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.

Stationary charges give rise to electric fields. Moving charges give rise to magnetic fields. In this chapter, we explore how this comes about, starting with currents in wires which give rise to a magnetic field wrapping the wire.

In this chapter, we rewrite the Maxwell equations yet again, this time in the language of actions and Lagrangians that we introduced in the first book in this series. This provides many new perspectives on electromagnetism. Among the pay-offs are a deeper understanding, via Noether’s theorem, of the energy and momentum carried by electromagnetism fields. This will also allow us to explore a number of deeper ideas, including superconductivity, the Higgs mechanism, and topological insulators.

Take an electric charge and shake it and it will produce light. The purpose of this chapter is to explain how this works.

There are four forces in our universe. Two act only at the very smallest scales and one only at the very biggest. For everything inbetween, there is electromagnetism. The theory of electromagnetism is described by four gloriously simple and beautiful vector calculus equations known as the Maxwell equations. These are the first genuinely fundamental equations that we meet in our physics education and they survive, essentially unchanged, in our best modern theories of physics. They also serve as a blueprint for what subsequent laws of physics look like.

This textbook takes us on a tour of the Maxwell equations and their many solutions. It starts with the basics of electric and magnetic phenomena and explains how their unification results in waves that we call light. It then describes more advanced topics such as superconductors, monopoles, radiation, and electromagnetism in matter. The book concludes with a detailed review of the mathematics of vector calculus.

Newtons laws of motion are not the last word in classical mechanics. In the 250 years after Newton, physicists and mathematicians found ways to reformulate classical mechanics, providing powerful tools to solve problems but, equally as importantly, giving us a new perspective on the laws that govern our universe. This chapter takes the first step in this direction. We will introduce the wonderful principle of least action, a simple rule that underlies all known laws of physics. This will give us new insights, not least the wonderful Noethers theorem, relating symmetries to conservation laws.