1A Calculation of the number of accessible states to an ideal gas

We consider an ideal gas enclosed in a container of volume 𝑉 at a temperature 𝑇. The gas consists of 𝑁 number of molecules, each of mass 𝑚. Suppose the total energy of the system lies in a narrow range from (𝐸 − 𝑑𝐸) to 𝐸. Any molecule of the ideal gas lying within this energy range is described by a state having an elementary volume

where 𝑞′𝑠 and 𝑝′𝑠 are, respectively, the position and momentum coordinates of the molecules of the gaseous system.

INTRODUCTION

Classical mechanics is mainly based on Newton's laws of motion and gravitation. Initially, it was thought that Newton's second law of motion was valid and applicable at all speeds. But new experimental evidence showed that Newton's second law of motion is valid and applicable at low speeds and invalid when the object is moving at high speeds comparable to the velocity of light. This failure of classical mechanics led to the development of the special theory of relativity by young physicist Albert Einstein in 1905, which showed everything in the universe is relative and nothing is absolute. Relativity connects space and time, matter and energy, electricity and magnetism, which are useful and remarkable to our understanding of the physical universe.

The special theory of relativity is applicable to all branches of modern physics, high-energy physics, optics, quantum mechanics, semiconductor devices, atomic theory, nanotechnology, and many other branches of science and technology.

The theory of relativity has two parts: the special theory of relativity and the general theory of relativity. The special theory of relativity deals with the inertial frame of references, while the general theory of relativity deals with the accelerated frame of references. Some common technical terms that are frequently used in relativistic mechanics are as follows:

• 1. Particle:A particle is a tiny bit of matter with almost no linear dimensions and is considered to be located at a single place. Its mass and charge define it. Examples include the electron, proton, and photon, among others.

Learning Outcomes

After reading this chapter, the reader will be able to

• Understand various thermodynamic potentials such as internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy and their applications

• Calculate the magnetic work done by a paramagnetic system and understand the process of creating low temperatures using the principle of adiabatic demagnetization

• Apprehend the idea of first and second-order phase transitions and Clausius–Clapeyron and Ehrenfest equations related to the phase transitions, respectively

• Derive Maxwell's thermodynamic relations

• Apply Maxwell's thermodynamic relations to derive energy equations, 𝑇 –𝑑𝑆 equations, and other thermodynamic relations connecting 𝐶𝑃 and 𝐶𝑉

• Derive Joule–Kelvin coefficient for ideal and real gases like Van der Waals gas

• Describe Joule's experiment in case of adiabatic expansion of ideal and real Gases

• Understand the Joule–Thomson effect for real and Van der Waals gases through porous plug experiment and the temperature of inversion

• Solve numerical problems and multiple choice questions on thermodynamic potentials, Maxwell's thermodynamic relations, and Joule–Kelvin coefficient

10.1 Introduction

The term thermodynamic potentials refers to a specific measure of the capacity of a thermodynamic system to perform work. It is a key concept in thermodynamics and encompasses four variables: internal energy 𝑈 , Helmholtz free energy 𝐹 , enthalpy 𝐻, and Gibbs free energy 𝐺. The choice of the suitable thermodynamic potential depends upon the specific conditions of the system—whether any isolated, closed, or open systems. This means each of these four potentials has its unique usage scenario and interpretation. These potentials are paramount in describing the energy changes within systems. These potentials are extensive state variables of dimensions of energy and are introduced to account for specific constraints such as isothermal, adiabatic, isochoric, and isobaric processes in a thermodynamic system. Their purpose is to allow for simple treatment of equilibrium for systems interacting with the environment. Starting from the first and second laws of thermodynamics, the differential form of four thermodynamic potentials are derived, and these are called fundamental equations.

Learning Outcomes

After reading this chapter, the reader will be able to:

• List the differences between ideal and real gas

• List the experiments that depicted the behavior of real gases over a large range of pressures and temperatures

• Demonstrate the meaning of liquid–gas interface, critical volume, critical pressure, and critical temperature

• Derive the equation of state of a real gas considering the effect of pressure and volume

• Obtain the reduced equation of state, the law of corresponding state, and the compressibility factor

• Compare and contrast the Van der Waals equation of state with experimental results on CO2 due to Andrews

• Solve numerical problems and multiple choice questions on the Van der Waals equation of state, reduced equation state, and critical constants of a gas

5.1 Introduction

The foundation of kinetic theory of gases (KTG) is based on two important assumptions: (i) the volume occupied by the molecules of the gas is negligible compared to the total volume of the container, and (ii) no appreciable intermolecular attractive or repulsive forces are present among the molecules. A gas is said to be an ideal one when it conforms exactly to these tenets of the KTG. According to the KTG, such an ideal gas of 𝑛 mole obeys the equation of state: 𝑃 𝑉 = 𝑛𝑅𝑇. It is the task of the experimental physicists to test the validity of this equation of state over the whole range of physical parameters such as pressure and temperature. There are a large number of direct and indirect experimental pieces of evidences which clearly indicate that in reality, gases do not behave ideally, that is, the equation 𝑃 𝑉 = 𝑛𝑅𝑇 is not satisfied by the real gases over the entire range of the above-mentioned physical parameters. Real gases deviate from ideal behavior, especially at high pressures and low temperatures.

1.1 Introduction

“Thermodynamics” is the branch of science that deals with the macroscopic properties of matter. In this branch of physics, concepts about heat and work and their inter-conversion, energy and energy conversion, and working principle of heat engines with their efficiency are mainly discussed. The name “thermodynamics” was originated from two Greek words: “therme” means “heat” and “dynamics” means “power” or “energy”. Thus, matter related to heat and energy is primarily paid attention in this subject. Further, it is believed that the term “thermodynamics” arises from the fact that the macroscopic thermodynamic variables used to describe a thermodynamic system depend on the temperature of the system.

Thermodynamics is the branch of physics in which the system under investigation consists of a large number of atoms and molecules contributing to the macroscopic matter of the system. The average physical properties of such a thermodynamic system are determined by applying suitable conservation equations such as conservation of mass, conservation of energy, and the laws of thermodynamics in equilibrium. The equilibrium state of a macroscopic system is achieved when the average physical properties of the system do not change with time and the system is not driven by any external driving force during the course of investigation. The interrelationships among the various physical properties are established with the help of associated thermodynamic relations derived from the laws of thermodynamics. These average (macroscopic) properties of thermodynamic systems are determined from the macroscopic parameters such as volume 𝑉 , pressure 𝑃 , and temperature 𝑇 , which do not depend on the detailed positions and ocities of the atoms and molecules of the macroscopic matter in the system. These macroscopic quantities are called thermodynamic coordinates, variables or parameters. Further, these macroscopic properties depend on each other. Therefore, from the measurements of a subset of these properties, the rest of them can be calculated using the associated thermodynamic relations.

5A Saha ionization equation

At a high enough temperature and/or density, the atoms in a gas suffer collisions due to their high thermal energy, and some of the atoms get ionized, making an ionized gas. In this process, a number of electrons that are normally bound to the atom in orbits around the atomic nucleus become free and thus form an independent electron gas cloud coexisting with the surrounding gas of atomic ions and neutral atoms. These ionized atoms and electrons generate an electric field that causes motion of the charges, and a current is generated in the gaseous medium. This current produces a localized magnetic field. The state of matter thus created is called plasma. In thermal equilibrium, the ionization state of such a gaseous system is related to the ionization potential, temperature, and pressure of the system. Thus, the Saha ionization equation. expresses how the state of ionization of any particular element in a star changes with varying temperatures and pressures. This equation takes into account the combined ideas of quantum mechanics and statistical mechanics for its derivation and is used to explain the spectral classification of stars. This equation was developed by the Indian astrophysicist Prof. Meghnad Saha in 1920. 5A.1 Derivation of Saha ionization equation According to Prof. M. N. Saha, the temperatures in the interior of stars are extremely high, and the elements present there are mostly in the atomic state. Saha argued that under the prevailing conditions inside the stars, atoms move very rapidly and undergo frequent collisions. In the process of such collisions, valence electrons are stripped off from their orbits. This is referred to as thermal ionization and is accompanied by electron recapture to form neutral atoms. The degree of such thermal ionization depends on the temperature of the star. Using the Saha ionization equation, a general relation between the degree of thermal ionization and the temperature can be obtained from the statistical description of plasma in thermodynamic equilibrium.

3A Temperature in Thermodynamics

For an infinitesimal reversible process, a combination of first and second laws of thermodynamics results

where 𝑌 𝑑𝑋 denotes the generalized expression for work done by the system, 𝑑𝑆 is the change in entropy, and 𝑑𝑈 is the change in internal energy of the system. Equation (16) leads to the definition of temperature 𝑇 as

Thus, equation (17) indicates that the temperature at any point depends on the slope of the 𝑆 − 𝑈 curve. If the slope of this curve (point 𝐴 in Figure 3A.1) is positive, the temperature will be positive. On the other hand, the temperature will be negative for the negative slope of the curve (point 𝐶 in Figure 3A.1).

The book Heat and Thermodynamics: Theory, Problems, and Solutions is an informal, readable introduction to the basic ideas of thermal physics. It is aimed at making the reader comfortable with this text as a first course in Heat and Thermodynamics. The basic principles and phenomenological aspects required for the development of the subject are discussed at length. In particular, the extremum principles of entropy and free energies are presented elaborately to make the content of the book comprehensive. The book provides a succinct presentation of the material so that the student can more easily determine the major objective of each section of a particular chapter. In fact, thermal physics is not the subject in physics that starts with its epigrammatic equations—Newton’s, Maxwell’s, or Schrodinger’s, which provide accessibility and direction. Instead, it (thermodynamics) can be regarded as a subject formed by the set of rules and constraints governing interconversion and dissipation of energy in macroscopic systems. Further, the syllabus of statistical mechanics for graduate students has changed significantly with the introduction of National Education Policy 2020.

Thermal physics has established the principles and procedures needed to understand and explain the properties of systems consisting of macroscopically large numbers of particles, typically of the order of 1023 or so. Examples of such collections of systems include the molecules in a closed vessel, the air in a balloon, the water in a lake, the electrons in a piece of metal, and the photons (electromagnetic wave packets) emitted by the Sun. By developing the macroscopic classical thermodynamic descriptions, the book Heat and Thermodynamics: Theory, Problems, and Solutions provides insights into basic concepts and relationships at an advanced undergraduate level. This book is updated throughout, providing a highly detailed, profoundly thorough, and comprehensive introduction to the subject. The laws of probability are used to predict the bulk properties like stiffness, heat capacity, and the physics of phase transition, and magnetization of such systems.