Properties of Materials

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Properties of Materials Equation of State

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

Mathematical Description of Material Behaviour..
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Second Level Thermodynamic Analysis

• If a solid is heated, strains and stresses
  develop.
• Conversely if body is strained rapidly, then heat
  is generated inside the body.
• The changes undergone by this system can be
  characterized by some functions.
• What kind functions can be used for correct
  recognition of these changes.

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The Functional Relation for Description of A
System

• What kind of Functional Relation?
• Assume that variables p, V, T are functionally
  related.
• Say F(p, V, T) Constant.
• Assume that each variable can be explicitly
  solved from this functional relation in terms
  of two other variables, which are allowed to vary
  freely.
• p to obtain an expression of the form p g(V,
  T), where V and T are chosen as free variables.
• Any function of p, V, T can be expressed as a
  function of any pair of free variables of your
  choice.
• F(p, V, T) F(g(V, T), V, T) is expressed as a
  function of a pair of free variables V and T.

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Functions of Several Variables

• Develop a function using these variables
  F(x,y,z,)
• If F(x,y,z,) Constant.
• This is called as Pfaffian function.
• Pfaffian function is denoted as F(.) and called
  as Point Function.

A total change in point function is expressed as
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Properties of A Point Function
Define functions M,N P such that
This is called as pfaffian differential equation.
As F(x,y,z) is a point function, differentiation
is independent of order.
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Creation of New Property Variable in
Thermodynamics
If we develop an equation for change in a new
characteristic of any thermodynamic system as
The necessary and sufficient condition for g to
be accepted as a property is.
The variable ? can provide a functional
relationship
? (p,v, ?) Constant
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Definition of A Thermodynamic Property

• Any Macroscopic variable, which can be written as
  a point function can be used as a thermodynamic
  property
• Thermodynamic properties are so related that F(.)
  is constant.
• Every substance is represented as F(.) in
  Mathematical (Caratheodory) Thermodynamics.
• This is shows a surface connectivity of Property
  of a substance.

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Mathematical description of A Substance
Identification of Phase of A Substance
ß 10-3/K for liquids ß 10-5/K for solids
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General Behaviour of Solids

• Incompressible Substance.
• Change in volume is infinitesimally small.
• Huge increase in temperature or pressure required
  for a finite change in volume/area/length.
• In an ideal (Hookean) solid, finite increase in
  pressure (stress) produces constant deformation
  (strain) at constant temperature.
• Thermal Expansion of Solids
• As the thermal energy in a solid increases, the
  mean separation of the atoms increases because
  the force curve is anharmonic.
• This causes the solid to expand.
• Linear, superficial and cubical Expansion
  coefficient.

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EOS for Solids

• The volume of A solid
• V f (p,T) p g (V,T)

Coefficient of volume or cubical expansion.
Bulk modulus
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Universal Equation of State for Solids
where
and
V0 is the volume of solid and B0 is bulk modulus
at reference pressure .
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Constants of EoS
Parameter Gold Nacl Xenon
B0 (1010 Pa) 16.6 2.35 0.302
?0(10-5 K-1) 4.25 12.0 60.0
(?B/?p)0 5.5 6.5 5.35 7.8
TR, K 300 298 60
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A common equation of state for Solid
Vm molar volume T temperature p pressure
C1, C2, C3, C4, C5 empirical constants The
empirical constants are all positive and specific
to each substance. For constant pressure
processes, this equation is often shortened to
Vmo molar volume at 00C A, B empirical
constants
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p-V-T Diagram of crystalline solid Phase
Pressure
Temperature
Volume