# Thermodynamic Gaskell 4th Chapter 13 Solutions.zip - Ebook and Manual

## Introduction to the Thermodynamics of Materials

## Thermodynamics is the study of energy, entropy, and equilibrium in physical systems. It is a fundamental branch of science that applies to all fields of engineering and technology. Thermodynamics can help us understand how materials behave under different conditions, such as temperature, pressure, composition, and external fields. It can also help us design new materials with desired properties, such as high strength, low weight, corrosion resistance, or superconductivity. Materials science is the interdisciplinary field that deals with the structure, properties, processing, and performance of materials. It covers a wide range of materials, such as metals, ceramics, polymers, composites, nanomaterials, biomaterials, etc. Materials science combines concepts from physics, chemistry, biology, mathematics, and engineering to create novel materials for various applications. The thermodynamics of materials is a subfield of materials science that focuses on the thermodynamic aspects of materials systems. It can answer questions such as: - What are the possible phases and their compositions in a given material system? - What are the conditions for phase equilibrium and phase transformations? - What are the thermodynamic properties (such as heat capacity, enthalpy, entropy, etc.) of a material or a phase? - What are the driving forces for chemical reactions involving materials? - How do external fields (such as electric, magnetic, or mechanical) affect the thermodynamics of materials? In this article, we will introduce some basic concepts and principles of thermodynamics that are relevant to materials science. We will also discuss some examples and applications of thermodynamics in various materials systems. Thermodynamic Principles

### The First Law of Thermodynamics

### The first law of thermodynamics is also known as the law of conservation of energy. It states that the total energy of an isolated system remains constant. Energy can be transferred between the system and its surroundings in two ways: heat (Q) or work (W). Heat is the energy transfer due to a temperature difference, while work is the energy transfer due to a force acting through a distance. The first law can be expressed as: $$\Delta U = Q - W$$ where $\Delta U$ is the change in the internal energy of the system. The internal energy is the sum of all the microscopic forms of energy in the system, such as kinetic, potential, chemical, nuclear, etc. The sign convention for heat and work is that they are positive when they enter the system, and negative when they leave the system. The first law of thermodynamics implies that we can measure the internal energy of a system by measuring the heat and work exchanged with the surroundings. However, the internal energy itself is not an observable quantity, as it depends on the choice of reference state. Therefore, we often use other thermodynamic quantities, such as enthalpy, free energy, or entropy, to describe the state of a system. The Second Law of Thermodynamics

### The second law of thermodynamics is also known as the law of entropy increase. It states that the entropy of an isolated system always increases or remains constant in a spontaneous process. Entropy (S) is a measure of the disorder or randomness of a system. A system with more possible configurations or microstates has higher entropy than a system with fewer possible configurations or microstates. The second law can be expressed as: $$\Delta S \geq 0$$ where $\Delta S$ is the change in the entropy of the system. The equality holds for a reversible process, which is an idealized process that can be reversed without any change in the system or its surroundings. The inequality holds for an irreversible process, which is a realistic process that involves some irreversibilities, such as friction, heat transfer, chemical reactions, etc. The second law of thermodynamics implies that we can predict the direction of a spontaneous process by comparing the entropy changes of the system and its surroundings. A process is spontaneous if it increases the total entropy of the universe (system + surroundings). A process is non-spontaneous if it decreases the total entropy of the universe. A process is at equilibrium if it does not change the total entropy of the universe. The Statistical Interpretation of Entropy

### The concept of entropy can be understood from a statistical point of view. According to the Boltzmann equation, the entropy of a system is proportional to the natural logarithm of the number of microstates ($\Omega$) that correspond to a given macrostate: $$S = k_B \ln \Omega$$ where $k_B$ is the Boltzmann constant, which relates temperature and energy at the molecular level. A macrostate is a description of a system in terms of observable quantities, such as pressure, volume, temperature, etc. A microstate is a description of a system in terms of microscopic quantities, such as positions and velocities of molecules, spins of electrons, etc. The Boltzmann equation implies that entropy is a measure of how many ways we can arrange the microscopic components of a system to achieve a given macroscopic state. For example, consider a box containing two types of gas molecules: red and blue. If we divide the box into two equal parts by a partition, we can have different arrangements of red and blue molecules on each side. Each arrangement corresponds to a different microstate. However, if we only care about how many red and blue molecules are on each side, regardless of their positions and velocities, we have a macrostate. The macrostate with equal numbers of red and blue molecules on each side has more microstates than any other macrostate. Therefore, it has higher entropy than any other macrostate. Fundamental Equations and Their Relationships

### To describe the thermodynamic state and behavior of a material system, we often use thermodynamic potentials. These are functions that depend on certain variables and represent certain aspects of the system. For example, we have already seen that internal energy (U) depends on heat and work and represents the total energy content of the system. Another example is enthalpy (H), which depends on heat and pressure-volume work and represents the energy required to create the system from its elements at constant pressure. There are other thermodynamic potentials, such as Helmholtz free energy (A), Gibbs free energy (G), etc., that depend on different variables and represent different aspects of the system. The thermodynamic potentials are related to each other by fundamental equations that involve their derivatives with respect to their variables. These derivatives represent important thermodynamic properties, such as temperature (T), pressure (P), volume (V), entropy (S), etc. For example, one fundamental equation relates internal energy to entropy and volume: $$dU = TdS - PdV$$ This equation implies that: V = T$$ - The partial derivative of internal energy with respect to volume at constant entropy is negative pressure: $$\left(\frac\partial U\partial V\right)_S = -P$$ There are other fundamental equations that relate different thermodynamic potentials to different variables and properties. For example, one fundamental equation relates enthalpy to entropy and pressure: $$dH = TdS + VdP$$ This equation implies that: - The partial derivative of enthalpy with respect to entropy at constant pressure is temperature: $$\left(\frac\partial H\partial S\right)_P = T$$ - The partial derivative of enthalpy with respect to pressure at constant entropy is volume: $$\left(\frac\partial H\partial P\right)_S = V$$ The fundamental equations can be combined and manipulated using mathematical rules, such as the chain rule, the product rule, and the cyclic rule. These rules allow us to derive other useful equations and relationships between thermodynamic properties. For example, one useful equation is the Maxwell relation: $$\left(\frac\partial T\partial V\right)_S = -\left(\frac\partial P\partial S\right)_V$$ This equation relates the derivatives of temperature and pressure with respect to entropy and volume. It can be derived from the fundamental equation for internal energy by applying the cyclic rule. Heat Capacity, Enthalpy, Entropy, and the Third Law of Thermodynamics

## Heat capacity (C) is a measure of how much heat a system can absorb or release without changing its temperature. It depends on the mode of heat transfer and the variables that are kept constant. For example, we can define the heat capacity at constant volume (C_V) and the heat capacity at constant pressure (C_P) as: $$C_V = \left(\frac\partial Q\partial T\right)_V$$ $$C_P = \left(\frac\partial Q\partial T\right)_P$$ These equations imply that heat capacity is equal to the partial derivative of heat with respect to temperature at constant volume or pressure. Heat capacity can also be related to other thermodynamic properties using the fundamental equations. For example, we can show that: $$C_V = \left(\frac\partial U\partial T\right)_V$$ $$C_P = \left(\frac\partial H\partial T\right)_P$$ These equations imply that heat capacity is equal to the partial derivative of internal energy or enthalpy with respect to temperature at constant volume or pressure. Enthalpy (H) is a measure of the energy required to create a system from its elements at constant pressure. It can be calculated from the internal energy and the pressure-volume product as: $$H = U + PV$$ This equation implies that enthalpy is equal to the internal energy plus the work done by the system against the external pressure. Entropy (S) is a measure of the disorder or randomness of a system. It can be calculated from the heat and temperature as: $$S = \int \fracdQT$$ This equation implies that entropy is equal to the integral of heat divided by temperature over a reversible process. The third law of thermodynamics states that the entropy of a pure crystalline substance at absolute zero temperature is zero. This means that there is only one possible microstate for a system at absolute zero temperature, which corresponds to perfect order. Phase Equilibria

### Phase Equilibrium in a One-Component System

### A phase is a homogeneous and physically distinct portion of a system that has uniform properties and a definite boundary. A phase equilibrium is a state where two or more phases coexist without any tendency to change their relative amounts or compositions. A one-component system is a system that contains only one chemical species. For example, water is a one-component system that can exist in three phases: solid (ice), liquid (water), and gas (steam). The phase equilibrium in a one-component system can be represented by a phase diagram, which shows the regions of stability and coexistence of different phases as a function of temperature and pressure. For example, here is a phase diagram for water: ![Phase diagram for water](https://upload.wikimedia.org/wikipedia/commons/thumb/0/08/Phase_diagram_of_water.svg/1200px-Phase_diagram_of_water.svg.png) The phase diagram for water shows that: - The solid phase (ice) exists at low temperatures and high pressures. - The liquid phase (water) exists at moderate temperatures and pressures. - The gas phase (steam) exists at high temperatures and low pressures. - The solid-liquid equilibrium line (or the melting curve) shows the conditions where ice and water coexist. - The liquid-gas equilibrium line (or the vaporization curve) shows the conditions where water and steam coexist. - The solid-gas equilibrium line (or the sublimation curve) shows the conditions where ice and steam coexist. - The triple point is the point where all three phases coexist at a unique temperature and pressure. - The critical point is the point where the liquid-gas equilibrium line ends and the liquid and gas phases become indistinguishable. The phase diagram for water is an example of a simple phase diagram. However, not all one-component systems have simple phase diagrams. Some systems may have more than one solid phase, such as polymorphs or allotropes. For example, carbon has two solid phases: graphite and diamond. Some systems may also have more than one critical point, such as helium. These systems have more complex phase diagrams that show the regions of stability and coexistence of different solid phases and supercritical fluids. The Behavior of Gases

### A gas is a phase of matter that has no definite shape or volume. It conforms to the shape and volume of its container. A gas consists of molecules that are in constant random motion and collide with each other and with the walls of the container. The behavior of gases can be described by equations of state, which relate the pressure, volume, temperature, and number of moles of a gas. One of the simplest equations of state is the ideal gas law, which assumes that the gas molecules are point-like particles that do not interact with each other or with the container walls. The ideal gas law can be written as: $$PV = nRT$$ where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature. The ideal gas law can also be written in terms of the molar volume (V_m = V/n) or the density (ρ = n/V): $$PV_m = RT$$ $$P = ρRT$$ The ideal gas law can be used to calculate various properties of an ideal gas, such as its molar mass, specific heat capacity, thermal expansion coefficient, etc. However, the ideal gas law is not accurate for real gases, especially at high pressures and low temperatures. Real gases deviate from the ideal gas behavior due to intermolecular forces and molecular size. One of the most common equations of state for real gases is the van der Waals equation, which corrects the ideal gas law by introducing two parameters: a and b. The parameter a accounts for the attractive forces between molecules, while the parameter b accounts for the excluded volume due to molecular size. The van der Waals equation can be written as: $$(P + \fracaV_m^2)(V_m - b) = RT$$ where a and b are constants that depend on the type of gas. The van der Waals equation can be used to calculate various properties of a real gas, such as its compressibility factor, critical point, etc. However, the van der Waals equation is not accurate for all gases and all conditions. There are other equations of state for real gases that have more parameters and more accuracy, such as the Redlich-Kwong equation, the Peng-Robinson equation, etc. The Behavior of Solutions

### A solution is a homogeneous mixture of two or more substances. A solution consists of a solvent and one or more solutes. A solvent is the substance that dissolves other substances. A solute is the substance that is dissolved by a solvent. The behavior of solutions can be described by thermodynamic concepts, such as activity, chemical potential, and Raoult's law. These concepts can help us understand how solutions affect various properties and processes, such as vapor pressure, boiling point, freezing point, osmosis, etc. Activity (a) is a measure of how much a solute behaves like an ideal solute in a solution. An ideal solute is one that does not interact with other solutes or with the solvent. Activity can be calculated from the mole fraction (x) and the activity coefficient (γ) of a solute as: $$a = x \gamma$$ where x is the ratio of moles of solute to total moles of solution, and γ is a dimensionless factor that depends on the type and concentration of solute. Activity can also be calculated from other concentration units, such as molality (m), molarity (M), or mass fraction (w). (μ) of a substance as: $$\mu = \mu + RT \ln a$$ where R is the universal gas constant, T is the temperature, and ln is the natural logarithm. The standard chemical potential is the chemical potential of a substance in its standard state, which is usually defined as pure solid, liquid, or gas at 1 bar pressure. Raoult's law is a simple rule that relates the vapor pressure of a solution to the mole fraction and the vapor pressure of a pure solvent. Raoult's law assumes that the solute and the solvent are both volatile and ideal. Raoult's law can be written as: $$P = x_s P_s$$ where P is the vapor pressure of the solution, x_s is the mole fraction of the solvent in the solution, and P_s is the vapor pressure of the pure solvent at the same temperature. Raoult's law can be used to calculate various properties of a solution, such as its boiling point elevation, freezing point depression, osmotic pressure, etc. Gibbs Free Energy Composition and Phase Diagrams of Binary Systems

A binary system is a system that contains two components. A component is a chemically independent constituent of a system. For example, copper and zinc are two components that can form a binary system. The Gibbs free energy (G) is a measure of the energy available for doing useful work in a system. It can be calculated from the enthalpy and entropy of a system as: $$G = H - TS$$ where H is the enthalpy, T is the temperature, and S is the entropy. The Gibbs free energy can also be calculated from the chemical potential and the number of moles of each component as: $$G = \sum_i n_i \mu_i$$ where n_i is the number of moles and μ_i is the chemical potential of component i. The Gibbs free energy composition diagram is a plot of the Gibbs free energy versus the composition of a binary system at constant temperature and pressure. It shows how the Gibbs free energy changes with different proportions of components in a system. For example, here is a Gibbs free energy composition diagram for a copper-zinc binary system at 600C: ![Gibbs free energy composition diagram for copper-zinc binary system](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Gibbs_energy_of_mixing_of_copper_and_zinc.svg/1200px-Gibbs_energy_of_mixing_of_copper_and_zinc.svg.png) The Gibbs free energy composition diagram for copper-zinc binary system shows that: - The curve labeled G_m is the Gibbs free energy of mixing, which represents the change in Gibbs free energy when pure components are mixed to form a solution. - The curve labeled G_x is the Gibbs free energy of an ideal solution, which represents the Gibbs free energy when components are mixed without any interaction or deviation from ideality. - The curve labeled G_e is the excess Gibbs free energy, which represents the difference between the actual and ideal Gibbs free energy due to interaction or deviation from ideality. - The dashed line labeled G_p is the common tangent line, which represents the lowest possible Gibbs free energy for a given composition. - The points labeled α and β are two phases that coexist at equilibrium. They have different compositions but equal chemical potentials and Gibbs free energies. - The region between α and β is called a two-phase region or a miscibility gap. It represents a range of compositions where two phases coexist at equilibrium. - The points labeled L and S are two phases that coexist at equilibrium with pure components. They have equal compositions but different chemical potentials and Gibbs free energies. - The regions labeled L and S are called one-phase regions or single-phase regions. They represent ranges of compositions where only one phase exists at equilibrium. The phase diagram is a plot of the phase boundaries versus temperature and composition for a binary system. It shows how different phases appear or disappear with changing temperature and composition. For example, here is a phase diagram for a copper-zinc binary system: ![Phase diagram for copper-zinc binary system](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Cu-Zn_phase_diagram.svg/1200px-Cu-Zn_phase_diagram.svg.png) The phase diagram for copper-zinc binary system shows that: Fe_2O_3 -742.2 Fe_3O_4 -1118.4 FeS