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.

Remote sensing involves measurements on a targetwithout getting in contact with it and it comprisestechniques for collecting, storing, and processinggeoreferenced and geospatial data to extractvaluable information. In this context, data refer torepresentations stored in computer memory, which canbe manipulated using computers to derive meaningfulinsights. Remote sensing imaging systems primarilywork with georeferenced images, capturing Earth'ssurfaces, environment, atmosphere, etc. Theseimaging systems may be carried by satellites orairborne platforms like airplanes or drones. Forsatellite-based imaging, revolution of the satellitearound continuously rotating Earth allows periodiccapture of images over the same area. There are twomain types of imaging systems: passive and active.In passive systems, sensors detect reflected andemitted electromagnetic (EM) waves from Earth'ssurface, from mainly two types of energy sources,namely sunlight during the day and terrestrial heatat night. These sensors operate within specificspectral bands, converting energy into electricalsignals stored as two-dimension (2-D) images. Theprinciple is similar to optical cameras.Additionally, energy from Earth's thermal emissioncontributes to night time imaging, particularly inthe thermal infrared (IR) or far IR bands. Passiveremote sensing involves capturing images acrossvarious spectral bands, resulting in multispectraland hyperspectral images of specific regions onEarth.

In active imaging systems, microwave radar (RAdioDetection And Ranging) technology is utilized. Aradar transmitter emits a pulse of an EM wave with aspecific wavelength (in the microwave band). Whenthis pulse strikes a target, some of its energyreflects back to the radar antenna to which theradar receiver is connected. The receiver capturesinformation about the location and geometry of thetarget by recording the phase and amplitude of thereturned signal. By scanning the radar beam over anarea, an image of that region is formed. One of thelimitations of radar imaging is that, the size ofthe transmitting antenna is required to be large forobtaining images of high spatial resolution.

An artificial neural network (ANN) is a network ofneural nodes or perceptual nodes.1 In a feed forwardneural network, each node is fed with a weightedinput vector and the net sum of the weighted vectorsfrom several such nodes is passed through anonlinear function, whose response is the output ofthat node. The layers formed by the input nodes andthe output nodes are known as the input layer andthe output layer, respectively. The layers formed byother nodes are known as hidden layers. A network ofseveral such layers (along with input and outputlayers) forms an ANN, as shown in Fig. 9.1.2 Theconventional ANNs have very few hidden layers,usually not more than three. Using only one or twohidden layers is also common in many applications.In contrast, deep neuralarchitectures have relatively more numberof layers. Even more than 100 hidden layers is notuncommon. This is one of the distinguishedcharacteristics of deep neural networks (DNN) froman ordinary ANN.

The concepts used in deep neural computations aredecades older. In fact, they involve the same neuralnetwork computations as in a simple ANN model. Theseconcepts were introduced in 1980's and their basicprinciples still remain the same. However, a boom inusing deep architectures after almost three decadesof their proposition is mainly attributed to theadvancement in technology and science.

Key Concepts

• • Electricity scenario in India and the need for transition to green energy in the country

• • Indian pledge at CoP-26 at Glasgow and the targets of 2030

• • National solar mission and major initiatives that led to exponential growth in solar energy installation in India

• • Net-zero target of India and road map for achieving it

• • Major solar power projects in India

• • Various policies and government organizations involved in achieving the target of solar PV deployment in India

• • Changes required in the grid in view of massive deployment of variable and uncertain sources of electricity

Introduction

India is now the most populous country in the world, with almost 18% of the global population. Being a developing country and one of the major evolving economies, the electricity demand in the country is also growing. India ranked fifth in terms of installed capacity and third in terms of electricity produced, in 2018, in the world. India's annual per capita electricity consumption, although, is about 1122 kWh, which is much lower than the world average of 2674 kWh per year, but this number too is one of the fastest-growing. In the last decade the installed capacity of the Indian grid has increased by more than 200 GW. At this rate India is set to become the biggest electric load centre in the world by 2030 with about 1.5 billion people.

Being the most populous developing country, Indian response to the climate crisis is key to the success of sustainable development and the climate protection mission. India historically accounted for less than 5% of the global emissions.

Directed graphs, also known as digraphs are a fundamental structure in graph theory, where the relations between two vertices are allowed to be asymmetrical. An edge as we have known so far, has always signified a two-way relationship between two vertices. But in directed graphs, a directed edge has a designated direction in which you may go from say a to b but not always from b to a. This directionality introduces the concept of ordered pairs where each edge has a distinct starting point (source) and an ending point (target). Understanding their properties and behavior is crucial for solving problems related to reachability, hierarchy and optimization. This chapter explores the core concepts of directed graphs including their representation, traversal techniques and properties while also providing a succinct introduction to the world of networks.

Trees form an important class of graphs, in graph theory. Trees hold a special place in graph theory, due to their simplicity, versatility and widespread applications. From representing hierarchical data structures like family trees and organizational charts, their unique properties, such as having a single path between any two vertices, make them indispensable in solving complex problems with efficiency and clarity. In this chapter, we will explore different types of trees such a spanning trees, rooted trees and binary trees.We will also study different algorithms to construct minimum spanning trees.

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.

3.1 Continuous Functions

3.1 Continuous Functions

We are familiar with the notion of continuous function in real analysis, in general, in a metric space. Recall Definition 2.1.7 of continuity of a function in a metric space. We know that a function is continuous on a set if it is continuous at all its points.

Now the question is, how can we define continuity of a function on a set in the absence of a distance function? In Subsection 2.1.2, we discussed that we need to define continuity of a function in terms of open sets in order to define it in a topological space. Let us see how this can be done. We recall the definition in a metric space below and try to see how we can generalize it to a topological space by bringing the open sets into picture.

Let (X, d) and (Y, ρ) be metric spaces and f : X → Y be a function. Then f is continuous if for every a ∈ X and ∈ > 0, there is δ > 0 such that ρ(f(x), f(a)) ∈ whenever d(x, a) δ.

That is,

f(Bd(a, δ)) ⊆ Bρ(f(a), ∈). (3.1)

Equivalently,

Bd(a, δ) ⊆ f−1(Bρ(f(a), ∈ )).

Thus, in order to have continuity of f, we must have Bd(a, δ) ⊆ f−1(Bρ(f(a), ∈ )). In other words, the set f−1(Bρ(f(a), ∈ )) must contain an open ball Bd(a, δ) containing a. That is, a is an interior point of the set f−1(Bρ(f(a), ∈ )). In fact, it can be easily shown that the set f−1(Bρ(f(a), ∈ )) is open in X. Note that the ball Bρ(f(a), ∈ ) is open in Y. Thus, if we have the condition that the “inverse image of every open subset of Y, under f, is open in X”, then we can conclude the continuity of f. This brings the open subsets of X and Y into picture.

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.

How do you pair individuals or objects in an optimal manner? This question was first framed in the context of marriage and other pairings, but spurred a great deal of research that had combinatorial optimization as its goal. Matchings are a cornerstone concept in graph theory, offering powerful solutions to optimization problems in network design, resource allocaton and scheduling. The concept of matchings play a significant role in finding perfect matches in partnership scenarios, solving complex assignment problems thus providing the framework for making optimal pairings. This chapter also deals with coverings that provide insights into efficient resource allocation. In addition to these concepts, we will explore an algorithm designed to find the maximum matching in bipartite graphs which culminates in the powerful Kuhn–Munkres algorithm for solving the personnel assignment and optimal assignment problems.