CHAPTER TWELVE
THERMODYNAMICS
12.1 INTRODUCTION
In previous chapter we have studied thermal properties of
matter. In this chapter we shall study laws that govern
thermal energy. We shall study the processes where work is
converted into heat and vice versa. In winter, when we rub
our palms together, we feel warmer; here work done in rubbing
produces the ‘heat’. Conversely, in a steam engine, the ‘heat’
of the steam is used to do useful work in moving the pistons,
which in turn rotate the wheels of the train.
In physics, we need to define the notions of heat,
temperature, work, etc. more carefully. Historically, it took a
long time to arrive at the proper concept of ‘heat’. Before the
modern picture, heat was regarded as a fine invisible fluid
filling in the pores of a substance. On contact between a hot
body and a cold body, the fluid (called caloric) flowed from
the colder to the hotter body! This is similar to what happens
when a horizontal pipe connects two tanks containing water
up to different heights. The flow continues until the levels of
water in the two tanks are the same. Likewise, in the ‘caloric’
picture of heat, heat flows until the ‘caloric levels’ (i.e., the
temperatures) equalise.
In time, the picture of heat as a fluid was discarded in
favour of the modern concept of heat as a form of energy. An
important experiment in this connection was due to Benjamin
Thomson (also known as Count Rumford) in 1798. He
observed that boring of a brass cannon generated a lot of
heat, indeed enough to boil water. More significantly, the
amount of heat produced depended on the work done (by the
horses employed for turning the drill) but not on the
sharpness of the drill. In the caloric picture, a sharper drill
would scoop out more heat fluid from the pores; but this
was not observed. A most natural explanation of the
observations was that heat was a form of energy and the
experiment demonstrated conversion of energy from one form
to another–from work to heat.
12.1 Introduction
12.2 Thermal equilibrium
12.3 Zeroth law of
Thermodynamics
12.4 Heat, internal energy and
work
12.5 First law of
thermodynamics
12.6 Specific heat capacity
12.7 Thermodynamic state
variables and equation of
state
12.8 Thermodynamic processes
12.9 Heat engines
12.10 Refrigerators and heat
pumps
12.11 Second law of
thermodynamics
12.12 Reversible and irreversible
processes
12.13 Carnot engine
Summary
Points to ponder
Exercises
2020-21
PHYSICS304
Thermodynamics is the branch of physics that
deals with the concepts of heat and temperature
and the inter-conversion of heat and other forms
of energy. Thermodynamics is a macroscopic
science. It deals with bulk systems and does not
go into the molecular constitution of matter. In
fact, its concepts and laws were formulated in the
nineteenth century before the molecular picture
of matter was firmly established. Thermodynamic
description involves relatively few macroscopic
variables of the system, which are suggested by
common sense and can be usually measured
directly. A microscopic description of a gas, for
example, would involve specifying the co-ordinates
and velocities of the huge number of molecules
constituting the gas. The description in kinetic
theory of gases is not so detailed but it does involve
molecular distribution of velocities.
Thermodynamic description of a gas, on the other
hand, avoids the molecular description altogether.
Instead, the state of a gas in thermodynamics is
specified by macroscopic variables such as
pressure, volume, temperature, mass and
composition that are felt by our sense perceptions
and are measurable*.
The distinction between mechanics and
thermodynamics is worth bearing in mind. In
mechanics, our interest is in the motion of particles
or bodies under the action of forces and torques.
Thermodynamics is not concerned with the
motion of the system as a whole. It is concerned
with the internal macroscopic state of the body.
When a bullet is fired from a gun, what changes
is the mechanical state of the bullet (its kinetic
energy, in particular), not its temperature. When
the bullet pierces a wood and stops, the kinetic
energy of the bullet gets converted into heat,
changing the temperature of the bullet and the
surrounding layers of wood. Temperature is
related to the energy of the internal (disordered)
motion of the bullet, not to the motion of the bullet
as a whole.
12.2 THERMAL EQUILIBRIUM
Equilibrium in mechanics means that the net
external force and torque on a system are zero.
The term ‘equilibrium’ in thermodynamics appears
in a different context : we say the state of a system
is an equilibrium state if the macroscopic
variables that characterise the system do not
change in time. For example, a gas inside a closed
rigid container, completely insulated from its
surroundings, with fixed values of pressure,
volume, temperature, mass and composition that
do not change with time, is in a state of
thermodynamic equilibrium.
Fig. 12.1 (a) Systems A and B (two gases) separated
by an adiabatic wall an insulating wall
that does not allow flow of heat. (b) The
same systems A and B separated by a
diathermic wall a conducting wall that
allows heat to flow from one to another. In
this case, thermal equilibrium is attained
in due course.
In general, whether or not a system is in a state
of equilibrium depends on the surroundings and
the nature of the wall that separates the system
from the surroundings. Consider two gases A and
B occupying two different containers. We know
experimentally that pressure and volume of a
given mass of gas can be chosen to be its two
independent variables. Let the pressure and
volume of the gases be (P
A
, V
A
) and (P
B
, V
B
)
respectively. Suppose first that the two systems
are put in proximity but are separated by an
(a)
(b)
* Thermodynamics may also involve other variables that are not so obvious to our senses e.g. entropy, enthalpy,
etc., and they are all macroscopic variables. However, a thermodynamic state is specified by five state
variables viz., pressure, volume, temperature, internal energy and entropy. Entropy is a measure of disorderness
in the system. Enthalpy is a measure of total heat content of the system.
2020-21
THERMODYNAMICS 305
adiabatic wall an insulating wall (can be
movable) that does not allow flow of energy (heat)
from one to another. The systems are insulated
from the rest of the surroundings also by similar
adiabatic walls. The situation is shown
schematically in Fig. 12.1 (a). In this case, it is
found that any possible pair of values (P
A
, V
A
) will
be in equilibrium with any possible pair of values
(P
B
, V
B
). Next, suppose that the adiabatic wall is
replaced by a diathermic wall a conducting wall
that allows energy flow (heat) from one to another.
It is then found that the macroscopic variables of
the systems A and B change spontaneously until
both the systems attain equilibrium states. After
that there is no change in their states. The
situation is shown in Fig. 12.1(b). The pressure
and volume variables of the two gases change to
(P
B
, V
B
) and (P
A
, V
A
) such that the new states
of A and B are in equilibrium with each other*.
There is no more energy flow from one to another.
We then say that the system A is in thermal
equilibrium with the system B.
What characterises the situation of thermal
equilibrium between two systems ? You can guess
equilibrium, the temperatures of the two systems
are equal. We shall see how does one arrive at the
concept of temperature in thermodynamics? The
Zeroth law of thermodynamics provides the clue.
12.3 ZEROTH LAW OF THERMODYNAMICS
Imagine two systems A and B, separated by an
adiabatic wall, while each is in contact with a third
system C, via a conducting wall [Fig. 12.2(a)]. The
states of the systems (i.e., their macroscopic
variables) will change until both A and B come to
thermal equilibrium with C. After this is achieved,
suppose that the adiabatic wall between A and B
is replaced by a conducting wall and C is insulated
from A and B by an adiabatic wall [Fig.12.2(b)]. It
is found that the states of A and B change no
further i.e. they are found to be in thermal
equilibrium with each other. This observation
forms the basis of the Zeroth Law of
Thermodynamics, which states that ‘two
systems in thermal equilibrium with a third
system separately are in thermal equilibrium
with each other’. R.H. Fowler formulated this
law in 1931 long after the first and second Laws
of thermodynamics were stated and so numbered.
The Zeroth Law clearly suggests that when two
systems A and B, are in thermal equilibrium,
there must be a physical quantity that has the
same value for both. This thermodynamic
variable whose value is equal for two systems in
thermal equilibrium is called temperature (T ).
Thus, if A and B are separately in equilibrium
with C, T
A
= T
C
and T
B
= T
C
. This implies that
T
A
= T
B
i.e. the systems A and B are also in
thermal equilibrium.
We have arrived at the concept of temperature
formally via the Zeroth Law. The next question
is :how to assign numerical values to
temperatures of different bodies ? In other words,
how do we construct a scale of temperature ?
Thermometry deals with this basic question to
which we turn in the next section.
Fig. 12.2 (a) Systems A and B are separated by an
adiabatic wall, while each is in contact
with a third system C via a conducting
wall. (b) The adiabatic wall between A
and B is replaced by a conducting wall,
while C is insulated from A and B by an
* Both the variables need not change. It depends on the constraints. For instance, if the gases are in containers
of fixed volume, only the pressures of the gases would change to achieve thermal equilibrium.
(a)
(b)
2020-21
PHYSICS306
12.4 HEAT, INTERNAL ENERGY AND WORK
The Zeroth Law of Thermodynamics led us to
the concept of temperature that agrees with our
commonsense notion. Temperature is a marker
of the ‘hotness’ of a body. It determines the
direction of flow of heat when two bodies are
placed in thermal contact. Heat flows from the
body at a higher temperature to the one at lower
temperature. The flow stops when the
temperatures equalise; the two bodies are then
in thermal equilibrium. We saw in some detail
how to construct temperature scales to assign
temperatures to different bodies. We now
describe the concepts of heat and other relevant
quantities like internal energy and work.
The concept of internal energy of a system is
not difficult to understand. We know that every
bulk system consists of a large number of
molecules. Internal energy is simply the sum of
the kinetic energies and potential energies of
these molecules. We remarked earlier that in
thermodynamics, the kinetic energy of the
system, as a whole, is not relevant. Internal
energy is thus, the sum of molecular kinetic and
potential energies in the frame of reference
relative to which the centre of mass of the system
is at rest. Thus, it includes only the (disordered)
energy associated with the random motion of
molecules of the system. We denote the internal
energy of a system by U.
Though we have invoked the molecular
picture to understand the meaning of internal
energy, as far as thermodynamics is concerned,
U is simply a macroscopic variable of the system.
The important thing about internal energy is
that it depends only on the state of the system,
not on how that state was achieved. Internal
energy U of a system is an example of a
thermodynamic ‘state variable’ – its value
depends only on the given state of the system,
not on history i.e. not on the ‘path’ taken to arrive
at that state. Thus, the internal energy of a given
mass of gas depends on its state described by
specific values of pressure, volume and
temperature. It does not depend on how this
state of the gas came about. Pressure, volume,
temperature, and internal energy are
thermodynamic state variables of the system
(gas) (see section 12.7). If we neglect the small
intermolecular forces in a gas, the internal
energy of a gas is just the sum of kinetic energies
associated with various random motions of its
molecules. We will see in the next chapter that
in a gas this motion is not only translational
(i.e. motion from one point to another in the
volume of the container); it also includes
rotational and vibrational motion of the
molecules (Fig. 12.3).
Fig. 12.3 (a) Internal energy U of a gas is the sum
of the kinetic and potential energies of its
molecules when the box is at rest. Kinetic
energy due to various types of motion
(translational, rotational, vibrational) is to
be included in U. (b) If the same box is
moving as a whole with some velocity,
the kinetic energy of the box is not to be
included in U.
Fig. 12.4 Heat and work are two distinct modes of
energy transfer to a system that results in
change in its internal energy. (a) Heat is
energy transfer due to temperature
difference between the system and the
surroundings. (b) Work is energy transfer
brought about by means (e.g. moving the
piston by raising or lowering some weight
connected to it) that do not involve such a
temperature difference.
2020-21
THERMODYNAMICS 307
What are the ways of changing internal
energy of a system ? Consider again, for
simplicity, the system to be a certain mass of
gas contained in a cylinder with a movable
piston as shown in Fig. 12.4. Experience shows
there are two ways of changing the state of the
gas (and hence its internal energy). One way is
to put the cylinder in contact with a body at a
higher temperature than that of the gas. The
temperature difference will cause a flow of
energy (heat) from the hotter body to the gas,
thus increasing the internal energy of the gas.
The other way is to push the piston down i.e. to
do work on the system, which again results in
increasing the internal energy of the gas. Of
course, both these things could happen in the
reverse direction. With surroundings at a lower
temperature, heat would flow from the gas to
the surroundings. Likewise, the gas could push
the piston up and do work on the surroundings.
In short, heat and work are two different modes
of altering the state of a thermodynamic system
and changing its internal energy.
The notion of heat should be carefully
distinguished from the notion of internal energy.
Heat is certainly energy, but it is the energy in
transit. This is not just a play of words. The
distinction is of basic significance. The state of
a thermodynamic system is characterised by its
internal energy, not heat. A statement like ‘a
gas in a given state has a certain amount of
heat’ is as meaningless as the statement that
‘a gas in a given state has a certain amount
of work’. In contrast, ‘a gas in a given state
has a certain amount of internal energy’ is a
perfectly meaningful statement. Similarly, the
statements ‘a certain amount of heat is
supplied to the system’ or ‘a certain amount
of work was done by the system’ are perfectly
meaningful.
To summarise, heat and work in
thermodynamics are not state variables. They
are modes of energy transfer to a system
resulting in change in its internal energy,
which, as already mentioned, is a state variable.
In ordinary language, we often confuse heat
with internal energy. The distinction between
them is sometimes ignored in elementary
physics books. For proper understanding of
thermodynamics, however, the distinction is
crucial.
12.5 FIRST LAW OF THERMODYNAMICS
We have seen that the internal energy U of a
system can change through two modes of energy
transfer : heat and work. Let
Q = Heat supplied to the system by the
surroundings
W = Work done by the system on the
surroundings
U = Change in internal energy of the system
The general principle of conservation of
energy then implies that
Q = U + W (12.1)
i.e. the energy (Q) supplied to the system goes
in partly to increase the internal energy of the
system (U) and the rest in work on the
environment (W). Equation (12.1) is known as
the First Law of Thermodynamics. It is simply
the general law of conservation of energy applied
to any system in which the energy transfer from
or to the surroundings is taken into account.
Let us put Eq. (12.1) in the alternative form
QW = U (12.2)
Now, the system may go from an initial state
to the final state in a number of ways. For
example, to change the state of a gas from
(P
1
, V
1
) to (P
2
, V
2
), we can first change the
volume of the gas from V
1
to V
2
, keeping its
pressure constant i.e. we can first go the state
(P
1
, V
2
) and then change the pressure of the
gas from P
1
to P
2
, keeping volume constant, to
take the gas to (P
2
, V
2
). Alternatively, we can
first keep the volume constant and then keep
the pressure constant. Since U is a state
variable, U depends only on the initial and
final states and not on the path taken by the
gas to go from one to the other. However, Q
and W will, in general, depend on the path
taken to go from the initial to final states. From
the First Law of Thermodynamics, Eq. (12.2),
it is clear that the combination Q W, is
however, path independent. This shows that
if a system is taken through a process in which
U = 0 (for example, isothermal expansion of
an ideal gas, see section 12.8),
Q = W
i.e., heat supplied to the system is used up
entirely by the system in doing work on the
environment.
2020-21