thermodynamics unit ii

8/12/2019 Thermodynamics Unit II

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UNIT-II

THERMODYNAMICS

2.1 Introduction:

Energy is manifested in nature in various forms. Most of the forms of energy are converted into

heat. Heat is converted into work, taking into account of all these changes, the matter of thermodynamics

deals with interaction of various forms of energy in a system. Subject literally, thermodynamics means

flow of heat. Over the years scientists have observed that the behaviour of bulk matter (macroscopic

systems) follows certain generalizations or restrictions. These generalizations arrived at are formulated as

three fundamental laws, viz., the zeroth, the first and the second law of thermodynamics. It provides

datum for the measurement of entropy at very low temperature. Despite having no direct proof for these

laws no contradictory proof has been given so far when the behaviour of matter is considered in bulk.

The importance of thermodynamics to chemists lies in its ability to predict the feasibility of a

chemical reaction and the extent of it under the given set of conditions.

2.2 Scope and Limitations:

The introduction of mathematical principles into the basic laws of thermodynamics enables us to

derive results which are widely used in physics, chemistry and engineering. For instance in the field of

chemistry the derived equations are applicable in the study of phase equilibrium, solutions, chemical

equilibrium, electrochemistry etc.,

Though the laws of thermodynamics are of wide applicability, there are certain limitations for

their use. These are concerned with the behavoiur of macroscopic matter and totally ignored of the nature

or even the existence of individual atoms and molecules. They do not provide any idea about the speed or

rate of a physical process or a chemical reaction.

2.3 Approaches to thermodynamics:

The study of thermodynamics can be approached in two ways viz., macroscopic and microscopic.

In the macroscopic approach the structure of the matter is not considered and only a few measurable

variables.

Classical thermodynamics is based on the macroscopic approach. In contrast with the

microscopic approach, where the knowledge of structure of matter is essential and large numbers of


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variables, whose values cannot be measured is used. Statistical thermodynamics rests upon this

microscopic approach.

2.4 Energy:

Energy is the most fundamental manifestation in nature. It accompanies all physical changes and

chemical transformations. Energy exists in nature in various forms such as potential energy, kinetic

energy, chemical energy, thermal energy, nuclear energy, electrical energy, mechanical energy etc. the

various forms of energy are inter-convertible. Experience shows that whenever one form of energy

disappears, an equivalent amount of another form appears. The law of thermodynamics governs the

principle of transformation of one kind of energy into another.

2.5 Zeroth law of thermodynamics or the law of thermal equilibrium:

The unusual name associated with this law is due to the belated realization of its importance to

the concept of temperature when the other two laws of thermodynamics were rather well developed.

Consider two system A and b separated by a wall. If the wall is an insulting one, the systems will

not have any influence on each other. On the other hand, if the wall is thermally conducting, the two

systems will be in thermal contact each other. After sometimes, they will attain the same degree of

hotness. When such a stage is reached, the system are said to be in thermal equilibrium (Fig 2.1) with

each other. Now, consider three systems A, B and C. let each of the systems, A and B in direct thermal

equilibrium with C. (Fig 2.2). It is found, by bringing systems A and B in direct thermal content and

finding no change in their macroscopic properties, that system A and B are also in thermal equilibrium

with each other. Zeroth law of thermodynamics is stated as follows, two systems which are in thermal

equilibrium with system are in thermal solid type equilibrium with each other.

The zeroth laws of thermodynamics form the basis of thermodynamic measurements. It gives an

operational definition of temperature i.e. system in thermal equilibrium with each other have the same

temperature.


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Figure2.2. Zeroth law of thermodynamics

2.6 Terminology of thermodynamics:

Before taking up the concepts of thermodynamics it is essential to learn the exact definitions and

significance of some of the terms used in it.

2.6.1 Thermodynamic system:

A thermodynamic system is that part of the universe, the properties of which are under

investigation. It is separated from the rest of universe by a real or an imaginary boundary. A system may

consist of one or more substances.

The rest of the universe which is in a position interact with the system is called the surroundings.

2.6.2 Types of system:

A thermodynamic system can be classified into three types namely. An open system is one that

can exchange both energy and matter with its surroundings.

A closed system is one that can exchange only energy but not matter with its surroundings. An

isolated system is one that can exchange neither energy nor matter with its surroundings. (Fig.2.3)

Depending on the nature of the substance a system is said to be homogeneous when it consists of only

phase e.g. a pure solid , a pure liquid, a solution and a mixture of gases. A system is said to be

heterogeneous when it consists of two or more phases. For example, a system of two immiscible liquids.

A phase is defined as a homogeneous, physically distinct and mechanically separable portion of a system.

For example, ice, liquid water and water vapour each constitutes a different phase of the same system.


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Fig.2.2.Types of thermodynamic systems

2.6.3. State of a system:

A system is said to be in a definite state when each of its observable macroscopic properties like

pressure, volume, and temperature and compost ion has a definite value. For a system of definite

composition, its state is completely definite by three observable properties viz. pressure (P), Volume (V)

and temperature (T) for instance the state of an ideal gas is specified by giving its temperature pressure

and volume. These three variables are related to the amount of gas by the relation PV= nRT, where n is

the number of moles of a gas and R is the universal gas constant.

2.6.4. State variables:

A thermodynamic system could be described by its state variables. A state variable is one that has

definite value in a given state of a system. The state of a system changes with change in values of the state

variables. Pressure, volume, temperature, energy etc., are some of the state variables.

2.6.5. Thermodynamic Equilibrium:

A system is said to be in thermodynamic equilibrium when the values of all its macroscopic

properties remain constant with time .Actually this implies the simultaneous existence of the following

three types of equilibrium.

Thermal equilibrium

A system is said to be in thermal equilibrium when the value s of the temperature remains constant

throughout the system.

2.6.5. Mechanical equilibrium:


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A system is said to be in mechanical equilibrium when the three is no macroscopic movement

within the system itself or of our system with respect to the surroundings.

2.6.5. Chemical equilibrium:

A system is said to be in chemical equilibrium when its composition does not change

with time.

2.6.6. Properties of a system:

The properties of a system are those physical qualities perceivable by the sense or can be

made perceptible by experiments. There are two types of physical properties.

i) An extensive property of a system is any property whose magnitude depends on theamount of the substance. Example of such properties are mass, volume and energy.

ii) An intensive property of a system is in dependent of the amount of substance. Examplesare temperatures, pressure, refractive index, surface tension, density and viscosity.

Specific property

The ratio of an extensive property to the mass of the system is called a specific property. For

instance, specific volume v=v/m where v= volume; m= mass

Molar property

The ratio of extensive property to the mole number of the system is called a molar property. For

Example, molar volume, vm= V/n, where v= volume; n= number of moles

Representative example of intensive and extensive properties are given in the following table (2.1)

Table.2.1.Illustrative examples of Extensive and Intensive properties

S.No Extensive properties Intensive property

1 Energy Density

2 Volume Dipole movement3 Internal energy Refractive index

4 Enthalpy Viscosity

5 Entropy Surface tension

6 Free energy Molar volume

7 Heat capacity Temperature


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8 Number of moles Specific heat capacity

9 Mass Chemical potential

2.6.7. Thermodynamic processes:

The operation by means of which a system changes from one state to another is called a process.

A thermodynamics process may be of one of the following types.

2.6.7.1. Isothermal process:

A process is said to be isothermal if the temperature of the system remains constant during the

entire operation.

2.6.7.2. Adiabatic process:

An Adiabatic process is one in which no heat enters or leaves the system at any stage of theprocess.

2.6.7.3. Isobaric process:

A process is said to be isobaric if the pressure of the system remains constant during operation.

2.6.7.4. Isochoric process:

If the volume of the system remains constant during each step of the process, it is said to be

Isochoric.

2.6.7.5. Cyclic process:

A process that returns the system exactly to its original state is called cyclic process or a cycle.

2.6.7.6. Reversible process:

A process in which the diving force is only infinitesimally greater than the opposing force is

called a reversible process. A reversible process takes place infinitesimally slowly so that it requires an

infinite amount or time for completion, as it lies between static and dynamic processes all reversibleprocesses are quasistatic but not vice versa.

2.6.7.7. Irreversible process:

A process in which the driving force is substantially greater than the opposing force is called an

irreversible process. An irreversible process is completed in a definite time interval.


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Example for reversible and irreversible process:

Consider a cylinder of gas filled with a weight less and frictionless piston. The entire

system is placed in a thermostat. (Fib.2.4)

(i) When the outside pressure on the piston is equal to the precision is assure of the gas, inside,the piston will neither move downward or upward, i.e., there is no change in volume.

(ii) When the outside pressure is increased by an infinitesimal amount, (dp), the piston will movedown and the gas will be compressed by an infinitesimal.

(iii) If the outside pressure is continuously maintained infinitesimally greater than the pressure ofthe gas the compression of the gas will continue infinitesimally slowly .i.e., in a

thermodynamically reversible manner.

(iv) If, however, the outside pressure is made substantially greater than the pressure of the gas thegas will be compressed rapidly. In this case, the compression of the gas is said to take place

irreversibly.

Fig.2.4. Reversible and irreversible expansion of a gas


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Difference between reversible and irreversible process

S.No. Reversible process Irreversible process

1 Driving force and opposite force differ by

small amount

Driving force and opposing force differ by large

amount

2 It is a slow process It is a rapid process

3 The work obtained is more The work obtained is less

4 It consists of many steps It has only two steps i.e, initial and final step

5 It is an imaginary process It is a real process

6 It occurs in both directions It occurs in only one direction

7 It can be reversed by changing

thermodynamic variables

It cannot be reversed

2.7. Complete differentials or exact differentials:

Before taking up the laws of thermodynamics it is essential to be familiar with the concept

of exact differentials. Consider any thermodynamic property such as G.G is a single-valued function of

certain variables x, y, z which completely determine the value of G.

G= f(x, y, z) .(2.1)

The change in C resulting from a change in the variables from x A,yA,zA ,. in the initial state to

xB,yB, zB in the final stage, is given by

G=f(xB , yB , zB,.) f (xA , yA , zA,..) ..(2.2)

For small changes (dG) in the property G, we can write the equation (2.2)

dG= , + , +

. (2.3)

In equation (2.3) the quantity , represents the rate of change of G with the variable x, while allother variables y ,z. remain constant. Any differential dG defined by equation (2.3), of a function G,

represented by (2.1) is known as a complete differential or an exact differential of that function.

Table 2.2 Differences between exact and inexact differential:


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S.No. Exact differential Inexact differential

1 An exact differential when integrated given a

finite difference which is independent of the

path of integration

An inexact differential when integrated give a

quantity whose value depends on the path of

an integration

2 A cyclic integral of an exact differential is zero

for any cycle

A cyclic integral of an inexact differential is

usually not zero

3 Examples; properties of a state of a system such

as T,P,V,E are exact differentials

Properties of a path such as Q(heat) and W

(work) are inexact differentials

2.8. Thermodynamics properties of complete differentials:

A thermodynamic property of a homogeneous system of constant composition is completely determined

by the three thermodynamic variables, pressure, volume and temperature.

If pressure and temperature are taken as the independent variables, then as

G=f (P, T)

Where G may be energy, volume or any other state property. It follows that,

dG= + .(2.5)

If pressure and volume are taken as the independent variables, then

G=f (P, V) .(2.6)

G= + .(2.7)

Alternatively, if the volume and temperature are chosen the independent variables, then,

G=f (V, T) (2.8)

dG= + ...(2.9)2.9. The physical significance of complete differentials:

Consider the differential equations (2.5)


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G= +

The physical significance of the complete differential may be understood in the following way. When the

pressure and temperature of the system are P and T, respectively, the value of the thermodynamic

property is G. and when the variables are changed to P + dP and T + dT, it becomes G+dG. Since the

value of the property is wholly determined by pressure and temperature, dG will be independent of the

path between the initial and final states. Hence, any method which brings about the indicated change of

pressure and temperature may be used for the purpose of calculating dG.

Let the change be carried out in two stages

(i) Temperature is kept constant, whereas pressure is changed from P+ dP

dG=dP(ii) Pressure is kept constant, whereas temperature is changed from T to T+ dT

dG=dT

The sum of (2.10) is (2.11) is identical with 2.5.

Any differential like dG will be a perfect differential when

i. G is a single-valued of function depending only on X and Y (where X and Y are statesvariables)

ii. dG between any two states of parts is independent of the path of transition.iii. dG for a complete cyclic process is equal to zero.iv. 2 . =

2 . i.e., the second differential of G with respect to X and Y carried out in either

order become equal to one another.

For instance, consider a system containing a gm-mole of an ideal gas. Then, V=f(P,t)

Hence, by definition dV is a perfect differential. It can be proved as follows:PV=RT

=

2 .= -

2


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Again,

2 .= -

2

2

. = 2

..2.12Hence, dV is a perfect differential.

2.10. Work and heat are inexact differentials:

It will be discussed in the following chapter that the differential du is an exact differential and its

values is given by, du= dq-dw, where dg and dw represents the heat and work term. The internal energy

du does not dependent on the path of the transformation and depends only on the initial and final state of

the system. The heat transferred, dq and work performed, dw on the other hand, are depend on the path

process. These two functions, accordingly, knows as path functions.

That the value of w depends on the manner in which the change is effect is illustrated as follows.

When a system expends from an initial volume V1 to final volume V2, the work done by the system is

given by w= 21 where p is external pressure. Depending on the values of P the value of W varies.This clearly shows that W depends on the path this transformation.

Similarly the value of Q associated with a given change in state depends upon the manner in

which the change is brought about. Consider the change in state:

H2O (1, 10.o g, 250c, 1 atm) H2O (1, 10.o g, 35

0c, 1 atm)

One can transfer 100 calories of heat from the surroundings into the system to effect this change,

in case Q = +100 calories. The same change also be brought about by dipping a stirrer in the water and

mechanically stirring it, in which case Q=0.

The differential of a state function is an exact differential and a path function is an inexact

differential, both dQ and DW are inexact differentials.

2.11 The First Law of Thermodynamics

First law of thermodynamics is also known as the law of conversation of energy which may be

stated as follows: Energy may be converted from one form to another, but cannot be created or be


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destroyed. There are many ways of enunciating the first law of thermodynamics. Some of the selected

statements are given below:

(i) Energy of an isolated system must remain constant although it may be changed from oneform to another.

(ii) The change in the internal energy of a closed system is equal to the energy that energy thatpasses through its boundary as heat or work.

(iii) Heat and work are equivalent ways of changing a systems internal energy.(iv) Whenever other forms of energies are converted into heat or vice versa there is a fixed ratio

between the quantities of energy and heat thus converted.

Significance of first law of thermodynamics is that, the law ascertains an exact relation between heat

and work. It establishes that ascertain quantity of heat will produce a definite amount of work or vice

versa. Also, when a system apparently shows no mechanical energy but still capable of doing work, it

said to possess internal energy or intrinsic energy.

Mathematically first law of thermodynamics is stated as

U=q-w 2.13

Where U the change in energy content, q is the amount of heat change involved and w is the

work involved in the system. In a system when it moves from an initial state one to final state two, the

change in energy content, due to a definite quantity

U=U2-U1 2.14

In the value of dU is given by the quantity of heat q absorbed or liberated by the system and the

work done by the system or the work done on the system (w). For an infinite small change in the system

the change in energy content is given by

dU= q-w 2.15

2.12 Enthalpy or Heat content (H) of a system:

Another thermodynamic property is the heat content or enthalpy of a system. It is the sum of

energy content and product of pressure-volume of a system. It is defined as,

H=U+PV 2.16


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When a system undergoes change from an initial state 1 and final state 2, the change in energy

content is given by

H=H2-H1 2.17

U=U2-U1 2.18

By definition, of a system involves change in the volume of the expansion of a gas, constant

pressure

H2=U2+PV2 2.19

H1=U1+PV1 2.20

Substituting on 2.18,

H= (V2+PV2)-(V1+PV1)

H= (V2-V1) +P (V2-V1)

H= U+PV 2.21

The above equation (2.21) is the well known relation between H and U.

2.13. The second law of thermodynamics

2.13.1. The need for the second law

The first law of thermodynamics deals with the conversation of the one form of energy into another.

It states that when one form of energy disappears an equivalent amount of another form must appear.

Thus, the conversion as well as the conservation of energy is dealt with by the first law. But it does not

indicate whether or not a particular change would occur and if it occurs, to what extent. It also does not

point to the direction in which a transformation would take place. It establishes a definite relationship

between the amount of heat absorbed and the work done but it does not tell us that heat cannot be

converted completely into work without producing changes elsewhere. Hence, the need for the secondlaw.

The following common examples would substantiate the inadequacy of the first law of

thermodynamics. Since continuous creation of energy is no possible without expecting equivalent amount

of energy, the first law of thermodynamics perpetual motion of the first kind (PMMFK).


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Suppose two bodies A and B brought in contact. We know from the first law, that if Q joules of

heat are lost by A; exactly Q joules of heat would be gained by B or vice versa. But the law does not say

whether A or B would lose the heat energy. To know the direction of flow of heat energy we need

information, namely, the temperatures A and B.

When two metal rods, say Cu and Zn, are placed in an electrolytic solution and connected by means

of wire, one would observe the flow of electrical energy in the process. To know the direction of flow

current in the internal circuit, we must know the electrode potentials of two metals.

It is clear from the above examples that to know the direction of a physical or chemical process,

we need more information or knowledge than that provided by the direction in which energy can be

transferred known by the second law. It also provides the criterion to the possibility rather the probability

of various processes.

2.13.2Spontaneous processes:

In order to understand the conditions under which a particular process will occur or not, it

necessary to know certain processes called spontaneous processes i.e. changes talking place in a system

without the aid of any external emergency. A few example of spontaneous process are -

A gas kept in a vessel at a higher pressure connected by a tube to another vessel where the

pressure is low. The gas would spontaneously flow from the high pressure vessel to the low pressure

vessel. This process would continue till the equilibrium is reached, the gas by itself will not go back to the

first vessel to increase the pressure to its original higher value.

When two solutions of different concentrations or a solution and a solvent are brought in contact,

the solute begins to diffuse from a region of higher concentration to a region of lower concentration

throughout becomes the same. This spontaneous process is also unidirectional. For substance a drop of

ink placed in a beaker containing water, the ink differs into all parts of water and because a uniform

concentration solution.

When a hot body comes in contract with a cold body, the beat will from the former to the later

until thermal equilibrium is reached. The spontaneous flow of heat is always unidirectional and

irreversible.

A liquid will always flow freely from a higher to a lower level until the levels are equalized i.e.,

mechanical equilibrium is reached. This is also an irreversible process.


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In all these spontaneous processes, it is seen that they are unidirectional and are irreversible.

Every system naturally moves towards the equilibrium state. The generalization arrival at may be

expressed as

All spontaneous processes are irreversible

All spontaneous processes tend towards equilibrium.

2.13.3. Reversal of spontaneous process

The fundamental characteristic of all the spontaneous process is that they never been observed to

reverse themselves without the intervention of an external agency. In other words, spontaneous processes

are not thermodynamically reversible. This fact forms the basis of the second law of thermodynamics. By

the use of an external agency it is possible to bring about the reversal of a spontaneous process. For

example, by introducing a piston into the vessel, the gas which has expanded into vacuum could restore to

its original volume by compression. Work would have to be done on the gas and at the same time an

equivalent amount of heat would be produced and the temperature of the gas would rise. If this could he

completely converted into work, then the original state of gas would have been restored and is this

process there could be no change in external bodies. It is a fundamental fact of experience that the

complete conversion of heat into work is impossible without leaving some effect elsewhere. This result is

in accordance with the thermodynamics irreversibility of spontaneous processes. Similarly, in any process

i.e. two metal rods of different temperatures in contact, a chemical reaction producing electrical energy

etc., the experience shows that the complete conversation of heat into work is not possible without leaving

some change elsewhere in the system or surroundings. To account for the fact, Clausius made the

following statement

It is impossible for a self acting machine, unaided by any external agency, to convey heat form a body

at a low one to at high temperature.

2.13.4 The second law of thermodynamics

From the experience, it is learnt that every form of energy has a natural tendency to be

transformed into thermal energy. The thermal energy shows least tendency to be transformed to other

form. By introducing some mechanism or machine, we convert heat into other forms of work. This

conversion of heat into work occurs into limited extent and not completely. The condition under which

the heat converted into work becomes the second law of thermodynamics. The following two conditions

must be satisfied for the conversion of heat into useful work.


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(i) A contrivance or machine, commonly known as heat engine is essential. Without the aid of anengine the conversion of heat into work is impossible. Besides, the engine must work in a

reversible cyclic process.

(ii) The engine must be operated between two temperatures. It will take up a heat from a body at ahigher temperature convert a portion of into work and give up the rest of the heat to a body at

lower temperature.

As discussed in the previous section, conversion of heat into work in a gaseous expansion will

result in a change in the volume of a gas. It is proved beyond doubt that the complete conversion of heat

into work is impossible; without producing a permanent change elsewhere.

In the same example, to bring back the gas under isothermal conditions to its original volume, a

gaseous work of compressions is to be performed. During this process, a certain amount of heat will be

given out into surroundings. Then the process is preformed and restored to its original state, the net heat

change involved is equal to that of work performed. We can, therefore, conclude that under isothermal

conditions no engine can convert heat into work. This is the reason why we cannot run our motor cars

with the heat of surrounding air. This forms another aspect of second law of thermodynamic. A

consequence of the impossibility of converting heat into work under isothermal conditions, is the

impracticability of what we called Perpetual motion machine of the second kind (PMMSK) i.e. the

utilizations of the vast stores of energy in the ocean and in the earth. For instance, we cannot move our

ships by utilizing the energy of ocean.

2.13.5. The macroscopic natural of second Law

By applying the kinetic theory of matter one would be to understand the basis of the second law

of thermodynamics. According to this theory, increases of temperature, as a result of absorption of heat

by the body, will in turn result in an increase of kinetic energy of the random motion of molecules. When

the energy of moving body is converted into heat, the directed motion of the body of a moving body as a

whole is transformed into the chaotic motion of individual molecules. The reversal of the process,

namely, the conversion of heat into work, it is required that all molecules must orient in a direction. The

probability of its occurring in a system consisting for a large number of molecules is very small.

Suppose the system consists of a small number of molecules, say, and five. Then at a point of

time, it could be probable to find that all five molecules spontaneously oriented in a preferred direction.

At that instant, the work is performed by the molecules. The greater the probability of reverse

spontaniety.


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For our observation and experiments, we take up systems consisting of a large number of

molecules. Therefore, the spontaneous reversal of a natural process is highly improbable or not possible.

Hence, the second law of thermodynamics is regarded to the macroscopic system.

2.13.6. Conversation of Heat into Work:

In any process, the conversation of heat into work is possible only when there exists a difference

of potential or there presents a directive influence. For example, when water falls from a higher to a lower

level, the work derived from the process is due to the potential called electronic force. In a heat reservoir

at constant temperature, there is no directive influence. Suppose we have two heat reservoirs at different

temperatures. It provides the difference of energy potential that is essential for conversation of heat into

work.

In order to effect this conversation, heat is absorbed form the reservoir at the higher temperature,

called source, a fraction of heat is conserved into work, and the remaining part of heat is returned to heat

reservoir at the lower temperature, called sink. It is clear, therefore, that a portion only of heat taken in

from the reservoir at the higher temperature can be converted into work.

The fraction of heat absorbed by a machine that it can transform into work is defined as the

efficiency of the machine. It heat Q is taken from the source and W is the work done; the efficiency is

equal to W/Q. it is known from the experience that W is invariable less than Q in a continuous

conversation process. This is in keeping with the second law of thermodynamics. The efficiency of a

machine is always less than unity. The first law is also applicable. Since the difference between Q and W

is returned to the reservoir at the lower temperature.

2.13.7. Definition of Entropy

The fact that dQrev/T=0 shows that dQrev/T is the differential of state function. Clausius called this

state function entropy and given the symbol S it is defined as,

dS = ..(2.22)(i) Entropy, S, like energy is a thermodynamic function whose value depends only on the

parameters of the system and can be expressed in terms of (P,V,T)

(ii) The differential ds, is a perfect differential. For any finite transformation, the entropychange depends only on the initial and final states of the system and is independent of the

path of change.


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(iii) Absorption of heat increases the entropy of the system. The rejection of heat by thesystem leads to a decreases in its entropy. During reversible adiabatic changes (dQrev=0),

and therefore the entropy change is also zero. Reversible adiabatic changes are also called

isoentropic changes.

(iv) Entropy is an extensive property. Its value depends upon the amount of matter in thesystem.

2.13.8. Entropy change and unavailable Heat

There is a simple relationship between entropy changes and heat which is returned at the lower

temperature in a heat engine. It is of practical importance. From Carnots theorem, it is seen, for a

reversible system,

1

1 = 2

2-Q1=T1 (Q2/T2)..(2.23)

ThereQ1is the heat returned to the reservoir at the lower temperature T1and hence it is not available for

conversation into work. Since heat is taken up reversibly, Q2/T2is the increase of entropy of the system in

the heat absorption change. Therefore it follows from the above equations,

-Q1= T1S2..(2,24)

This gives the quantity of heat returned to the reservoir at lower temperature in a reversible cycle in terms

the entropy change. For an irreversible cycle the heat available is greater since the efficiency is less.

2.13.11.CLAUSIUS INEOALITY

Consider the following cycle. The system is transformed irreversibly from state 1 to state

2 and restored reversibly from state 2 to state 1. For this irreversible cycle,(fig 2.4) we have

=

2

1

+

1

2

< 0

= 21 + < 012 { = }

= 21 + < 021 (or)


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21

< 2

1

dS > ..(2.25)The above equation is called Clausius inequality. It means that the energy change in reversible process is

not equal to entropy change in a irreversible process.

Fig.2. Clausius Inequality

2.13.12.Entropy change in an ideal gas:

For an infinitesimal change, the first law of thermodynamics is expressed as,

dU = dQdW ..(2.26)

(or)


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dQ = dU + dW (2.27)

Suppose the work involved is one of work of expansion only

dQ =dU + PdV ..(2.28)

If the process is reversible,

dQrev=dU + PdV (2.29)

By definition, dS = dQrev/T

TdS = dQrev

Substituting the above result in equation.(5.36)

dS = + ..(2.30)

For an ideal gas, we can write ,

Cv= at constant volume

CVdT= dU .(2.31)

Combining equations. (5.37) and (5.38),

dS = + (2.32)

If the process involves an appreciable change, then equation (5.39) takes the form,

S = S = + .

For one mole of an idle gas,

PV = RT

P = RT/V

Substituting the result in equation.(5.41)


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21

S = +

S = + .(2.33)

Assuming that CVis constant over the temperature range,

S = Cv lnT + Rln V + S0.(2.34)

Where S0is the integration constant.

Suppose the system moves from volume V1to volume V2when the temperatures changes from T1to T2,

the equation is given as,

S = Cv + 2121

S = Cvln21 + R ln

21 (2.35)

Converting ln into log,

S = Cv2.303 log21 + 21 (2.36)

2.13.12 Another From

For 1 mole of an ideal gas V2= 2

2 V1= 21 (2.37)

(2.37) in (2.36)

S = Cv 21 + 12 (2.38)Converting ln into log,

S = Cv2.303 21 + 2.303 12 (2.39)2.14.The concept of free Energy

In a system the maximum work a process may yield is not necessary the amount of energy

available for doing useful work, even though the process is conducted reversibly. Of the total amount of


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work available a certain amount has to be utilized for the performance of pressure volume work against

the atmosphere due to contraction of expansion of the system during the process. Since this work is

accomplished at the expense of the maximum work given by the process, the net amount of energy

available for work other than pressure-volume against the comprising atmosphere must be

Wmax- PV.

Entropy can be employed to measure the tendency of systems to undergo change, under

conditions most frequency used; entropy is not a convenient quantity. As a result, a new thermodynamic

property called free energy is introduced. A latest discussion can be seen in following sections.

2.14.1. Helmholtz free energy / work function (A)

Helmholtz free energy or the work function, a, is defined by

A = UTS ..(2.40)

Where U is the energy, T, the temperature and S, the entropy of the system.

We know that U, T and S area thermodynamic functions. Hence A is also a thermodynamic

function. Hence, for any transformation the value of U depends only on the initial and final states of the

system. As U and S are extensive properties, A is also an extensive property.

2.14.2. NATURE OF WORK FUNCRTION:

Consider a system which moves from state 1 to 2 at a given temperature. For such a change, the

change in work function, A, is given by

=2 1 .(2.41)Where 2 = 2 2

1 = 11

The equation (2.14) becomes,

= 2 2 (1 1)= 2 1 (2 1)

=


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From the definition of entropy,

=

= Introducing in equation (2.41)

= Constant temperature,

= From the first law of thermodynamics,

= =

Introducing in eqn. (2.43)

= = .. (2.44)

where Wrev refers to the maximum work involves in the system. The decrease in the function results in

maximum work which may be mechanical or any other external work.

2.14.3. Free energy or Gibbs free energy (G/F)

By definition, G = HTS ... (2.45)

We know,

H = U + PV

G = UPVTS (2.46)

By definition, A = UTS

Introducing in equation (2.46)


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G = A + PV (2.47)

As H, T, S, A, P and V are all thermodynamic functions, G is also a thermodynamic function. Since H

and S are extensive properties, G is also extensive property.

2.14.4. Significance of the term G, or the nature of the Gibbs free energy

G = A + PV (2.48)

For a free energy change at constant pressure and temperature

, = + p (2.49)From the previous discussion,

This equation is applicable to all processes taking place at a constant pressure.

(i) It is used to calculate the heat change, , for a process or a reaction taking place at constantpressure, if free energies at two different temperatures are known.

(ii) It is used to calculate the e.m.f. of reversible cells.(iii) If the rate of change of free energy with temperature remains. Practically constant, then, the

known at one temperature can be used to calculate at a different temperature.(iv) Various equations such as clausiuschaperon equation, Vant Hoff equation can be deduced.

2.15. Gibbs Helmholtz equation

We know that G = HTS and H = UPV

G=U+PV-TS

upon differentiation we get

dG=U+PdV+VdP-TdS-SdT ..(2.51)

dq=U+Dw = U+PdV (2.52)

dG=dq+VdP-TdS-SdT .(2.53)

For a reversible process,

dS=dq/T or TdS=dq.(2.54)

From equation (2.53) &(2.54)

dG= VdP-SdT

If pressure remains constant i.e., dP=0, then

dG= -SdT .(2.55)


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Let G1= initial free energy of a system at T; and G1+dG1= initial free energy of the system at T+dT.

Where dT is infinitesimally small and pressure is constant,

dG1= -S1dT .(2.56)

Where S1is the entropy of the system in the initial state.

Now suppose that the free energy of the system in financial state is G2at T. Let G2+dG2 is the free

energy of the system at T+dT in the final state, then

dG2= - S2dT .(2.57)

Where S2is the entropy of the system of the final state.

Subtracting Eq. (ii) from Eq.(iii), we get

dG2-dG1= -(S2-S1) dT

d(G)= -SdT .(2.58)

(or)

At constant pressure,

() = -S ..(2.59)We know that, G=H-TS

S = GHT

..(2.60)

From the equations (2.59)&(2.60) we get,

G

H

= (

)

= + () .(2.61)

This equation is called the Gibbs Helmholtz equation in terms of free energy and enthalpy changes at

constant pressure.

Applications of Gibbs-Helmholtzs equation

This equation is applicable to all process taking place at constant pressure.

i. It is used to calculate the heat change, H, for a process or a reaction taking place at aconstant pressure, if free energies at two different temperatures are known.

ii. It is used to calculate emf of a reversible cellsiii. It the rate of change of free energy remains, particularly constant, then the known G at

one temperature can be used to calculate G at different temperature.

iv. Various equations such as Clausius Clapeyron equation, Vant Hoff equation arededuced.


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2.15.1. Applications of concept of free energy

Phase equilibrium in one component system:

Consider a closed system containing a single substance i,e., one component system. The following two

phase equilibria are possible for the system.

Liquid Vapour

Solid Vapour

Solid Liquid

A equilibrium GT,P= 0

If GA is free energy of phase A and GBis the free energy of phase B then at equilibrium as

G = GB - GA = 0

Or GB = GA

2.16.Clapeyron equation:

In the above section, we have seen that a given temperature and pressure, the free energy of substance

is the same in the different phases that are in equilibrium with each other.

When there is a change in temperature and pressure,

Say,

TT + dT and P P + dP, the free energy changes from G G + dG.For the equilibrium between phases A & B, let dGAand dGB be the changes in free energy of phase A &

B respectively.

Now, G = 0

d(G) = d(GB- GA) = dGB- dGA= 0

Or dGB= dGA .( 2.62)

We know that for a reversible process,

dG = V dPS dT (2.63 )

Hence,

dGA= VAdPSAdT .(2.64 )

and dGB= VBdP - SBdT .(2.65)

From equation (2.62)

dGA= dGB


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VAdPSAdT = VBdP - SBdT

SBdTSAdT = VBdP - VAdP

(SA- SB) dT = (VB- VA) dP

dP

dT

=SA SBVB VA

=

.(2.66)

Rewriting

=

S

V (2.67)

By definition,

S = =

H

.(2.68)

Where H = enthalpy change for the phase transformation. Equation becomes, =

(2.69)

This is a general equation applicable to all one component two phase equilibria.

2.17. CLAUSIUSCLAPEYRON EQUATION:

The chaperon equation is given by,

=

(2.70)

Consider the equilibrium

Liquid vapour

The chaperon equation is applicable to the above equilibrium.

In this case,

H = HV= Heat of vaporization

T = temperature at which vaporization occurs.

P = vapour pressure at a given temperature.

V = VVV1

Where VV= volume in vapour phase.

V1= volume in liquid phase.

Since volume in vapour phase is considerably greater than the volume in liquid phase, the latter term is

omitted.

i.e. VV> V1

Hence


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V = VV

Introducing all these equations in equation , it becomes

=

H v,p

T(nRT)

= H v,p(nRT)2Rearranging the above equation

=

H v

(nRT )2dT.(2.71)

This is the differential form of Clausius chaperon equation

Integral form

Suppose P1 and P2are the equilibrium vapour pressure at T1and T2, then

Then the equation becomes

21 = 2 21 .(2.72)Assuming Hv is constant over the temperature ranges T1T2 the equation becomes,

ln21 = 221 .(2.73)

Since n and R are constants

ln21 =

Hv

2

2

1

ln21 =Hv

1

12

ln21 =

Hv

1

2 1

1

ln21 =

Hv

1

11

2

2.303 log21 =

Hv

2112 (2.74)ln X(logeX) = 2.303 log10X

log21 =

Hv

2.303 2 112


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ln21 =

Hv

2.303 2112 .(2.75)

By taking any three variables among P1, P2, T1& T2fourth variable is calculated.

2.18. Thermodynamic relations or Maxwells relations

The aim of Maxwells relation the experimentally indeterminable quantities like energy, entropy,

etc., to experimentally measurable quantities like pressure, volume and temperature. For this purpose,

consider closed system, where gravitational, magnetic and electrical forces are absent.

2.18.1 First equation

Helmholtz free energy is determined by

A = UTS

Differentiating the above equation

dA =dUTdSSdT

From the first law of thermodynamics,

dU = dQPdV

Introducing in the equation

dA = (dQ - PdV)TdSSdT

Since,

dS =

; TdS = dQ

Substituting in the equation

dA = (dQ - PdV)dQSdT

dA = - PdVSdT .. .

It means that, A = f (V,T)

i. When temperature remains constant, the equation becomes,dAT= - PdVT

Or = Differentiate we know that volume

,=

2 =


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ii. When the volume is constant the equation becomes,

dAV = - SdTv

Or

=

Differentiate we know that volume at constant temperature

= -

T

By comparing the equation

Gives change in entropy with change in volume at constant temperature i.e., entropy change

during isothermal expansion or compression: it is given by V.This suggests a method of measuring entropy change during isothermal process.

2.18.2 SECOND EQUATION

Gibbs free energy is defined by

G = HTS

Since, H = U + PV

G = UTS + PV

Differentiate of the equation

dG = dUTdSSdT + PdV + VdP

For an infinitesimal change in a reversible process involving only work of expansion,

TdS = dU + PdV

The equation dG=VdPSdT

i. when temperature constant, the equation takes the form,dGT=VdPT

(or) = .Differentiate w.r.t. temperature at constant pressure

T=V


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TP= 2 .= P

ii. At constant pressure, the change in pressure, dP = 0

Hence the equation becomes,

= Rewriting,

= Differentiate we know that pressure, at constant temperature,

=

2

. = Comparison of equation

_______ (II)

Since is positive, the above equation shows that at constant temperature when the pressureincreases, the entropy get degreased (-ve sign)

2.18.3 THIRD EQUATION

The energy content, U, of a homogeneous system may depend on the two variables entropy and

volume which is given by,

U = f (S, V).

The first law of thermodynamics may be stated as,

dU = dQPdV

From the 2nd

law of thermodynamics, it is know that

Substituting in equation

TdS = dQ

Rearranging,

= Differentiate w, r, t, S at constant volume,

=

=


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Or

2. =

ii. When the volume remaining constant, dv = 0Hence the equation dU = TdSPdV becomes

dUV= (TdS)V

Or

= Differentiating the above equation with respect to volume at constant entropy,

= Or

2 . =

By comparing the equations

_______ (III)

2.18.4 FOURTH EQUATION

The heat content H, of a homogenous system of constant composition may depend on two variables

entropy and pressure. It is given by

H = f (S, P)

By definition,

H = U + PV

dH = dU + P dV +V dP

From the first law of thermodynamics

dQ = dU + P dV

Introducing in equationdH = dQ + VdP

The second law of thermodynamics defines,

dS =

TdS = dQ

Substituting the above equation with respect to entropy at constant pressure,

=


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dH = TdS + VdP

=

2. =

iii. When the pressure is kept constant, dP = 0The equation becomes

dHP= T dSP

Or =

Differentiate with respect to pressure, at constant entropy,

, =

2. =

Comparison of equation gives the result,

=

________ (IV)

The four equations I, II, III and IV are generally known as Maxwell relations or Maxwell operations.

These equations are applicable to systems of all types, homogeneous or heterogeneous, provided the

following two conditions are satisfied.

i. The mass of the system is assumed to be constant. It means that there is no loss or gain ofmatter in the course of thermodynamics change, i.e., the system is closed systems.

ii. The work involved is work of expansion only and that is equal to P dV where P is thepressure of the system. This means that the system is always remain in equilibrium with the

external pressure.

2.9.THE REACTION ISOTHERM (Vans Hoff):

Again, consider the general reaction,

aA + bB + .. lL + mM +

The free energy mixture of a moles of A, b moles of B etc., is given in terms of the chemical

potentials.

G(reactants) = a A= b B +


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And the free energy of a mixture l moles of L, m moles of M, etc

G(products)= l L= m M

At constant temperature, pressure and composition in each case.

This equilibrium are applicable to the system of reactants and products at any arbitrary

concentration, not necessarily the equilibrium values. The free energy increases G, accompanying the

reaction at constant temperature and pressure, is given by,

GT,P= G(products)G(reactants)

=(l Lm M+) (a Ab B+)

Using the general equation = + RT ln a

Where is the chemical potential of any substance being the value in the standard state.

GT,P = l ( L + RT ln aL) + m (M+ RT ln aM) +.-a (A+RT ln aA)b (B+ RT ln aB) +

= (l L+ m M +) (aA+ m B + )+.+ RT ln

GT,P = GT + RT ln It is know that

G = RTlnKSubstituting in equation

GT,P= - RT ln K + RT + GT,P= -RT ln K + RT ln Ja

NOTE: (i) If, Ja the activity are arbitrarily chosenIf the arbitrarily activities were the equilibrium values, RT ln Ja would be identical with RT ln K then

GT,P would be zero

The equations ( ) and ( ) are forms of reaction isotherm derived by JH Vant Hoff.

(ii) It gives the increases of free energy accompanying the transfer of reactant at any arbitrary

concentrations (activities) to products at arbitrary concentrations (activities).

At a given temperature, when the activity is equal to 1, =

It is a constant term, independent of pressure because of the choice of the standard state. a = 1 or p = 1

atm. When the gas behaves ideally at a given temperature. It is a constant. The variation of this term w.r.to pressure is zero.

Hence, (G) T= 0(ln ) T = 0


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The equilibrium constant is unaffected by change in pressure. The equations may shift in one

direction due to the applied pressure, while the equilibrium constant will remain the same.

Variation of Equilibrium Constant with temperature:

According to Vant Hoff Isotherm,

G = RTlnKRearranging,

G

= ln,Differentiate with respect to temperature at constant pressure,

G P= (ln ) P 1 () (G) = ln P

According to Gibbs Helmholtz eqn. under standard conditions

(G)/

P=

H

2

Substituting this in equation

-1

H2 = ln PH2 = ln P

Where H =(L HL+ m HM +) - (a HA+ b HB + )

Or In general,

H = This eqn. is the rigorous form of a relationship originally postulated by Vant Hoff. It is

sometimes referred to as Vant Hoff Equation. / Vant Hoffs Isochore.

Integrating within limits,

22

1 = ln

2

1

1

1 1

2 = ln21

Or

2

1

21 = ln2

1If remains constant over a range of temperature it can be taken outside the integral and get

integrated as given in equation.

If is given as a function of temperature, it should be integrated by terms after dividing by T2Eqn. (8.50) may be written more commonly written as


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log21=

2.303 2112

This shows the equilibrium value changes as the temperature changes. Using this equation it is

possible to evaluate the standard enthalpy change. For an ideal system, is almost equal to, upto a temperature 100

. The above equation is referred to as Vant HoffsIsochore.

2.19.1. Vant Hoffs Isotherm

Let us consider the general reaction

aA + bB cC + dD

We know that G = HTS

G = U + PVTS [H = U + PV]

Differentiating the above equation

dG = dE + PdV + VdPTdSSdT

But dq = dE + PdV and dS = dq / T

dG = VdPT * dq / TSdT

dG = VdPSdT

At constant temperature

(dG)T= VdP

Free energy change for 1 mole of an ideal gas at constant temperature is given by,

= = . [ = ]Integrating the above equation

=

G = G+ RT ln P

Where Gis integration constant and is known as standard free energy, i.e., = GWhen P = 1 atm

Let the energy per mole of A, B, C and D at their respective pressure PA, PB, PCand PDare the GA, GB, GC

and GDrespectively, now the free energy change for the reaction 1 is given by

= (Products)G(reactants)= [cGC+ dGD]- [aGA- bGB]

Substituting the value of GA, GB, GC and GD in the above equation

=[cGC+ cRT ln PC+ dGD+ dRT ln PD][aGA+ aRT ln PA+ bGB+ bRT ln PB]= [cGC+ dGD][aGA+ bGB] + RT ln

()()( )( )

Where = the standard free energy of the reaction 1We know that at equilibrium = 0


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+ ln = 0 + ln = 0 Since = ()()( )( )

Or

=

ln

From the equation we get

= ln + ln ()()

()() Or = ln ln ()( )( )( )

The above equation called as Vant Hoffs Isotherm. This gives quantitative relation for free energy

change accompanying a chemical reaction.

thermodynamics unit ii

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