
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 A 
 A 
Thus, P   
 and S   
 .
 V T
 T V
Since the theory of partial differentiation tells us that for a state function, higher-order derivatives are
independent of the order in which the differentiation is carried out, i.e.,

xi
  
   

 

,
 x j  x j  xi 
 P   S 

 
 (Maxwell equation).
 T V  V T
 A 
 A 
Application:  A  0  
 V1  
 V2 .
 V1 
 V2 
Since V  V1  V2 , V1  V2 ,
 A   A  
 A 
 A 

 
   V1  0, i.e., 
 
 or P1  P2 .

V

V
 V1 T  V2 T 
 1 T  2 T
4.2 The Gibbs thermodynamic potential (or Gibbs free energy)
For a PVT system, G  A  PV with dG  dA  PdV  VdP .
Since dW  dA , dW  dA  PdV  dA  dG  0 (dP  0) .
Theorems:
For a system kept at constant T and P,
a) the Gibbs potential never increases;
b) the state of equilibrium is the state of minimum Gibbs potential.
The change in the Gibbs potential is the maximum amount of work that can be extracted from a system
through a process at fixed T and P.
Consider G  G(T , P) . For an infinitesimal reversible transformation
dG  dA  VdP  PdV  SdT  VdP , since dA  SdT  PdV .
 G 
 G 
Thus, S   
 and V  
 .
 T  P
 P T
 S   V 
And     
 (Maxwell relation).
 P T  T  P
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4.3 The Maxwell relations
We have seen two other thermodynamic potentials already, namely, the internal energy U and the
enthalpy H.
a) The first law gives us, in an infinitesimal reversible transformation dU  PdV  TdS .
 U 
 U 
Thus, T  
 and P   
 .
 S V
 V S
b) Consider the enthalpy H  U  PV for a PVT system.
For an infinitesimal reversible transformation
dH  dU  PdV  VdP  TdS  VdP since dU  TdS  PdV .
 H 
 H 
Thus, V  
 and T  
 .
 P S
 S  P
To summarize, for a PVT system, from U  U ( S ,V ), H  H ( S , P), A  A(T ,V ) and G  G(T , P) ,
we have
Therefore,
 U 
 U 
dU  TdS  PdV  
 dS  
 dV ,
 S V
 V  S
 H 
 H 
dH  TdS  VdP  
 dS  
 dP,
 S  P
 P  S
 A 
 A 
dA   SdT  PdV  
 dT  
 dV ,
 T V
 V T
 G 
 G 
dG   SdT  VdP  
 dT  
 dP.
 T  P
 P T
 U 
 U 
T 
 and P   
 ,
 S V
 V  S
 H 
 H 
T 
 and V  
 ,
 S  P
 P  S
 A 
 A 
S  
 and P   
 ,
 T V
 V T
 G 
 G 
S  
 and V  
 .
 T  P
 P T
Since all the four thermodynamic potentials are exact differentials, we obtain the Maxwell relations
 T 
 P   T   V   S   P   S 
 V 

    , 
 
 ,
 
 ,    
 .
 V S
 S V  P S  S  P  V T  T V  P T
 T  P
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Figure 0.4.1 conveniently summarize the Maxwell relations.
Figure 0.4.1 Summary of the Maxwell relations. The functions A, G, H, U are flanked by their
respective natural arguments. The derivative with respect to one argument, with the other held
fixed, may be found by going along a diagonal line, either with or against the direction of the
arrow. Going against the arrow yields a minus sign.
5.
Response Functions
A great deal can be learned about a macroscopic system through its response to various changes in
externally controlled parameters. Important response functions for a PVT system include
 dQ 
 S 
 dQ 
 S 
a) CV  
 T 
 and CP  
 T 
 .
 dT V
 T V
 dT  P
 T  P
b) The isothermal and adiabatic compressibilities
1  V 
1  V 
T   
 and  S   
 .
V  P T
V  P  S
1  V 

 .
V  T  P
Intuitively, we expect that the response functions to be positive and that CP  CV and  T   S .
c)
Thermal expansion coefficient  
However, there are important exceptions, e.g., water (see Fig. 0.5.1).
Figure 0.5.1 Volume of water
versus temperature.
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 S 
 S 
Consider a PVT system: S  S (T ,V ). Then dS  
 dT  
 dV .
 T V
 V T
 S 
 S 
 S   V 
 S   V 
And T 
 T
 T 
 
 
 .
 , CP  CV  T 
 T  P
 T V
 V T  T  P
 V T  T  P
 z   y   x 
 S   P 
Using the Maxwell relation 
 
 and chain rule  x   z   y   1 [f(x,y,z)=0] ,
  y  x  z
 V T  T V

 S   P 
 P   V 
We obtain 
.
 
  
 
 
 V T  T V
 V T  T  P T
  V  TV 2
Therefore CP  CV  T

 .
T  T P T
This shows that (CP  CV )  0 if  T  0 . Experience shows that for most substances (This is implied
by neither the first nor the second laws). This can be proven in Statistical Mechanics where use is made
of the nature of intermolecular forces and where it is known as van Hove’s theorem.
Consider an ideal gas. The equation of state PV
Nk
 V 
Since PdV  VdP  NkdT , 
and
 
P
 T  P
 NkT ( PV  nRT Boyle’s law).
Nk
 P 
.

 
 T V V
 S   V 
Thus, CP  CV  T 
 
  Nk , i.e., it is more efficient to heat an ideal gas by keeping the
 V T  T  P
volume constant than to heat it by keeping the pressure constant.
 V 
 V 
Assume V  V ( S , P) . Then dV  
 dP  
 dS and
 P S
 S  P
1  V 
1  V  1  V   S 
 
  
  
  
V  P T
V  P S V  S  P  P T
Using the Maxwell relations, one can show
1  V   S 
2 TV
.
T   S   
   
V  S  P  P T
CP
And CP T   S   T  CP  CV    2TV and
CP T
.

CV  S
Lecture notes on statistical mechanics by Guang-Yu Guo
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6. Thermodynamics of Phase Transitions
6.1 Phase diagrams of a typical substance
Figure 0.6.1 Surface of equation of state of a typical substance. The shaded areas are cylindrical
surfaces, representing regions of phase transition.
Figure 0.6.2 P-V and P-T diagrams of a typical substance. The curves represent the coexistence of
two phases (coexistence curves).
At the critical point the properties of the fluid and vapor phase become identical. At the triple point, all
three phases (solid, liquid and gas) coexist.
6.2 The liguid to gas phase transition (a first-order phase transition)
a) During the phase transition, both P and T remain constant.
b) In the gas-liquid mixture, the liquid exists in the same state as at 1 and the gas exists in the same
as at 2 (see Fig. 0.6.3).
Total volume expands since the gas has a smaller density , i.e., a “first-order phase transition”.
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Figure 0.6.3 (Left) An isotherm exhibiting a phase transition and (right) schematic illustration of
the total volume of the system changes as the relative amount of the substance in the two phases
changes, because the two phases have different densities.
6.3 The condition for the gas-liquid mixture equilibrium
Consider a gas-liguid mixture in equilibrium at T and P(T). The Gibbs potential of this state must be
at a minimum, i.e., if any parameters other than T and P are varied slightly, we must have G = 0.
Let us vary the composition by converting an amount of liquid to gas so that  m1   m2   m
where m1 is the mass of the liquid and m2 is the mass of the gas.
G  m1 g1  m2 g 2 where g1 and g2 are the Gibbs potential per unit mass in phases 1 and 2 or chemical
potentials (state functions). Thus  G  0  ( g1  g 2 ) m.
Therefore, the condition for equilibrium is g1 = g2.
6.4 Derivation of vapor pressure P(T)
We now apply the second law of thermodynamics and use the condition for equilibrium to determine
the vapor pressure. Since, for each phase,
 G 
 G 
 g 
 g 
S  
   s,    v
 , V 
 , we have 
 T  P
 P T
 T  P
 P T
where s and v are entropy and volume per unit mass, respectively.
We see from Fig. 1.6.4 that the first derivatives of g1 and g2 may differ at the TC and PC:
  ( g 2  g1 ) 

  ( s2  s1 )  0,
T

P
  ( g 2  g1 ) 

  v2  v1  0.
P

T
Lecture notes on statistical mechanics by Guang-Yu Guo
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This is why the transition is called
“first-order” phase transition.
Figure 0.6.4 Chemical potentials,
volumes and entropies for the
two phases in a first-order
transition. In (a) and (b), the solid
and dashed lines represent stable
and unstable g-lines.
g  g 2  g1
Let s  s2  s1
v  v2  v1
being evaluated at the transition temperature T and vapor pressure P.
 g 


 T  P   s .
v
 g 


 P T
Since g is a function of T and P, i.e., f (T , P, g )  0 , we can use chain relation
 g   T   P 

 
 
  1. Thus
 T  P  P  g  g T
 g 


 P 
 T  P ,




 g 
 T  g


 P T
dP(T )  P 
s
l



.

dT
 T  g 0 v T v
where l  T s is the latent heat of transition. This is known as the Clapeyron equation. It governs the
vapor pressure in any first-order transition.
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6.5 The second-order and higher-order phase transitions
If at a phase transition, s1  s2  0 and v1  v2  0 , the first derivatives of the thermodynamic
chemical potentials are continuous. Such a transition is not of the first-order and its isotherm would not
have a horizontal part in the P-V diagram.
If, at the transition point,
 n g1  n g 2
 n g1  n g 2

and

whereas all lower derivatives are equal,
T n
T n
P n
P n
it is an nth-order transition (e.g., superconductivity: the second-order transition) (Ehrenfest definition).
Other examples of the second-order transition include the Curie point transition in ferromagnets, orderdisorder transition in binary alloys (CuZn) and the transition in liquid helium. In these cases, the
specific heat diverges logarithmically at the transition point, since the specific heat is related to the
second derivative of g.
Modern usage distinguishes only between first-order and higher-order transitions, and usually the latter
are all called “second-order” transitions.
Figure 0.6.5 Diagram of a second-order phase transition. In (a), the first derivative of g is
continuous at TC. In (b) s is continuous at TC while its first derivative is discontinuous. (c) is an
example of discontinuous CP, e.g., superconductivity. (d) is an example of divergent CP at , e.g., a
ferromagnetic phase transition.
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