Physical chemistry vol II dr j n gurtu, aayushi gurtu (pragati prakashan, 2008)

p~~--------------~-

PHYSICAL CHEMISTRY
I Vol. II)

Dr. J. N. Gurtu
M.Sc., Ph. D.

Former Principal
Meerut College, MEERUT.

Aayushi Gurtu

~

PRAGATIPRAKASHAN


PRAGATIPRAKASHAN
Heat! Office:

Educatiollal Publishers

Second Edition 2008

PRAGATl BHAWAN,

240, W. K. Road, MeerUl-250 001
Tele Faxe' 0121-2643636, 2640642
SMS/Ph()ne~: 0121-6544642. 6451644

Regd. Office:
New Market, Begum Bridge, Meerut-250 001
Phone:0121-2651907

ISBN: 978-81-8398-511-6

Published by K.K. Mittal for Pragati Prakashan, Meerut - 250 00 I and Photocomposing by : Pragati Laser Type Setters
Pvt. Ltd., Meerut (Phone 2(i(i 1657) Pri'lted at Urvashl Offset Press, Meerut.


CONTENTS--------------1.

THERMODYNAMICS-I
Basic definitions 1
Energy 11
Internal energy 11

1- 38

12

Zeroth law of thermodynam:cs

13

First law of thermodynamics
Heat changes

14

Heat content or enthalpy
Heat capacity

15

15

17

Applications of first law of thermodynamics
Joule-Thomson effect

26

Miscellaneous numerical problems
Exercises

2.

31

35

THERMOCHEMISTRY

39-73

Heat of reaction or enthalpy of reaction

39

Variation of enthalpy of reaction with temperature, i.e., Kirchoffls equation

45

Heat of formation or enthalpy of formation
Enthalpies of compounds

46

Heat of combustion or enthalpy of combustion

47

Heat of neutralisation or enthalpy of neutralisation
Heat of transition or enthalpy of transition

50

50

Heat of solution and heat of dilution or enthalpy of solution and dilution
Intrinsic energy

Laws of thermochemistry
Flame temperature
Resonance energy

55

55

58

Bond energy and dissociation energy

59

61

Miscellaneous numerical problems

3.

52

54

Exothermic and endothermic reactions and compounds

Exercises

42

62

71

74-134

THERMODYNAMICS-II
Spontaneous and non-spontaneous processes
Reversible processes
Carnotls cycle

74

74

75

Second law of thermodynamics

80

Thermodynamic or Kelvin scale of

temperatu~e 81

Concept of entropy

85


(vi)
Criteria of spontaneity (irreve,jibility) and conditions of equilibrium
Prediction of direction or occurrence of a process

102

104

105

Nernst's heat theorem

11 2
Third law of thermodynamics 11 2
Free energy and work function 123
Exercises 129
Concept of residual entropy

4.

CHEMICAL EQUILIBRIUM
Chemical equilibrium 1 35
Law of mass action 1 36
Le Chatelier's principle

135-164

1 41

van't Hoff isotherm or maximum work obtained from gaseous reactions
van't Hoff isochore or van't Hoff equation
Clausius-Clapeyron equation

5.

1 51

1 54

Clapeyron equation
Exercises

1 58

1 62

PHASE RULE

165-229

165

Introduction
Phase

148

165

Component

1 66

Degree of freedom or variance

168

Criterion of phase equilibrium

1 69

Statement of phase rule

169

Thermodynamic derivation of phase rule

170

1 72

One component system

1 72

Water system

Carbon dioxide system
Sulphur system

1 75

1 77

Two component systems
Lead-silver system

1 81

1 82

Potassium iodide-water system
Bismuth-cadmium system

1 84

186

Binary systems with formation of compounds with congruent melting point
Binary systems with formation of compounds with incongruent melting point
Solid-gas systems

197

Determination of solid-liquid equilibria
Henry's law

202

Systems of liquid in liquid

204

1 99

1 87

194


(vii)

Solubility of partially miscible liquid pairs
Exercises 224
6.

7.

8.

21 9

DISTRIBUTION LAW
Distribution in liquid-liquid systems 230
Thermodynamic derivation of distribution law
Different cases of distribution law 232
Applications of distribution law 234
Numerical problems 239
Exercises 2 4 3

230-247

231

ELECTROCHEMISTRY-I
Electrical transport-conduction in metals 248
Conduction in electrolytic solutions 250

Arrhenius theory of electrolytic dissociation 260
Migration of ions 263
Transference number or transport number 266
Kohlrausch's law or law of independent migration of ions
Ostwald's dilution law 277
Applications of conductivity measurements 279
Conductometric titrations 282
Anomaly of strong electrolytes 283
Exercises 2 86
ELECTROCHEMISTRY-II
Electrochemical cells 291
Reversible and irreversible cells 294
Notations used in cell diagrams 295
Electromotive force 296
EMF of a cell and cell reaction 298
Weston standard cell 299
Reversible electrodes 300
Electrode potential 301
Electromotive series or potential series 308
Calculation of thermodynamic constants 31 2
Polarisation 3 1 7
Oxidation-reduction potential 31 9
Overvoltage or overpotential 323
Liquid junction potential 328

248-290

273

291-364

(viii)
Electrode concentration cells

01

Electrolyte concentration cells

amalgam cells
331

Applications of concentration cells
Fuel cells

9.

337

350

Potentiometric titrations
Exercises

330

353

359

HYDROGEN ION CONCENTRATION, BUFFERS
AND HYDROLYSIS
pH values

365

Buffer solutions
Ionic product

371

of water 375

Salt hydrolysis

376

Degree of hydrolysis
Exercises

of a sdlt 380

388

10. CORROSION
Corrosion

394-418

394

Theories of corrosion

394

Factors influencing corrosion
Corrosion inhibitors
Passivity

400

402

403

Types of corrosion

403

Protection from corrosion or corrosion control
Exercises

o
o

365-393

408

4 12

SUBJECT INDEX
LOG AND ANTILOG TABLES

419-420
(i)-(iv)


Chapter

~

""I

THERMOIj~rAMICS-I
BASIC DEFINITIONS

[I] Thermodynamics, Objectives and Limitations
(a) ,!,bermodynamics means the study of flow of heat. It deals with energy
changes accompanying all types of physical and chemical processes. It is based on
three generalisations, known as first, second and third laws of thermodynamics.
(b) Objectives : Thermodynamics is of great importance in physical
chemistry. Most of the important generalisations of physical chemistry such as van't
Hoff law of dilute solutions, law of chemical equilibrium, phase rule etc. can be
deduced from the laws of thermodynamics. It also lays down the criteria for
predicting spontaneity' of a process, i.e., whether a given process is possible or not
under given conditions of pressure, temperature and concentration. It also helps us
to determine the extent to which a process can proceed before obtaining the state
of equilibrium.
(c) Limitations : The laws of thermody-namics apply to the behaviour of

assemblages of a large number of molecules and not to individual atoms or
molecules. It does not tell us about the rate at which a given process may occur,
i.e., it does not tell whether the reaction will be slow or fast. It concerns only with
the initial and final states of a system.

[II] Thermodynamic System
A thermodynamic system is defined as the specified portion (>f matter which is
separated from the rest of the universe with a bounding surface.
It may consist of one or more substances. Boiling water in a beaker is an
example of a thermodynamic system.

[III] Surroundings
The rest of the universe which might be in a position to exchange matter and
energy with the system is termed as surroundings.
Consider a reaction between :linc and dilute H 2S0 4 in a test tube. Here the test
tube forms a system. Everything also around this system is called surroundings.

[IV] Types of Thermodynamic Systems
(i) Closed system : In a closed system, exchange of energy with the
surroundings is possible, while matter can neither enter into nor leE.ve the system.
(ii) Isolated system: In this system, there is no exchange of matter or energy
b~tween the system and the surroundings.


2

PHYSICAL CHEMISTRY-II

(iii) Open system: In an open system, both matter and energy can enter into
or leave the system and thus there can be an exchange of matter and energy between

the system and the surroundings.

[V] Types of Thermodynamic Systems
If a system is kept at constant temperature, it is called an isothermal system.
If the system is so insulated from its surroundings that no heat flows in or out of
the system, it is called an adiabatic system.
A system is said to be homogeneous when it consists of only one phase, i.e.,
when it is completely uniform throughout. For example, a solution of salt is a
homogeneous system. A system is said to be heterogeneous when it consists of two
or more phases, i.e., when it is not uniform throughout. For example, a mixture of
two immiscible liquids is a heterogeneous system.
Besides the above, thermodynamic system can be open, closed or isolated as
discussed above.
[VI] Nature of Work and Heat
Whenever a system changes from one state to another, there is always a change
in energy, which may appear in the form of heat, work etc.
The unit of energy is erg. It is defined as the work done when a resistance of
1 dyne is displaced through a distance of 1 cm. As erg is a small quantity, a bigger
unit, called joule (1 joule =: 107 ergs) is used. Joule (1850) observed a definite
relationship between mechanical work done (W) and heat produced (II), i.e.,
WocH or W=J.H
where J is a constant, known as mechanical equivalent of heat. Its value is
4.185 x 107 ergs or 4.185 joules. Thus, for the use of 4.185 x 107 erg of mechanical
energy, 1 calorie of heat is produced.
1 Joule = 4.;85 calorie = 0.2389 calorie.

..

Work can be defined as the product of an intensity factor (force, pressure
etc.) and a capacity factor (distance, electrical energy etc). Work is done in

various ways.
(i) Gravitational work : The work done is said to be gravitational work if a
body is moved through a certain height against the gravitational field. If m gm be
the mass of a body and h cm be the height of the gravitational field of acceleration
gem sec-2 , then the force used to overcome gravity is mg, i.e., the intensity factor
is mg dynes. The capacity factor is the height h cm. The wor:;: done is, therefore,
mgh ergs.
(U) Electrical work: The work done is said to be electrical, if a current flows
in an electrical circuit. If a potential difference causing the flow is E volts (intensity
factor) and the quantity of electricity that flows in a given time is Q coulombs
(capacity factor), then electrical work done is EQ volt coulombs or EQ joule.
(iii) Mechanical work : The work done is said to be mechanical whenever
there is a change in the volume of the system. As seen in latter articles, the work

done is given by

fv

2

VI

PdV.


THERMODYNAMICS-I

3

(iv) Maximum work: The magnitude of work done by a system on expansion

depends upon the magnitude of the external pressure. Maximum work is obtained
when the gaseous pressure and the external pressure differ only by an
infinitesimally small amount from one another. It is obtained in an ideal reversible
process.

[VII] Thermodynamic Variables or State Variables
The quantities whose values determine the state of a system are called its
thermodynamic variables or state variables. The most important state variables
are mass, composition, temperature, pressure and volume. It is, however, not

necessary that we should always specifY all the variables, because some of them
are inter-dependent. For a single pure gas, composition may not be one of the
variables, as it remains only 100%. For one mole of an ideal gas, the gas equation
PV = RT is obeyed. Evidently, if only two out of the three variables (P, V, T) are
known, the third can be easily calculated. The two variables generally specified are
pressure and temperature. These two variables are known as independent
variables. The third variable, viz., volume is known as dependent variable, as its
value depends upon the values of pressure and temperature.

[VIII] Extensive Variable
The variable of a system which depends upon the amount of the substance or
substances present in the system is known as an extensive variable. In other words,

those variables whose values in any part of the divided system are different from
the values of the entire system are called extensive variables. Examples of extensive
variables are volume, energy, heat capacity, entropy, enthalpy, free energy, length
and mass.

[IX] Intensive Variable
The variable of a system which is independent of the amount of the substance

present in the system is known as intensive variable. In other words, those variables

whose values on division remains the same in any part of the system are called
intensive variables. Examples of intensive variables are temperature, pressure,
concentration, dipole moment, density, refactive index, surface tension, viscosity,
molar volume, gas constant, specific heat capacity, specific gravity, vapour pressure,
emf of a dry cell, dielectric constant etc.

[X] State Functions and Path Functions
State variables which are determined by the initial and final states ofthe system
only and not by the path followed are called state functions. These depend upon

how the change from initial to the final state is carried out.
State variables, on the contrary which are determined or depend on the path
followed are called path functions.
Consider the expansion of a gas from PI, VI, Tl toP2, V2 and T2 : (i) in steps
and (ii) adiabatically. In adiabatic expansion, let the work done by the system be
W and heat absorbed is zero. In stepwise expansion, heat absorbed is Q and work
done is W. Here W *" Q and the heat absorbed in the two cases are also different
even though the system has undergone the same net change. Thus, W and Q are


4

PHYSICAL CHEMISTRY-I!

path functions and not the state functions. However, the change in internal energy
(heat absorbed-work done) in the two cases will be seen to be constant. 'rhis is
possible only if internal energy has the same values in the two states ofthe system.
Internal energy of a system is thus a state function. Entropy, free energy, enthalpy

are other state functions.

[XI] Thermodynamic Equilibrium
A system in which the macroscopic properties donot undergo any change with
time is said to be in thermodynamic equilibrium. When an isolated system is left

to itself and the pressure and temperature are measured at different points of the
system, it is seen that although these quantities may initially change with time,
the rate of change becomes smaller and smaller until no further observable change
occurs. In such a state, the system is said to be in thermodynamic equilibrium.
Thermodynamic equilibrium means the existence of three kinds of equilibria in
the system. These are termed as thermal equilibrium, mechanical equilibriu m and
chemical equilibrium.
(a) Thermal equilibrium : A system is said to be in thermal equilibrium if
there is no flow of heat from one part of the system to another. This is possible
when the temperature remains the same throughout in all parts of the system.
(b) Mechanical equilibrium : A system is said t.o be in mechanical
equilibrium if there is no mechanical work done by one part of the system or the
other. This is possible when the pressure remains the same throughout in all parts
of the system.
(c) Chemical equilibrium: A system is said to be in chemical equilibrium if
the concentration of the various phases remain the same throughout in all parts of
the system.

[XII] Thermodynamic Process
A thermodynamic process is a path or an operation by which a system changes
from one state to another. Following different thermodynamic processes are known
which are explained as fellows :
(a) Isothermal process: A process is said to be isothermal ifthe temperature
of the system remains constant throughout the whole process. This is obtained by

making a perfect thermal contact of the system with a thermostat of a large heat
capacity.
(b) Adiabatic process: A process is said tv be adiabatic ifno heat is allowed
to enter or leave the system during the whole process. In such a nrocess, therefore,
the temperature gets altered because the system is not in a po::,ition to exchange
heat with the surroundings. It is obtained by having the wall of the system made
of perfect heat insulating substance.
(c) Isobaric process: A process is said to be isobaric if the pressure remains
constant throughout the whole process.
(d) Isochoric process: A process is said to be isochoric if the volume remains
constant throughout the whole process.
(e) Cyclic process: A process in which a system undergoes a series ofchanges
and finally comes back to the initial state is known as a cyclic process.


THERMODYNAMICS~-~I

_________________________________________________--=5

[XIII] Reversible and

Irrevers~ble

Process

A process which is carried Dut infinitesimally slowly so that the driving force
is only infinitesimally greater than the opposing force is called a reversible process.
In a reversible process, the direction of the process can be reversed at any point by
making a small change in a variable like pressure, temperature etc.
Any process which does not take place in the above way, i.e., a process which

does not occur infinitesimally slowly, is called an irreversible process.
A reversible process cannot be realised in practice, it would require infinite time
for its completion. Hence, all those reactions which occur in nature or in laboratory
are irreversible. A reversible process is thus theoretical and imaginary. The concept
of reversibility can be understood as follows :
Consider a gas cylinder provided with a frictionless and weight1ess piston upon
which is kept some sand. At the beginning, when an equilibrium exists between the
inside pressure ofthe gas and outside pressure of atmosphere plus sand, the piston
is motionless. If we remove a grain of sand, the gas will expand slightly, but the
equilibrium will be restored almost instantaneously. If the same grain of sand is
replaced, the gas will return to its original volume and the equilibrium remains
unchanged.

[XIV] Exact and Inexact Differentials
An exact differential is one which integrates to a finite difference, e.g.,

f:

dE:::: EB - E A , where EA and EB are the internal energy of the system in the

initial and final states, respectively. An exact differential is independent of the path
of integration. In a cyclic process, the final state is the same as the initial state,
i.e., E B :::: EA, the cyclic integral of an exact differential, i.e.,

f

dE:::: O. The

difference, EB - EA is denoted by t;E.
An inexact differential is one which integrates to a total quantity depending

upon the path of integration, e.g.,

where W is the total quantity of work appearing during the change from initial
state A to the final state, B. Small changes in path independent state functions are
represented by symbols like dE, dG, etc., while small changes in path dependent
functions are represented by symbols like Sq, oW etc. Finite changes in the former
are represented by symbols like ~,AG etc, but symbols like Aq, AWare
meaningless when we deal with path dependent functions. This is so, because the
system in the two states is not associated with any heat or work. Heat and work
appear during the process only. Moreover, the cyclic integral of an inexact
differential is generally not zero.
[XV] Total Differential
An exact differential serves as a strong tool for deducing quantitative
conclusions about heat, work and energy. Consider a system of definite mass. The


6

PHYSICAL CHEMISTRY-II

state function, E is dependent on volume, V, temperature, T and pressure, P.
However, these variables are related to one another through an equation of state
PV = nR1'; i.e., P is a function of V and T and, therefore P, is a dependent variable.
We could thus write E (T, P) or E (P, V). We can now find the change in internal
energy, E when V is changed to V + dV and T to T + dT, where dV and dT are very
small changes in volume and temperature. This is done as follows.
First consider a change in temperature from T to T + dT, keeping the volume
constant. If we know the slope of E with respect to T, at constant V, i.e., value of

( ~~ )v ,we can write the change in internal energy as,

E (V, T+dT) =E (V, T) +( ~~ )vdT
The coefficient ( ~~ )v is termed as the partial derivative of E with respect to
T. It is a measure of change in E per unit change in T, keeping the volume constant.
If volume is changed from V to V + dV, keeping temperature constant, we get

E (V +dV, T) =E (V, 1')

+( ~~ )TdV

When both V and T change to V + dVand T + dT, simultaneously, we obtain,
E (V + dV, T + dT)

or

=E (V, 1') + ( ~~ )v dT + ( ~~ )T dV

E (V + dV, T + d1') - E (V, T) = (

~~

1

dT + (

~~ )T dV

The left hand side of the above equation gives the difference in internal energy
at (V + dV) and (T + dT) and that at (V, T) and is denoted by dE. Therefore,
dE = (

~~

1

dT + (

~~ )T dV

... (1)

Equation (1) gives the total change in internal energy, dE for a system of fixed
concentration. It shows that dE is the sum of the changes accompanying change in
temperature (

~~ Jv dT and change in volume ( ~~ )T dV.

Thus, dE is called the

total differential of the function.
In general, any function \jI dependent on two variables x, y, has a total
differential d\jl given by
d

\jI

=(

~; )y dx + ( ~

l

dy

... (2)

[XVI] Properties of Exact Differentials
(i) Cross derivative rule : By means of thIS rule, we can find out whether a
given differential equation is an exact differential or not as follows :
Let \jI be a state function of two independent variables x and y of the system,
i.e.,
\jI =f(x,y)
As \jI is a state function, so differential of \jI is an exact differential and can be
written as


7

THERMODYNAMICS-I

d\jl=(~)
dx+(~)
dy
aXy
ayx
or

d\jl =M (x,y)

whereM(x,y)

dx +N (x,y) dy

... (3)

=( ~ 1andN=(~ l

Taking mixed second derivatives, we have

-~
( alvl)
iJy x - iJy . ax

and

( ~)-(~)
ay . ax - ax. iJy
(

~: ) x =( ~~ )y

... (4)

Equation (4) is called the Euler reciprocal (or reciprocity) relation or
cross-derivative rule. It is applicable for state functions only.
As \jI is a state function, the finite change, A\jI, as the system passes from initial
state A to final state B, is given by
A\jI = \jiB - \jI A

Also

~

d\jl = 0

... (5)

where cyclic integral ~ means that the system is in the same state at the end of
its path as it was at the beginning, i.e., it has traversed a closed path. Thus, d\jl is
an exact differential.
(ii) The cyclic rule: It is another useful relation between partial derivatives.
Consider a quantity, x = f (y, z). The total differential of x may be written as
dx = (

~;)z dy + (~~ )y dz

... (6)

At constant x, i.e., when y and z vary in such a way that x remains constant,
the above equation takes the form,

0=( ~; )z (dy)x + ( ~~ )y (dz)x

... (7)

Dividing equation (7) by (dz)x, we get

or
or

0=( ~;)z (~)x + (~~ )y

Multiplying equation (8) by ( ~~ )y' we get
0=( ~)z (~ )x ( ~= )y + ( ~~ )y (~= )y
0=( ~;)z (~ )x (~= )y + 1
( ~; )z (~ )x ( ~~ )y =- 1

... (8)

... (9)


8

PHYSICAL CHEMISTRY-II

This is the cyclic rule. It is so called because the variables X,}" z in the
numerator are related to y, z and X in the denominator and to the subscripts
z, X andy by cyclic permutations. In general, wherever any three variables are
related to Ol1e another, the three partial derivatives follow the cyclic rule.

Problem 1. Given that P :; RJ +

;2'

where R and a are constants, show that dP

is an exact differential.
Solution : It is seen that, P = f (1', V). So, dP will be an exact differential if,
'iPp
a2p

aT . av =av . aT

a
y2

RT

p=-+--

We have,

V

... (i)

Differentiating equation (i) with respect to T. at constant V, we get

=11.
( ap)
aT v V

. .. (ii)

Differentiationg equation (ii) with respect to V, at constant T, we get
a2p
R

aT .av=-Y2

... (iii)

Now differentiating equation (i) with respect to Vat constant T, we get

(~~1=-~ -~

... (iv)

Differentiating equation (iv) with respect to T, at constant V, we get
a2p
R

aV.Clr=-\r 2

... (v)

Comparing equations (iii) and (v), we see that
a2p
a2p

aT . av = av . aT

Therefore, dP is an exact differential.

Problem 2. Given that P =

R:

+ ~, where

R

and a are constants, show that:

(~~ )v (~~ )p n~ )T = - 1.
R1' a
p = - + -2

Solution: We have,

V
V '
Differentiating equation (i) with respect to Vat constant 1', we get
(

or

~~ ') T = - ~~ - ~ = - ( ~; + ~~ )
av,
1
1

( ap ) T = ( ~~

1

= - (RT/y2 + 2a1V3 )

Differentiating equation (D with respect to l' at constant V, get
(ap) _fi
\. aT v- V

Differentiating equation (i) with respect to Vat constant P, we get

... (i)

... (ii)

... (iii)


9

THERMODYNAMICS-I

O=~(~~)p -~ -~
or
or

Ii(3I.')
V

==!f1:+~

av)p V2 V.3
( dT \ V ( RT 2a 'I
l )p -= R " V2 + V 3 )

av

... (iv)

Multiplying equations (ii), (iii) and (iv), we get

B.) (YI ( RT2 + 2a3 )\ x r____
1 -

( gP '\ (aT 1 (av) == _ (
\ aT Jvl av)p dP T \ V

R). v

v

1(~; + ~~ ) 1t::: _ .
1

Problem 3. Prove that dT is a perfect differential for an ideal gas.
Solution: For an ideal gas,
PV==RT

... (i)

So, dT will be a perfect differential, if

a2T

iPT

av. ap =- oP . av
Differentiating equation (i) with respect to T at constant P. we get
'I

P

aT
( av)p
=li

... (li)

Differentiating equation (i) with respect to P at constant V, we get
T j =~
( 3ap /v
R

... (iii)

Differentiating equation (ii) with respect to P, at constant V, we get
a aT
a p

ap (av )p == ap (Ii )

... (iv)

and
Now differentiating equation (iii) with respect to Vat constant P, we get

a (aT)

av ap
or

v=

a2T

a fV)

avlli
1

av, ap ==li

... (v)

From equations (iv) and (v) it follows that

a2T

ap-:-av ==

cPT
av.ap

Thus, dT is a perfect differential.

Problem 4. Verify the cyclic rule for one mole of an ideal gas.
Solution: For 1 mole of an ideal gas,
PV=RT

Differentiating the above equation, we get

PdV + VdP::. RdT
At constant temperature, dT::: 0, so
ap 'I
P

( aV)T=-"

... (i)


10

PHYSICAL CHEMISTRY-II

At constant pressure, dP = 0, so

(~~)p =~

... (ii)

At constant volume dV = 0, so

( aT)
ap y=Iiv

... (iii)

From equations (i), (ii) and (iii), we get
(

~~ ) T ( ~~) p ( ~~ ) y = - ( ~ ) ( ~ ) (

Hence, cyclic rule is verified.

*)

=- 1

Problem 5. Show that for an ideal gas, the work differential dw is not an exact
differential.
Solution: We know that the work differential is given by
dw =PdV

... (i)

As volume (V) is a state function IV = f (T, P)], so dV is an exact differential
..

dV = (

~~ ) pdT +( ~~ ) T dP

... (ii)

For an ideal gas, PV = RT and so
V=RT
P

av ) =!iP
(aT

p

(av)
aT T -_- RT:
p2

and

... (iii)

Combining equations (i), (ii) and (iii), we get
dw = P [ (
=P[

~~ )p dT + ( ~~ )T dP ]

~ . dT - ~; . dP ]

=RdT_

RT

P

dP

dw=RdT- VdP

or

... (iv)

Applying Euler's reciprocal relation to equation (iv), we have
(

~~ ) T = - n~) p

As R is gas constant, we have (

~~

1 o.
=

... (v)

So, from equation (v),

av) = °
(aT
'P

This is not true because at constant pressure, there is always a change in volume with
temperature. So, we can say that dw is not an exact differential.

Problem 6. If V is a function of T and P, show that for an ideal gas, dV is an
exact differential.
Solution:

V=f(T,P)

dV=(

For an ideal gas, PV = RT or V =

~~ )pdT +( ~~)TdP

RT

p- . So,

... (i)


11

THERMODYNAMICS-I

( av')
aT
Substituting the values of (

and

_fi
p

p -

(~P~)T=_RpT2
u

... (ii)

~~ )p and ( ~~)T in equation (i), we get
dV=!idT- RT dP
P
p2

... (iii)

According to Euler's reciprocal relation, dV will be an exact differential, if

[
or

~R/P) ]

ap

= _[
T

R

2J!!!/P~ ]

aT

p

R

-p2=- p2
As this is true, so dV is an exact differential.

Energy is defined as 'the work and all else that can arise from work and can
be converted into work'. In other words, it can also be stated as, 'any property which
may be generated (rom or be cOnl)erted into work, including work itselr. In simpler
words, energy may be defined as 'capacity to do work'. If the energy is due to the
motion of the body, it is known as kinetic energy. If on the contrary, energy is
possessed by a body by virtue of its position in a field of force, then it is called
potential energy. All forms of energy have the dimensions :
ML2T-2, i.e., mass x (length)2/(time)2.
Three important forms of energy are : mechanical work, heat energy and
electrical energy. Energy is composed of two factors; (i) capacity factor, and (ii)
intensity factor. In fact, energy is the product of capacity factor and intensity factor.
Capacity factor is the extent to which the force or resistance is overcome when a
body performs work or expends energy, whereas intensity factor is the force or
resistance which is overcome. Suppose, a mass (m) is lifted against gravity (g) to a
certain height (h); then capacity factor is equal to h and the intensity factor will be
equal to mg. Therefore, work performed or energy expended is mg x h.

lEI INTERNAL ENERGY
Every substance is associated with a certain amount of energy which depends
upon its chemical nature, as well as volume, temperature, pressure, and mass. This
energy is known as internal, intrinsic or chemical energy. The exact magnitude of
the internal energy is unknown, because the chemical nature includes such
indeterminant factors such as translational, rotational and vibrational motion of
the molecules, the manner in which the molecules are arranged together and the
energy possessed by the nucleus. Inspite of this, internal energy of the substance

or the system will be a definite quantity. It is completely determined by the state
of the substance or system itself and is independent of the previous history of the
system. Internal energy is also known as 'energy of the system' and is usually
denoted by E or U. It is not possible to ascertain the absolute value of internal
energy, but in thermodynamics this value is of no significance. However, it is
possible to measure the change in internal energy which accompanies a certain
process.


12

PHYSICAL CHEMISTRY-II

If EA and EB be the internal energy ofthe system in the initial and final states,

respectively, then the difference (ME) between the final and the initial energy is
given by:

We know that temperature is one of the quantities necessary to define the state
of a thermodynamic system. In other words, every thermodynamic system must
possess a temperature. This postulate of the existence of temperature is sometimes
called the zeroth law of thermodynamics or law of thermal equilibrium. This law
was first developed and defined by R.H. Fowler (1931). In other words, zeroth law
of thermodynamics states that, if body A is in equilibrium with body C and body B
is also in equilibrium with body C, then bodies A and B are in equilibrium with
each other.

Energy manifests itself not only in the form of mechanical work, but also as
heat energy, electrical energy and chemical energy. Energy is composed of two
factors viz., intensity factor and capacity factor. The product of these two factors

gives the energy.
Heat energy is measured by the product of temperature (intensity factor) and
heat capacity (capacity factor) of the system. The product gives the energy of the
system. If a substance of mass m kg and specific heat s kJ per kg is heated through
t, the heat energy involved is given by mst kJ.
The precise experimental basis for this assumption is tbe law of thermal
equilibrium from which it can be shown that in a thermodynamical system for every
participant. in equilibrium, there exists a certain single valued function f of the state
variables, P and V, which have the same values for all participants.

[I] Mathematical Treatment of Zeroth Law of Thermodynamics
Suppose there are three fluids A, B and C. Suppose PA and VA are the pressure
and volume of A, PB and VB are the respective values of Band Pc and Ve are the
respective values of C. Suppose A and B are in thermal equilibrium with one
another, then we can write

(i>! (PA, VA) = <1>2 (PB, VB)
or

FdPA, VA,P B, VB)=O
On solving equation (1), we get
PB =fI (PA, VA, VB)

... (1)

... (2)

Suppose Band C are in thermal equilibrium, then we can write
<1>2 (PB' VB)

or

= <1>3 (Pc, Vo)

F 2 (PB , VB, Pc, Vo)=O

On solving,
P B = f2 (VB, pc. V 0)
... (3)
On equatjng equations (2) and (3), we get the following equation for A and C to
be in thermal equilibrium separately.
fl (FA, VA, VB) = f2 (VB, Pc, Vo)

... (4)


THERMODYNAMICS-I

13

If A and C are in thermal equilibrium with B separately then A and C in
accordance with zeroth law of thermodynamics, are also in thermal equilibrium
with one another. So,
... (5)

As equation (4) has a variabie VB, while equation (5) does not have a variable
VB' it means that
... (6)

In general,

... (7)

From equation (7), it follows that all the three functions <1>1, <1>2 and <1>3 have the
same numerical value but the parameters of P and V are different. This numerical
value is termed as the temperature (T) of the body, i.e.,
</> (P, V) = T
So, temperature of a system is the property which decides whether the body is
in thermal equilibrium with neighbouring systems or not.
It is clear from the above definition that if two systems are not in thermal
equilibrium with one another, they are at different temperatures.

The principle of c:mservation of energy which results from a ,vide range of
experience ca.'} be expressed as that, in an isolated system the sum total of all forms
of energy remains constant, although it may change from one form to another.
Attempts have been made from time to time to disprove the energy conservation
principle, but every attempt proved a failure. No person succeeded in inventing a
machine to produce perpetual motion (continuous production of mechanical work
without putting in an equivalent amount of energy from outside). This failure led
to the universal acceptance of the law of conservation of energy. It was considered
to be a basic law of nature and was given the name 'first law of thermodynumics'.
According to this law, whenever a quantity of one kind of energy is produced, an
exactly equivalent amount of some other kind of energy must disappear.
The law of conservation of energy has been partly modified. It is now known
that energy can be produced by the destruction of mass also, and the two quantities,
viz., energy (E) and mass (m) are connected by the relation,

E=mc 2
where c is the velocity of light. The modified law, therefore, states that the total
mass and energy of an isolated system remain unchanged.

[I] Mathematical Formulation of the First Law of Thermodynamics
Suppose a system in a state A having an internal energ-j EA, undergoes a
change to another state B. During this change, let the system absorbs a small
quantity of heat, q. The net amount of energy of the system would be EA + q. During
this transformation, the system might have also performed some work, W
(mechanical, electrical or any other type). If EB be the internal energy of the system


14

PHYSICAL CHEMISTRY-I!

in state B, the net energy of the system after the change would be EB + W. From
the principle of conservation of energy, we have
EA+q=EB+W
... (1)
or
t.E =q - W
... (2)
In the differential form,
dE = dq - dW
Equation (2) is the mathematical statement of the first law, i.e., the heat taken
up by a system would be equal to the internal energy increase of the system and
the work done by the system.
In an isolated system, there is no heat exchange with the surroundings,
i.e., dq = O. Therefore, from equation (2),
dE+dW=O or dW=-dE,
i.e., in an isolated system, the work performed will be equal to the decrease in
internal energy.
In a non-isolated system, the system gains heat (dq) and performs work (dW).

For such a system, we have
dE = dq - dW
If we consider the surroundings, it loses dq amount of heat, but receives dW
units of work. The internal energy change of the surroundings will then be
dE'=dW-dq
Thus,
dE = - dE'
or
dE + dE' =0
i.e., net change in the internal energies of the system and the surroundings taken
together would be nil.
•

HEAT CHANGES

(a) At constant volume (qv) : When a process is carried out at constant
volume, then there is no mechanical work done either by the system or on the
system. In other words, W = O. Hence, at constant volume, equation (1) reduces to
qv = t.E.
... (3)
(b) At constant pressure (qp) : Suppose a process is carried out at constant
pressure say P, then in this case the volume is allowed to change say from
VI to V 2, then the work done is given by,
W =p (V2 - VI)'

Hence, equation (1) reduces to,
t.E = qp - P (V2 - VI)

... (4)

where qp is the heat absorbed in the change at constant pressure.
From equation (4),

qp = t.E + P (V2 - VI) = t.E + Pt:N,

... (5)

where 11V ie the change in volume at constant pressure.
From equations (3) and (4), we get, qp - qv =PI1V
or
qp = qv + PI1V = qv + I1n.RT
(because I1nRT = PI!. V for an ideal gas).

... (6)


15

THERMODYNAMIC,':;-I

III HEAT CONTENT OR ENTHALPY
If!ill is replaced by EB - EA, and the increase in volume (11 V) by VB - VA (where
A and B represent the initial and the final states of the system, respectively), then
equation (5) becomes
qp = (EB -EA) +P (VB - VA) = (EB -PVB) - (EA +PVA)

... (7)

The factors P and V are the properties of the state of the system. Hence, it is
evident that the quantities (EB + PVB) or (EA + PVA), like the energy E must be

dependent only on the state of the system and not on the previous history of the
system. Therefore, we may detine a new quantity by the equation,
H=E+PV,
where H is known as heat content or enthalpy of the system. Therefore,
HB =EB+PVB and HA =EA +PVA
qp=HB-HA=m,

... (8)

where m is the increase in the heat content of the system. It (L1H) may, therefore,
be defined as the amount of heat absorbed at constant pressure.
If the system is composed of ideal gases and if nl and nz are the number of
moles before and after a chemical change, then
PVI = nl RT and PVz = nzRT
PI1V

=P (Vz -

VI)

=(nz -

nl) RT = I1nRT

... (9)

From equations (7) and (8),
qp = m = !ill + PI1 V.

Hence, from equation (9), we have,

... (10)
m =!ill + I1nRT
The internal energy of an ideal gas does not depend upon the volume, but
depends upon temperature, therefore, at constant temperature, (

~~ )

T o.
=

At constant temperature, the pressure varies with volume. Since for ideal gases,
(

Further, we know H

~~ ) T = 0, hence ( ~~ ) T = 0
=

E + PV, i.e.,

(~~)T=(~~)T+[~~JT

... (11)

Since ideal gases obey Boyle's law, therefore,

[a(:x) JT

=

o.

Equation (11) now becomes (~~)T =:: 0 + 0 =:: 0 and also

(~~)T = o.

When a definite amount of material is heated in a specified way, then it is found
that the heat added is approximately proportional to the temperature change. Since


16

PHYSICAL CHEMISTRY-II

the quantity of heat required to produce a certain temperature change is of
importance, it is, therefore, convenient to define a new quantity, known as heat
capacity of the system. The heat capacity, or more accurately the mean heat
capacity of a system between any two temperatures, is defined as the quantity of
heat required to raise the temperature of the system from the lower t"J the higher
temperature divided by the difference. It is usually represented by the letter, c. It

has two values one, heat capacity at constant volume (c v ) and the other, heat capacity
at constant pressure (cp ).
Thus, the heat capacity of a system between two temperatures Tl and T2 is
given by,
q
c(T2 ,T1)=-T-T
2-

1

In general, since heat capacity varies with the tempprature, the true heat
capacity is defined by the differential equation
dq
c =--dT
The heat capacity of a system at constant volume is different from that
calculated at constant pressure. In the former case, no external work is done by the
system or on the system, as there is no change in volume. So, from equation (10),
we have
q = L'lE
cu (T 2 ,T1)=(T q T )

1 v

2--

or

C

v

=

(aE)
aT T

=:

=(T~T

)
2- 1

!J

0

... (12)

So, the heat capacity of a system at constant volume is defined as the increase
in internal energy of the system per degree rise of temperature.
At constant pressure, there is a change of volume and some work is done. Let
the increase in volume be ilV, then from equation (1),
... (13)
q=L'lE+w=6.E+PilV

q
)
( LlE + PilV)
cp (T2 ,T1)= ( T -T p= T -T
P
2

1

2

( aH ) (aH)

= T 2 - TIP = ~ aT P

... (14)

1

... (15)

So, the heat capacity of a system at constant pressure is defined as the increase
in enthalpy of the system per degree rise in temperature.
For one mole of a gas, heat capacities at constant volume and constant pressure
are represented by C v and Cp ' respectively. These are then known as molar heat
capacities. Thus, for one mole of a gas,

Cv = ( aE
aT )v

... (16)
... (17)


17
THERMODYNAMICS-I
~--------------------------------------------APPLICATIONS OF FIRST
LAW OF
.
. .THERMODYNAMICS
..
c

[I] Difference in the Two Heat Capacities

(a) For an ideal gas : The internal energy of a system is a function of
temperature and volume. Thus,
E =f(T, V)
or
or

dE

=( ~~ hdT +( ~~ )TdV

... (18)

... (19)
( ~~ )p = ( ~~ )v + ( ~~ )T ( ~~ )p
/ 'OR) (aE)
[From equations (12) and (16)]
Cp-Cv=l aT 'P - aT v

=(
=

~~ )p + P ( ~~ )p -( ~~ )v

(AsR=E +PV)

aE)T (av)
av)'P - (aT
aE ) v
( aE
aT )v + ( av

aT 'P + P ( aT

~~ )T + P ] ( ~~ )p
For 1 mole of an ideal gas, ( ~~ )T = 0
=[ (

p- Cu =P ( ~~)p
(av)
aT =RP (As V =RT/P)

From equation (20),

C

'P

From equations (21) and (22), we get,
R
Cp-Cv=Px P =R

... (20)

... (21)
... (22)

... (23)

(b) For a real gas: We have,
dq = (

or

(

fr )p

~ )T dV + ( ~ )v dT

= T(

~~ )TdV + CVdT

=T(

~~ )T ( ~~)p + Cv

From equation (16), we have,

p- Cv = T( ~~ )T( ~~)p
We also know, ( ~~ )T = ( ~~ )v
(From Maxwell's relation)
..
C p- Cv= T ( ~~ )v ( ~~)p [From equation (24)]
C

For a real gas, the vander Waals equation is

... (24)

... (25)

18

PHYSICAL CHEMISTRY-II

... (26)
... (27)

a
V

or

ab

a
V

RT=PV -Pb +---""PV -Pb +-

From equation (26),

".-2

(~~)p= P~~

... (28)

y2

R

Cp-Cv=T·(V_b)·

R

a

(p- ~)

[From equations (25), (27) and (28)]
RT
R
--V - b· RT
2a (V - b) ]
V-b
RTv2

[1 _

"" R [ 1 + R~ ] (Neglecting b in comparison to V)
=

R [ 1 + R2rf2
2aP ]

... (29)

[II] Work done in Isothermal Expansion of an Ideal Gas

When a system undergoes a change in such a way that its temperature remains
constant throughout the change, then it is spoken of as an isothermal change.
During such a change, there is an exchange of heat between the system and the
surroundings and the heat exchanged corresponds to the work done either by the
system against external pressure or upon the system by some external pressure.
Imagine any chemical system contained in a cylinder fitted with a frictionless
and weightless piston. Suppose that the cylinder has a cross sectional area A and
that the pressnre exerted by the system at the piston face is P. The force on the
piston is, therefore, PA. If the piston moves out an infinitesimal distance dx, the
system performs an infinitesimal amount of work dW equal to
dW = PA . dx = PdV
(where V is the volume)
Now imagine that the piston is made to move a finite distance, the volume
changes from VI to V 2 . The amount of work done by the system is then

W=

V2

J

PdV

VI

where P is the force acting on the piston per unit area at any moment. The
evaluation of the integral can be made once the behaviour of the temperature
coordinate is specified, because then P can be expressed as a function of V only. In
addition, the change in volume must be performed quasistaticall)~ i.e., the system
must be at all times during the expansion infinitesimally near a state of