An Overview of Thermodynamics-I

P.K. Ahluwalia

Learning Outcomes

After studying this module, you shall be able to

• know the broad learning goals of this course on Statistical Mechanics for PG students

• know the overall place of studying Statistical Mechanics in the study o f physics

• get an overview of representative models of statistical mechanics and their application in various areas of physics

• know the importance of learning statistical methodology beyond physics in the new and emerging areas in biology, geology, ecology, economicsand financial markets

1. INTRODUCTION

Thermodynamics embodies a systematic empirical knowledge about thermal behavior of the macroscopic systems in terms of the state of the system. Therefore, it is a phenomenological theory with well- established terminology. The aim of this module and module VI is to have a recapitulation of the various concepts, which are needed in statistical mechanics wherever a link with the state of the system is to be made experimentally. These states most often shall be in a state of equilibrium as described in module III.

2. Thermodynamical Terms Introduction

State Variables: State of the thermodynamic system which is a macroscopic system is described in terms of a minimum number of variables known as state variables/ parameters which can be measured and for each of these we have an intuitive feeling. These are volume  V , pressure P , temperature T , magnetic field H etc. These are also known as thermodynamic variables or macroscopic variables.

Equation of State: The functional relationship between these minimum numbers of state variables is called equation of state.

For example to describe the state of a gaseous system we require three thermodynamic variables,
P, V, T, the equation of state of such system can be expressed as f (P. V. T.) = 0. Ideal gas equation for an ideal gas,  PV = RT  and van der Waal’s equation of state for a real gas, (P + a/v² ) (V-b) = RT are two most well-known examples of an equation of state.

Table 1 below gives some examples of thermodynamic systems, the state variables normally used to describe their state and the corresponding equation of state. Some of these equations of state are familiar and some are not familiar. However, we must remember there are numerous equations of state available to describe the variety of thermodynamic systems, involving solids liquids and gases.

Table 1 System, System Variables and Equation of State

 3. Thermodynamic Transformation Types

Thermodynamic Transformation Types : It is nothing but change of state by change in external conditions of the system. These changes can be classified as quasi static, reversible and irreversible.

In an irreversible process, change occurs so fast that during the process, thermodynamic variables P, V, T, can not be defined. This actual means that during this process system is not in equilibrium because of number of unexpected reasons, such as friction, turbulence or such situations, or the observation time being much smaller than the time intervals involved at the molecular/atomic level. A non-equilibrium process is always irreversible.

A quasi static process is the process, when it occurs so slowly that at each step it is in equilibrium described by thermodynamic variables.

If the process on removal of external conditions comes back to its original state via the same path or intermediate states, it is said to be reversible. Here it needs to be emphasized that if time intervals involved in going from non-equilibrium state to equilibrium state in each step of the reversible process is much smaller than the characteristic observation time scale of the system, it can be treated at equilibrium in each of the steps of the process.

4. Inte rnal Energy(U), Heat (Q) and Work (W)

When a system is in a state of equilibrium, which happens when it is isolated, energy of the system is a conserved quantity, i.e. Total energy E  is a well-defined quantity. If a system is in thermal equilibrium with its surroundings say a heat reservoir, its energy may fluctuate, but its average energy denoted by <E>  is again well defined. In certain systems this total energy may contain a part which may be termed as mechanical energy, for example part of the energy possessed by the system in a gravitational field or kinetic energy of the system as a result of its overall motion and these energies can be used easily, need to be separated from the thermodynamic equilibrium, The rest of the energy possessed by the system is then called internal energy U. If there is no mechanical energy of the say center of mass of the system in thermal equilibrium, the total energy of the system is the internal energy.

The question which then arises is  can we change the internal energy of  the system? Answer is  yes, by performing work (W) on the system or by supplying heat (Q) to it. Interestingly, energy which you supply to the system by placing it on a hot plate and can not be considered as work is heat. Work can be performed on the system by mechanical (say for example by compressing a gas) means or electromagnetic means.

Heat and work can be given to the system and extracted from it by variety of methods. This may lead to change in its potential energy. This is what first law of thermodynamics tells us. If we supply a system Q  amount of heat and perform W amount of work, the change in internal energy U= Q + W. It needs to noted that Q and W are not state variables, you cannot describe these when a system is in thermal equilibrium. For a system in equilibrium it is meaningless to say that there is this much amount of heat and there is this much amount of work. Heat and work come into picture when we go from one state of equilibrium to another.

In differential form, first law of thermodynamics can be written as

5. Notion of Thermo-dynamical or Macroscopic Generalized Co-ordinates and Gene ralised Forcestroduction

6. Basic Postulates of Equilibrium Thermodynamics: Phenomlogical Way of Looking at Macroscopic Systems

The fundamental problem which needs to be formulated for a given macroscopic system in equilibrium thermodynamics is how to determine the final equilibrium state of a macroscopic system and describe it. This problem is addressed through a set of basic postulates of Thermodynamics, which cannot be proved from some known laws but must be taken as fundamental axioms whose validity lies in their applications predictions matching with experimental results pointing towards its phenomenological methodology. Incidentally, thermodynamic or macroscopic state of a system can be fully described in terms of a very small number of thermodynamic variables with introduction of additional number of thermodynamical variables to take care of complexities as and when these arise.

These postulates are:

First Postulate: The equilibrium states of a macroscopic system can be fully described in terms of a set of extensive variables including its internal energy U. The other extensive parameters being volume V  and number of particles in the system N or more, as per the typical thermodynamic system.

Second Postulate: There exists an extensive parameter/variable called entropy S, which is a function of theses extensive parameters i.e.  S = S (U, V, N) which is defined for all equilibrium states of the system. This function contains all the information of the system and is a fundamental equation of the system. It has following properties:

Third Postulate: If a macroscopic state is not in a state of equilibrium, its entropy will be less than the entropy of the equilibrium state with same values of extensive variables on which it depends. However, left to itself for

7. Fundamental Thermodynamic Relations and Intensive Variables:

8. Euler Relation and Gibbs Duhem Relation:

9. Summary

In this module we have learnt

• That state of a system is described in terms of thermodynamic variables also called macroscopic variables.
• That equation of state is a functional relation ship between minimum number of state parameters needed for describing the state of the system.
• That there are variety of thermodynamic processes which may occur in a themodynamic system and can be classified as quasistatic, reversible and irreversible processes.
• That first law of Thermodynamics is law of conservation of energy which relates internal energy, a perfect differential, with heat and work which are inexact differentials of heat and work done under various conditions.
• About processes undergone by an ideal gas under different conditions leading to different equations of state, work done and heat capacity of the system.  Basic postulates of equilibrium thermodynamics provide us a phenomenological way of looking at macroscopic systems such that second postulate defines entropy and third postulate leads to Nernst’s theorem
• Understand the entropic representation, energy representation, number representation and volume representation of thermodynmamic relations.
• How to get intensive variables in terms of extensive variables starting from entropic representation and energy representation
• The derivation of Euler relation and Gibbs-Duhem relation using extensive property of internal energy.

References :

• Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008.
• Matveev A.N., “ Statistical Physics,” Moscow: Mir Publishers, 1985.
• Rao Y.V.C., “ Postitutional and Statistical Thermodynamics,” New Delhi: Allied Publishers, 1994.
• Fermi E., “ Notes on Thermodynamics and Statistics,” Chicago, The University of Chicago Press, Phonix edition, 1966.
• Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008
• Greiner W., Neise L., Stocker H., “ Thermodynamics and Statistical Mechanics,” New York, Springer Verlag, 1995.