Properties of Materials & Equation of State Mathematical Description of Material Behaviour….. P M V Subbarao Professor Mechanical Engineering Department I I T Delhi

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.

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.

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:

Define functions M,N & P such that: As F(x,y,z) is a point function, differentiation is independent of order. This is called as pfaffian differential equation. Properties of A Point Function

The necessary and sufficient condition for g to be accepted as a property is. Creation of New Property Variable in Thermodynamics If we develop an equation for change in a new characteristic of any thermodynamic system as The variable  can provide a functional relationship  (p,v,  )= Constant

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.

Mathematical description of A Substance Identification of Phase of A Substance: β ≈ 10 −3 /K for liquids β ≈ 10 −5 /K for solids

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.

EOS for Solids The volume of A solid: V = f (p,T) & p = g (V,T) Bulk modulus : Coefficient of volume or cubical expansion.

Universal Equation of State for Solids where and V 0 is the volume of solid and B 0 is bulk modulus at reference pressure.

Constants of EoS ParameterGoldNaclXenon B 0 (10 10 Pa)  0 (10 -5 K -1 ) (  B/  p) – T R, K

A common equation of state for Solid V m = molar volume T = temperature p = pressure C 1, C 2, C 3, C 4, C 5 = empirical constants The empirical constants are all positive and specific to each substance. For constant pressure processes, this equation is often shortened to V mo = molar volume at 0 0 C A, B = empirical constants

p-V-T Diagram of crystalline solid Phase Volume Pressure Temperature