Equilibrium, Thermodynamic Parameters and Response Functions

P.K. Ahluwalia

Learning Outcomes

After studying this module, you shall be able to

• Understand and analyse the way term equilibrium is used in thermodynamics and statistical mechanics.

• Understand the importance of long observation time while describing equilibrium

• Learn that the value of thermodynamic parameters is a mean or average value which remains constant over the observation time

• Distinguish between extensive and intensive parameters and that these are homogeneous functions of order 1 and 0 respectively.

• See how intensive parameters are related to extensive parameters

• Understand relaxation time in the context of a perturbed system going to equilibrium

• Learn about the concept of generalized forces, generalized coordinates and response functions.

1. INTRODUCTION

State of equilibrium and non-equilibrium are intuitively the most subtle concepts in the study of thermodynamics and statistical mechanics. It is one of the very fundamental questions to pose that when can we say that a system is in equilibrium and when is it not. In this module we shall explore the meaning of equilibrium as applied to thermodynamics and statistical mec hanics in the context of macroscocopic systems encountered in the physical world. Indeed thermodynamic equilibrium is a macroscopic phenomenon, it has no meaning in the case of an isolated particle. A macroscopic system consists of a large number of particles ( N >> 1 ) under a given physical condition in a particular equilibrium state described by various parameters such as Pressure P, temperature T, magnetic field H etc. which can be tuned from outside as external conditions. As the external conditions change the state of the system may change from one equilibrium state to another. Following table 1 describes state of matter in equilibrium at specified external conditions. If the external conditions change the system may go from one state of equilibrium to another,

Table 1 States of Matter in Equilibrium

External condition	Matter	State
373 K at normal pressure	Water	Boils
273 K at normal pressure	Water	solidifies
2.17 K at normal pressure	Helium	Superfluid
Below 1043 K at normal pressure	Iron	Ferromagnet
At 170nK at normal pressure	Gas of rubidium atoms	Bose Einstein Condenstate
Below 7.19oK at normal pressure	Lead	Superconductor

2. Meaning of Equilibrium vis-a-vis observation time

An intuitive and plausible way to understand the meaning of equilibrium is with relation to observation time. It will be nice to look at a few examples. Let us take the case of warm water in a thermos flask, it remains warm over a long period of time implying that if you record the temperature of water over a period of an hour you will find it has not changed. Pour the same water in a cup. For a couple of minutes you may find that water remains at the same temperature, but if you wait for half an hour you will find that water has cooled down and it is not in the same state when it was poured into the cup.

Also take a cup of water at room temperature, you can notice it has a definite volume and this volume stays the same after a few hours, however, if you observe the level of water say after three or four days you will notice that, may be half of water has evaporated. This has happened because evaporation caused removal of the water molecules from the cup, though temperature may have stayed the same.

Therefore, one can conclude that equilibrium has a meaning for a given system provided observation time is not too large.

By keeping water in a thermos flask, you have been able to prolong the state of equilibrium through isolation, making cooling of water a very slow process. In simple words we can say that if we observe a system over a short enough time that its state remains unchanged, system is in equilibrium. It must be underlined here that no system is in a state of absolute equilibrium i.e. in equilibrium over an infinite period of observation time. Absolute equilibrium is in a sense idealization, which does not normally happen in reality because no system can be practically made completely isolated.

3. Extensive and Intensive Parameters

Before we further discuss the meaning of thermodynamic equilibrium, it will be nice to understand the meaning of thermodynamical parameters e.g. Temperature, Pressure, Volume, Entropy etc. which may be needed to identified for a system in equilibrium. Infact by specifying these parameters, we specify the state of a system. Broadly, thermodynamical parameters can be classified into two types:

(i) Extensive parameters
(ii) Intensive parameters

Extensive parameters: As the name suggests these are the parameters whose value depends on the extent of the system or simply put the size of the system. These parameters are diretly proportional to the size of the system. If size of the system becomes times the original size the corresponding parameter should also become times the original value There is another way of visualizing the meaning of an extensive parameter. Imagine we have a macroscopic system having two parts as shown in the figure 1 below

_{Figure 1 A macroscopic system made up of two parts, Part I and Part II}

If an extensive thermodynamic parameter of the composite system is Y and of part I and two are respectively YI and YII then Y is said to be extensive if Y=YI+YII.

For example mass, volume, entropy, Energy are extensive parameters. Therefore, an extensive parameter of a system in equilibrium has a value which is sum of the values of each part of the system.

Intensive parameters: Intensive thermodynamical parameters of a macroscopic system in equilibrium is independent of the size of the system. With reference to figure 1, if Y is an intensive parameter than Y=YI=YII. For example, temperature, pressure, chemical potential. Here, again it is worth noting that intensive parameters are homogeneous functions of zero degree of the variables on which they depend. By checking the degree of thermodynamic parameters, we can keep an eye o n their intensive and extensive nature.

For example knowing equation of state PV= NkBT and internal energy U=3/2 NkBT of the classical ideal gas we can write entropic forms of the equation of state and internal energy and hence classical entropy of the monoatomic gas and deduce functional form of entropy as a function of U,V,N.

4. Thermodynamic Equilibrium: Three Types of Equilibrium

For a macroscopic system to be in thermodynamic equilibrium in fact it has to be in three types of equilibrium:

(i) Thermal equilibrium

(ii) Mechanical equilibrium

(iii) Chemical equilibrium

Thermal equilibrium means that every part of the system and the surroundings are not experiencing any change in temperature with time. Mechanical equilibrium means that net external forces and net external torques acting on the system are zero. Chemical equilibrium means that there are no chemical reactions taking place in the system and the concentration of the constituent particles of the system is constant with respect of time. Thermal equilibrium is related with constancy of temperature. Mechnical equilibrium means constancy of pressure and chemical equilibrium means constancy of chemical potential.

5. Statistical Equilibrium and Relaxation time

If a macroscopic system and its macroscopic subparts have macroscopic physical quantities equal to their mean values, the system is in Statistical equilibrium. In other words if a macroscopic system is observed over a long enough time than for most part of the observation time it will be in a state of statistical equilibrium. It is possible that a macroscopic system may by some external interaction at a given instant of time not in statistical equilibrium, it will after the disturbance ceases to act shall come back to the state of equilibrium. This time which it takes to come back to the state of equilibrium is known as relaxation time.

6. Non-equilibrium and Irreversible Processes

When a system is not in equilibrium there is some action from outside the system. Sometimes we wonder that most the phenomenon which we see around us are not in a state of thermodynamic equilibrium. For example weather pattern. If we analyse closely we shall find that the cause of weather patterns is from outside the earth in the form of solar radiations. Also energy flows out of the earth also. As long as this exchange of energy shall continue system will not be in a state of equilibrium. When by applying a potential difference across a wire current flows, we have created a state of non-equilibrium. As long as the system is in this state of energy input the non-equilibrium process will last.

Another simple example that brings in non-equilibrium, is the case of two cups of water at different temperatures brought in thermal contact with each other. In this case system shall evolve from a state of non-equilibrium to an equilibrium and the two cups of water will come to the same temperature. Now if we desire that the water in the two cups comes back to their original temperatures it will not be possible because the process is irreversible. In this course we are only interested in systems in systems in equilibrium only

7. Generalised Forces and Response Function

Whenever an external force is applied on a macroscopic system it shows a response to it and do appropriate work on the body. The response to the external force is a measurable quantity and is described by a response function. Such forces are many and are appropriately called generalized forces. Each generalized force has a corresponding conjugate variable parameter. Change in the value of generalized parameter multiplied by generalized force gives the work done of the system. Table 1 below gives the list of generalized forces and co-ordinates of interest to us in thermodynamics and statistical mechanics

Table 2 Generalized forces and co-ordinates

Generalized Force	Generalised Co-Ordinates (Conjugate parameter or Variable)	Work done
Mechanical force, f	Position, x	f dx
External Magnetic Field, h	Magnetic Moment, M	h dm
Torque, n	Angle, θ	n d0
Pressure P	Volume, V	– PdV
Chemical potential u	Number of particles, N	udN
Temperature T	Entropy, S	TdS

It is interesting to note that entropy is a generalized coordinate which is a response to temperature in a collective way of small scale microscopic changes. Generalised co-ordinates are large scale co-ordinates or thermodynamic co-ordinates. From the third column in the Table 2 it is clear that each of the generalized force causes a change in the corresponding generalized co-ordinate. Thus chemical potential is a generalized force which causes a change in number of particles in t he system. Temperature is a generalized force which causes a change in entropy of the system.

8. Summary

In this module we have learnt

• The meaning of equilibrium for a macroscopic system and that it may involve thermal equilibrium or mechanical equilibrium or chemical equilibrium or all three.

• That observation time decides whether we are looking at a system in state of equilibrium or not. Large observation time may allow system to change the state.

• That thermodynamic parameters are of two types, e xtensive and intensive. Extensive parameters are homogeneous functions of order 1 and intensive parameters are homogeneous functions of order 0. Therefore, extensive parameters scale with size, wheras intensive parameters are independent of the size of the system.

• The meaning of relaxation time vis-à-vis the transition of a perturbed system into equilibrium state once the perturbation is removed.

• The notion of irreversible process in the transition of a system from non-equilibrium system.

• The concept of generalized forces and corresponding generalized co-ordinates encountered in thermodynamics related to the response of the system in terms of a measurable quantity.

References:

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2. Greiner W., Neise L., Stocker H., “ Thermodynamics and Statistical Mechanics”, New York, Springer Verlag, 1995.

3. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008.

4. Matveev A.N., “ Statistical Physics,” Moscow: Mir Publishers, 1985.

5. Pathria R.K. and Beale P. D., “Statistical Mechanics”, 3rd ed., Elsevier, 2011.

6. Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rd Edition, Oxford: Pergamon Press.,1982.

7. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008.