### Introduction to the Thermodynamics of Materials, Fourth Edition

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Like this document? Why not share! Embed Size px. Start on. Show related SlideShares at end. The most important thermodynamic potentials are the following functions:. Thermodynamic systems are typically affected by the following types of system interactions. The types under consideration are used to classify systems as open systems , closed systems , and isolated systems.

Common material properties determined from the thermodynamic functions are the following:. The following constants are constants that occur in many relationships due to the application of a standard system of units. The behavior of a Thermodynamic system is summarized in the laws of Thermodynamics , which concisely are:. The first and second law of thermodynamics are the most fundamental equations of thermodynamics.

They may be combined into what is known as fundamental thermodynamic relation which describes all of the changes of thermodynamic state functions of a system of uniform temperature and pressure. The fundamental thermodynamic relation may then be expressed in terms of the internal energy as:. Some important aspects of this equation should be noted: Alberty , Balian , Callen By the principle of minimum energy , the second law can be restated by saying that for a fixed entropy, when the constraints on the system are relaxed, the internal energy assumes a minimum value.

This will require that the system be connected to its surroundings, since otherwise the energy would remain constant. By the principle of minimum energy, there are a number of other state functions which may be defined which have the dimensions of energy and which are minimized according to the second law under certain conditions other than constant entropy.

These are called thermodynamic potentials. For each such potential, the relevant fundamental equation results from the same Second-Law principle that gives rise to energy minimization under restricted conditions: that the total entropy of the system and its environment is maximized in equilibrium.

The intensive parameters give the derivatives of the environment entropy with respect to the extensive properties of the system. After each potential is shown its "natural variables". These variables are important because if the thermodynamic potential is expressed in terms of its natural variables, then it will contain all of the thermodynamic relationships necessary to derive any other relationship.

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In other words, it too will be a fundamental equation. For the above four potentials, the fundamental equations are expressed as:. The thermodynamic square can be used as a tool to recall and derive these potentials. Only one equation of state will not be sufficient to reconstitute the fundamental equation. All equations of state will be needed to fully characterize the thermodynamic system. Note that what is commonly called "the equation of state" is just the "mechanical" equation of state involving the Helmholtz potential and the volume:.

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:. Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that:. The Gibbs-Duhem is a relationship among the intensive parameters of the system.

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 Willard Gibbs and Pierre Duhem. There are many relationships that follow mathematically from the above basic equations. See Exact differential for a list of mathematical relationships. Many equations are expressed as second derivatives of the thermodynamic potentials see Bridgman equations. 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:. The thermodynamic square can be used as a tool to recall and derive these relations. Second derivatives of thermodynamic potentials generally describe the response of the system to small changes. The number of second derivatives which are independent of each other is relatively small, which means that most material properties can be described in terms of just a few "standard" properties. For the case of a single component system, there are three properties generally considered "standard" from which all others may be derived:.

## Introduction to the thermodynamics of materials

These properties are seen to be the three possible second derivative of the Gibbs free energy with respect to temperature and pressure. Properties such as pressure, volume, temperature, unit cell volume, bulk modulus and mass are easily measured. Other properties are measured through simple relations, such as density, specific volume, specific weight. Properties such as internal energy, entropy, enthalpy, and heat transfer are not so easily measured or determined through simple relations.

Thus, we use more complex relations such as Maxwell relations , the Clapeyron equation , and the Mayer relation. Maxwell relations in thermodynamics are critical because they provide a means of simply measuring the change in properties of pressure, temperature, and specific volume, to determine a change in entropy. Entropy cannot be measured directly.

The change in entropy with respect to pressure at a constant temperature is the same as the negative change in specific volume with respect to temperature at a constant pressure, for a simple compressible system. Maxwell relations in thermodynamics are often used to derive thermodynamic relations. The Clapeyron equation allows us to use pressure, temperature, and specific volume to determine an enthalpy change that is connected to a phase change. It is significant to any phase change process that happens at a constant pressure and temperature.

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One of the relations it resolved to is the enthalpy of vaporization at a provided temperature by measuring the slope of a saturation curve on a pressure vs. It also allows us to determine the specific volume of a saturated vapor and liquid at that provided temperature. The Mayer relation states that the specific heat capacity of a gas at constant volume is slightly less than at constant pressure. This relation was built on the reasoning that energy must be supplied to raise the temperature of the gas and for the gas to do work in a volume changing case.

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According to this relation, the difference between the specific heat capacities is the same as the universal gas constant. This relation is represented by the difference between Cp and Cv:. From Wikipedia, the free encyclopedia. For a quick reference table of these equations, see: Table of thermodynamic equations Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process.

Thermodynamics The classical Carnot heat engine. Classical Statistical Chemical Quantum thermodynamics. Zeroth First Second Third. System properties. Note: Conjugate variables in italics. Work Heat. Material properties. Carnot's theorem Clausius theorem Fundamental relation Ideal gas law. Free energy Free entropy.