Materials Science and Engineering/Equations

Thermodynamics edit

Notation edit

Conjugate variables
   p Pressure V Volume
   T Temperature S Entropy
   μ Chemical potential N Particle number
Thermodynamic potentials
   U Internal energy A Helmholtz free energy
   H Enthalpy G Gibbs free energy
Material properties
   ρ Density
   CV   Heat capacity (constant volume)
   Cp   Heat capacity (constant pressure)
     Isothermal compressibility
     Adiabatic compressibility
     Coefficient of thermal expansion
Other conventional variables
   δw infinitesimal amount of Mechanical Work
   δq infinitesimal amount of Heat
   kB Boltzmann constant
   R Ideal gas constant

Equations edit

Laws of Thermodynamics edit

First Law of Thermodynamics:


Second Law of Thermodynamics:


Fundamental Equations edit

The Fundamental Equation:


The equation may be seen as a particular case of the chain rule:


from which the following identifications can be made:


These equations are known as "equations of state" with respect to the internal energy.

Thermodynamic Potentials edit

Thermodynamic Potentials:

Name Formula Natural variables
Internal energy    
Helmholtz free energy    
Gibbs free energy    

For the above four potentials, the fundamental equations are expressed as:


Euler Integrals:

Because all of natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that


Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials:


Note that the Euler integrals are sometimes also referred to as fundamental equations.

Gibbs Duhem Relationship:

Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that:


which is known as the Gibbs-Duhem relationship. The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Josiah Gibbs and Pierre Duhem.

Materials Properties edit

Compressibility: At constant temperature or constant entropy


Heat Capacity at Constant Pressure:


Heat Capacity at Constant Volume:


Coefficient of Thermal Expansion:


Maxwell Relations:

Maxwell relations are equalities involving the second derivatives of thermodynamic potentials with respect to their natural variables. They follow directly from the fact that the order of differentiation does not matter when taking the second derivative. The four most common Maxwell relations are:


Processes edit

Incremental Processes:





Equation Table for an Ideal Gas ( ):

Constant Pressure Constant Volume Isothermal Adiabatic
Heat Capacity,         or     or  
Internal Energy,