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THERMODYNAMICS LECTURE NOTES
June 14,
2015
UNIT I
Introduction to Thermodynamics: Thermodynamics is the science that deals with heat and
work and those properties of substance that bear a relation to heat and work. Thermodynamics is
the study of the patterns of energy change. Most of this course will be concerned with
understanding the patterns of energy change. More specifically, thermodynamics deals with (a)
energy conversion and (b) the direction of change. Basis of thermodynamics is experimental
observation. In that sense it is an empirical science. The principles of thermodynamics are
summarized in the form of four laws known as zeroth, first, second, and the third laws of
thermodynamics. The zeroth law of thermodynamics deals with thermal equilibrium and
provides a means of measuring temperature. The first law of thermodynamics deals with the
conservation of energy and introduces the concept of internal energy. The second law of
thermodynamics dictates the limits on the conversion of heat into work and provides the yard
stick to measure the performance of various processes. It also tells whether a particular process is
feasible or not and specifies the direction in which a process will proceed. As a consequence it
also introduces the concept of entropy. The third law defines the absolute zero of entropy.
System, Surrounding, Boundaries, Universe, Types of Systems:
A system is defined as a quantity of matter or a region in space chosen for study. The mass or
region outside the system is called the surroundings. The real or imaginary surface that
separates the system from its surroundings is called the boundary. The boundary of a system can
be fixed or movable. Note that the boundary is the contact surface shared by both the system and
the surroundings. Mathematically speaking, the boundary has zero thickness, and thus it can
neither contain any mass nor occupy any volume in space. System and surroundings when put
together result in universe. These terms are illustrated in figure 1.1.
Figure 1.1: System, surroundings and boundary
Two types of exchange can occur between system and surroundings: (1) energy exchange (heat,
work, friction, radiation, etc.) and (2) matter exchange (movement of molecules across the
boundary of the system and surroundings).
Based on the types of exchange which take place or don't take place, there are three types of
systems:
A closed system (also known as a control mass) consists of a fixed amount of mass, and no
mass can cross its boundary. That is, no mass can enter or leave a closed system, as shown in
figure 1.2. But energy, in the form of heat or work, can cross the boundary; and the volume of a
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closed system does not have to be fixed. Closed system is the system having only energy
interactions at its boundary, for example, boiling water in a closed pan etc.
Figure: 1.2: Closed System
An open system, or a control volume, as it is often called, is a properly selected region in space.
It usually encloses a device that involves mass flow such as a compressor, turbine, or nozzle.
Flow through these devices is best studied by selecting the region within the device as the control
volume. Both mass and energy can cross the boundary of a control volume as shown in figure
1.3. The open system is one in which the energy and mass interactions take place at the system
boundary, for example automobile engine etc.
Figure: 1.3: Open System
Isolated system refers to the system which has neither mass interaction nor energy interaction
across system boundary, for example Thermos Flask etc. Thus, the isolated system does not
interact with the surroundings/systems in any way.
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Control Volume:
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Control volume is defined as a volume which encloses the matter and the device inside a
control surface.
Everything external to the control volume is the surroundings with the separation given
by the control surface.
The surface may be open or closed to mass flows and it may have flows from energy in
terms of heat transfer and work across it.
The boundaries may be moveable or stationary.
In the case of a control surface that is closed to the mass flow, so that no mass can enter
or escape the control volume, it is called a control mass containing same amount of
matter at all times.
Concept of Continuum:
Matter is made up of atoms that are widely spaced in the gas phase. Yet it is very convenient to
disregard the atomic nature of a substance and view it as a continuous, homogeneous matter with
no holes, that is, a continuum.
In Macroscopic approach of thermodynamics the substance is considered to be continuous
whereas every matter actually comprises of myriads of molecules with intermolecular spacing
amongst them. In engineering thermodynamics where focus lies upon the gross behaviour of the
system and substance in it, the statistical approach is to be kept aside and classical
thermodynamics approach be followed. In classical thermodynamics, for analysis the atomic
structure of substance is considered to be continuous.
For facilitating the analysis this concept of continuum is used in which the substance is treated
free from any kind of discontinuity. As this is an assumed state of continuum in substance so the
order of analysis or scale of analysis becomes very important. Thus, in case the scale of analysis
is large enough and the discontinuities are of the order of intermolecular spacing or mean free
path then due to relative order of discontinuity being negligible it may be treated continuous. In
the situations when scale of analysis is too small such that even the intermolecular spacing or
mean free path are not negligible i.e. the mean free path is of comparable size with smallest
significant dimension in analysis then it cannot be considered continuous and the microscopic
approach for analysis should be followed.
For example, whenever one deals with highly rarefied gases such as in rocket flight at very high
altitudes or electron tubes, the concept of continuum of classical thermodynamics should be
dropped and statistical thermodynamics using microscopic approach should be followed.
Thus, in general it can be said that the assumption of continuum is well suited for macroscopic
approach where discontinuity at molecular level can be easily ignored as the scale of analysis is
quite large. The concept of continuum is thus a convenient fiction which remains valid for most
of engineering problems where only macroscopic or phenomenological information are desired.
For example, let us see density at a point as a property of continuum. Let us take some mass of
fluid Δm in some volume ΔV enveloping a point „P‟ in the continuous fluid. Average mass
density of fluid within volume ΔV shall be the ratio (Δm/ΔV). Now let us shrink the volume ΔV
enveloping the point to volume ΔV′. It could be seen that upon reducing the volume, ΔV′ may be
so small as to contain relatively few molecules which may also keep on moving in and out of the
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considered very small volume, thus average density keeps on fluctuating with time. For such a
situation the definite value of density cannot be given. Therefore, we may consider some limiting
volume ΔVlimit such that the fluid around the point may be treated continuous and the average
density at the point may be given by the ratio (Δm/ ΔVlimit).
Thus, it shows how the concept of continuum although fictitious is used for defining density at a
point as given below,
Average density at the point = limΔV→ΔVlimit(Δm/ΔV)
Macroscopic and Microscopic Approaches:
Microscopic approach uses the statistical considerations and probability theory, where we deal
with “average” for all particles under consideration. This is the approach used in the disciplines
known as kinetic theory and statistical mechanics.
In the macroscopic point of view, of classical thermodynamics, one is concerned with the timeaveraged influence of many molecules that can be perceived by the senses and measured by the
instruments.
The pressure exerted by a gas is an example of this. It results from the change in momentum of
the molecules, as they collide with the wall. Here we are not concerned with the actions of
individual molecules but with the time-averaged force on a given area that can be measured by a
pressure gage.
From the macroscopic point of view, we are always concerned with volumes that are very large
compared to molecular dimensions, and therefore a system (to be defined next) contains many
molecules, and this is called continuum.
The concept of continuum loses validity when the mean free path of molecules approaches the
order of typical system dimensions.
The State Postulate
The state of the system is described by its properties. Once a sufficient number of properties are
specified, the rest of the properties assume some values automatically. The number of properties
required to fix a state of a system is given by the state postulate:
The state of a simple compressible system is completely specified by two independent, intensive
properties.
The system is called a simple compressible system in the absence of electrical, magnetic,
gravitational, motion, and surface tension effects.
The state postulate requires that the two properties specified be independent to fix the state. Two
properties are independent if one property can be varied while the other one is held constant.
Temperature and specific volume, for example, are always independent properties, and together
they can fix the state of a simple compressible system. Thus, temperature and pressure are not
sufficient to fix the state of a two-phase system. Otherwise an additional property needs to be
specified for each effect that is significant. An additional property needs to be specified for each
other effect that is significant.
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Equilibrium
 When the property of a system is defined, it is understood that the system is in
equilibrium.
 If a system is in thermal equilibrium, the temperature will be same throughout the system.
 If a system is in mechanical equilibrium, there is no tendency for the pressure to change.
In a single phase system, if the concentration is uniform and there is no tendency for
mass transfer or diffusion, the system is said to be in chemical equilibrium.
 A system which is simultaneously in thermal, mechanical, and chemical equilibrium is
said to be in thermal equilibrium.
Property
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In thermodynamics a property is any characteristic of a system that is associated with the
energy and can be quantitatively evaluated.
The property of a system should have a definite value when the system is in a particular
state.
Thermodynamic property is a point function.
Properties like volume of a system that depend on the mass of a system are called
extensive properties.
Properties like pressure or temperature which do not depend on the system mass are
called intensive properties.
The ratio of extensive property to the mass of the system are called specific properties
and therefore become intensive properties.
Substance can be found in three states of physical aggregation namely, solid, liquid and
vapor which are called its phases.
If the system consists of mixture of different phases, the phases are separated from each
other by phase boundary.
The thermodynamic properties change abruptly at the phase boundary, even though the
intensive properties like temperature and pressure are identical.
Any characteristic of a system is called a property. Some familiar properties are pressure P,
temperature T, volume V, and mass m. The list can be extended to include less familiar ones such
as viscosity, thermal conductivity, modulus of elasticity, thermal expansion coefficient, electric
resistivity, and even velocity and elevation.
Properties are considered to be either intensive or extensive.
Intensive properties are those that are independent of the mass of a system, such as
temperature, pressure, and density.
Extensive properties are those whose values depend on the size or extent of the system. Total
mass, total volume, and total momentum are some examples of extensive properties.
An easy way to determine whether a property is intensive or extensive is to divide the system
into two equal parts with an imaginary partition. Each part will have the same value of intensive
properties as the original system, but half the value of the extensive properties. Generally,
uppercase letters are used to denote extensive properties (with mass m being a major exception),
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and lowercase letters are used for intensive properties (with pressure P and temperature T being
the obvious exceptions). Extensive properties per unit mass are called specific properties. Some
examples of specific properties are specific volume (v= V/m) and specific total energy (e= E/m).
Process:
Any change that a system undergoes from one equilibrium state to another is called a process,
and the series of states through which a system passes during a process is called the path of the
process as shown in figure 1.4. To describe a process completely, one should specify the initial
and final states of the process, as well as the path it follows, and the interactions with the
surroundings.
A process is path followed by a system in reaching a given final state of equilibrium state
starting from a specified initial state. An actual process occurs only when the equilibrium state
does not exist. An ideal process can be defined in which the deviation from thermodynamic
equilibrium is infinitesimal. All the states the system passes through during a quasi-equilibrium
process may be considered equilibrium states. For non-equilibrium processes, we are limited to a
description of the system before the process occurs and after the equilibrium is restored. Several
processes are described by the fact that one property remains constant. The prefix iso- is used to
describe such processes. A process is said to be reversible if both the system and its surroundings
can be restored to their respective initial states by reversing the direction of the process.
 Reversible: if the process happens slow enough to be reversed.
 Irreversible: if the process cannot be reversed (like most processes).
 Isobaric: process done at constant pressure
 Isochoric: process done at constant volume
 Isothermal: process done at constant temperature
 Adiabatic: process where q=0
 Cyclic: process where initial state is equal to final state
Figure 1.4: Process
A system is said to have undergone a cycle if it returns to its initial state at the end of the process.
That is, for a cycle the initial and final states are identical.
Quasi-static Process:
When a process proceeds in such a manner that the system remains infinitesimally close to an
equilibrium state at all times, it is called a quasi-static, or quasi-equilibrium, process. A quasiequilibrium process can be viewed as a sufficiently slow process that allows the system to adjust
itself internally so that properties in one part of the system do not change any faster than those at
other parts.
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This is illustrated in figure: 1.5. When a gas in a piston-cylinder device is compressed suddenly,
the molecules near the face of the piston will not have enough time to escape and they will have
to pile up in a small region in front of the piston, thus creating a high-pressure region there.
Because of this pressure difference, the system can no longer be said to be in equilibrium, and
this makes the entire process non quasi-equilibrium. However, if the piston is moved slowly, the
molecules will have sufficient time to redistribute and there will not be a molecule pileup in front
of the piston. As a result, the pressure inside the cylinder will always be nearly uniform and will
rise at the same rate at all locations. Since equilibrium is maintained at all times, this is a quasiequilibrium process.
Figure 1.5: quasi static process
It should be pointed out that a quasi-equilibrium process is an idealized process and is not a true
representation of an actual process. But many actual processes closely approximate it, and they
can be modeled as quasi equilibrium with negligible error. Engineers are interested in quasi
equilibrium processes for two reasons. First, they are easy to analyze; second, work-producing
devices deliver the most work when they operate on quasi equilibrium processes. Therefore,
quasi-equilibrium processes serve as standards to which actual processes can be compared.
Note that the process path indicates a series of equilibrium states through which the system
passes during a process and has significance for quasi equilibrium processes only. For non quasiequilibrium processes, we are not able to characterize the entire system by a single state, and thus
we cannot speak of a process path for a system as a whole. A non quasi-equilibrium process is
denoted by a dashed line between the initial and final states instead of a solid line.
Concept of Temperature:
Although we are familiar with temperature as a measure of “hotness” or “coldness,” it is not easy
to give an exact definition for it. Based on our physiological sensations, we express the level of
temperature qualitatively with words like freezing cold, cold, warm, hot, and red-hot. However,
we cannot assign numerical values to temperatures based on our sensations alone. Furthermore,
our senses may be misleading. A metal chair, for example, will feel much colder than a wooden
one even when both are at the same temperature.
Fortunately, several properties of materials change with temperature in a repeatable and
predictable way, and this forms the basis for accurate temperature measurement. The commonly
used mercury-in-glass thermometer, for example, is based on the expansion of mercury with
temperature. Temperature is also measured by using several other temperature-dependent
properties.
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It is a common experience that a cup of hot coffee left on the table eventually cools off and a
cold drink eventually warms up. That is, when a body is brought into contact with another body
that is at a different temperature, heat is transferred from the body at higher temperature to the
one at lower temperature until both bodies attain the same temperature as shown in figure 1.6. At
that point, the heat transfer stops, and the two bodies are said to have reached thermal
equilibrium. The equality of temperature is the only requirement for thermal equilibrium.
Figure 1.6: Thermal Equilibrium
Zeroth Law of Thermodynamics:
We cannot assign numerical values to temperatures based on our sensations alone. Furthermore,
our senses may be misleading. Several properties of material changes with temperature in a
repeatable and predictable way, and this forms the basis of accurate temperature measurement.
The commonly used mercury-in-glass thermometer for example, is based on the expansion of
mercury with temperature. Temperature is also measured by using several other temperature
dependant properties. Two bodies (eg. Two copper blocks) in contact attain thermal equilibrium
when the heat transfer between them stops. The equality of temperature is the only requirement
for thermal equilibrium.
The Zeroth Law of Thermodynamics: If two bodies are in thermal equilibrium with a third body,
they are also in thermal equilibrium with each other. This obvious fact cannot be concluded from
the other laws of thermodynamics, and it serves as a basis of temperature measurement. By
replacing the third body with a thermometer, the zeroth law can be restated two bodies are in
thermal equilibrium if both have the same temperature reading even if they are not in contact.
Temperature Scales:
All temperature scales are based on some easily reproducible states such as the freezing and
boiling point of water, which are also called the ice-point and the steam-point respectively.
A mixture of ice and water that is in equilibrium with air saturated with water vapour at 1atm
pressure is said to be at the ice-point, and a mixture of liquid water and water vapour (with no
air) in equilibrium at 1atm is said to be at the steam-point.
Celsius and Fahrenheit scales are based on these two points (although the value assigned to these
two values is different) and are referred as two-point scales.
In thermodynamics, it is very desirable to have a temperature scale that is independent of the
properties of the substance or substances.
Such a temperature scale is called a thermodynamic temperature scale.(Kelvin in SI)
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Ideal gas temperature scale:
The temperatures on this scale are measured using a constant volume thermometer. Based on the
principle that at low pressure, the temperature of the gas is proportional to its pressure at constant
volume. The relationship between the temperature and pressure of the gas in the vessel can be
expressed as
T = a + b.P
Where the values of the constants a and b for a gas thermometer are determined experimentally.
Once a and b are known, the temperature of a medium can be calculated from the relation above
by immersing the rigid vessel of the gas thermometer into the medium and measuring the gas
pressure.
Ideal gas temperature scale can be developed by measuring the pressures of the gas in the vessel
at two reproducible points (such as the ice and steam points) and assigning suitable values to
temperatures those two points. Considering that only one straight line passes through two fixed
points on a plane, these two measurements are sufficient to determine the constants a and b in the
above equation. If the ice and the steam points are assigned the values 0 and 100 respectively,
then the gas temperature scale will be identical to the Celsius scale. In this case, the value of the
constant a (that corresponds to an absolute pressure of zero) is determined to be –273.150C when
extrapolated.
The equation reduces to T = bP, and thus we need to specify the temperature at only one point to
define an absolute gas temperature scale. Absolute gas temperature is identical to
thermodynamic temperature in the temperature range in which the gas thermometer can be used.
We can view that thermodynamic temperature scale at this point as an absolute gas temperature
scale that utilizes an ideal gas that always acts as a low-pressure gas regardless of the
temperature.
At the Tenth international conference on weights and measures in 1954, the Celsius scale has
been redefined in terms of a single fixed point and the absolute temperature scale. The triple
point occurs at a fixed temperature and pressure for a specified substance. The selected single
point is the triple point of water (the state in which all three phases of water coexist in
equilibrium), which is assigned the value 0.01 C. As before the boiling point of water at 1 atm.
Pressure is 100.0 C. Thus the new Celsius scale is essentially the same as the old one. On the
Kelvin scale, the size of Kelvin unit is defined as “the fraction of 1/273.16 of the thermodynamic
temperature of the triple point of water, which is assigned a value of 273.16K”. The ice point on
Celsius and Kelvin are respectively 0 and 273.15 K.
Constant volume gas thermometer:
The constant volume gas thermometer is the most accurate laboratory thermometer and is used
for the calibration of other thermometers. When we heat a gas keeping the volume constant, its
pressure increases and when we cool the gas its pressure decreases. The relationship between
pressure and temperature at constant volume is given by the law of pressure. According to this
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law, the pressure of a gas changes by (1/273) of its original pressure at 00C for each degree
centigrade (or Celsius) rise in temperature at constant volume.
If Po is the pressure of a given volume of a gas at 0oC and Pt is the pressure of the same volume
of the gas (i.e., at constant volume) at toC, then
--------------------eq(1)
Where
is constant and is known as coefficient of increase of pressure.
It consists of a glass bulb B connected to a tube A, through a capillary glass tube „C‟. The tube A
is connected to a mercury reservoir R which is clamped on the board and can be lowered or
raised whenever required to keep the volume of the air constant. The capillary tube C is provided
with a three way stopper S and can be used to connect capillary and bulb as well as to disconnect
tube from bulb B. A pointer is provided such that the end P is projecting inside from the upper
part of A. A scale calibrated in 0oC is provided between A and R. It is as shown in figure 1.7.
Figure 1.7: Constant Volume Gas Thermometer
The whole apparatus is leveled by adjusting the leveling screws. By adjusting the stopper, the
bulb „B‟ is filled with air or some gas and the pointer is adjusted so that tip of the pointer just
touches the level of mercury in the tube A. After filling the bulb, it is kept in an ice bath for
some time till the air inside the bulb attains the temperature of ice at which the mercury level
becomes stationary. Now the reservoir R is adjusted so that the level of mercury in the tube A
just touches the tip of the pointer P. The difference between the mercury levels in the two tubes
is noted and let it be ho. If Po is the pressure exerted by the air in the bulb, then
--------------------eq(2)
Now ice bath is removed and the bulb B is surrounded with steam. Again when the mercury level
in both the tubes becomes stationary, tube R is adjusted so that the mercury column in this
position in tube A touches the tip of the pointer.
Let the difference between the mercury columns levels in A and R be h100. The pressure of the
enclosed gas in B will be given by,
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--------------------eq(3)
--------------------eq(4)
Finally the bulb is brought in contact of the body whose temperature is to be found out. After the
mercury level becomes stationary, the tube R is adjusted so that the mercury level touches the tip
of the pointer again. Let ht be the difference between the heights of mercury levels and if Pt is
the pressure of the air inside the bulb B, then we have,
--------------------eq(5)
--------------------eq(6)
Dividing equation (6) by equation (4), we get
--------------------eq(7)
Substituting Pt and P100 in equation (7), we get,
--------------------eq(8)
Joule’s Experiments:
Let us take a closed system which permits work interaction and heat interaction both, as in case
of stirring in a container, as shown in figure 1.8. As a result of stirring it is seen that the
temperature of water gets raised up. This rise in temperature can be accounted by quantifying the
amount of heat supplied for raising this temperature. Thus, it is obvious that for any closed
system undergoing a cycle
i.e., the net heat interaction is proportional to the work interaction. Also the constant is known as
“Joule‟s mechanical equivalent of heat”. Joule‟s constant is described as;
Thus, J is a numerical conversion factor which could be unity if the heat is also given in joules.
For any cyclic process in the closed system the relationship between heat and work shall be, (if
the consistent units are used)
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Figure 1.8: Joule‟s Experiment
Energy in State and in Transition, Types:
Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential, electric,
magnetic, chemical, and nuclear, and their sum constitutes the total energy E of a system.
In thermodynamic analysis, it is often helpful to consider the various forms of energy that make
up the total energy of a system in two groups: macroscopic and microscopic.
The macroscopic forms of energy are those a system possesses as a whole with respect to some
outside reference frame, such as kinetic and potential energies. The microscopic forms of energy
are those related to the molecular structure of a system and the degree of the molecular activity,
and they are independent of outside reference frames. The sum of all the microscopic forms of
energy is called the internal energy of a system and is denoted by U.
The macroscopic energy of a system is related to motion and the influence of some external
effects such as gravity, magnetism, electricity, and surface tension. The energy that a system
possesses as a result of its motion relative to some reference frame is called kinetic energy (KE).
The energy that a system possesses as a result of its elevation in a gravitational field is called
potential energy (PE).
Internal Energy:
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The molecule as a whole can move in x, y and z directions with respective components
of velocities and hence possesses kinetic energy.
 There can be rotation of molecule about its center of mass and than the kinetic energy
associated with rotation is called rotational energy.
 In addition the bond length undergoes change and the energy associated with it is called
vibrational energy.
 The electron move around the nucleus and they possess a certain energy that is called
electron energy.
 The microscopic modes of energy are due to the internal structure of the matter and
hence sum of all microscopic modes of energy is called the internal energy.
Bulk kinetic energy (KE) and potential energy (PE) are considered separately and the other
energy of control mass as a single property (U).
The total energy possessed by the body is given by:
E = KE + PE + U
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Work:
Whenever a system interacts with its surroundings, it can exchange energy in two ways- work
and heat. In mechanics, work is defined as the product of the force and the displacement in the
direction of the force. Work done when a spring is compressed or extended: According to
Hooke's law
Spring force = - k (x – x0)
Where k is the spring constant, x0 is the equilibrium position, and x is the final position. The
negative sign shows that the direction of the spring force is opposite the direction of the
displacement from x0. The external force is equal in magnitude but opposite in sign to the spring
force, so
External force (force of your hands) = k (x –x0).
Now, we want to calculate the work done when we stretch the spring from position 1 to position
2.
W = F dx = k (x – x0) d(x-x0) = 1/2 k [(x2-x0)2 - (x1-x0)2]
Work done when a volume is increased or decreased
Consider a gas in a container with a movable piston on top. If the gas expands, the piston moves
out and work is done by the system on the surroundings. Alternatively, if the gas inside
contracts, the piston moves in and work is done by the surroundings on the system. Why would
the gas inside contract or expand? It would if the external pressure, Pex, and the internal
pressure, Pin, were different.
To calculate the work done in moving the piston, we know that the force = pressure times area
and then work equals pressure times area times distance or work equals pressure times the
change in volume.
So, W = the integral of (Pex) dV
The differential work done (dW) associated with a differential displacement (dl) is given by
dW = F dl
For a piston cylinder assembly,
dW = F dl = PA (dl) = P dV
If the gas is allowed to expand reversibly from the initial pressure P to final pressure P, then the
work done is given by
W = ∫ p dV
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The integral represents the area under the curve on a pressure versus volume diagram.
Therefore the work depends on the path followed and work is a path function and hence
not a property of the system.
The above expression does not represent work in the case of an irreversible process.
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The thermodynamic definition of work is “ Work is said to be done by a system on the
surrounding if the sole effect external to the system could be reduced to the raising of a
mass through a distance”.
Heat:
Heat like work, is a form of energy. The energy transfer between a system and its surroundings is
called heat if it occurs by virtue of the temperature difference across the boundary. The two
modes of energy transfer work and heat depend on the choice of the system. Heat energy moves
from a hotter body to a colder body upon contact of the two bodies. If two bodies at different
temperatures are allowed to remain in contact, the system of two bodies will eventually reach a
thermal equilibrium (they will have the same temperature). A body never contains heat. Rather
heat is a transient phenomenon and can be identified as it crosses the boundary.
Note that a quantity that is transferred to or from a system during an interaction is not a property
since the amount of such a quantity depends on more than just the state of the system. Heat and
work are energy transfer mechanisms between a system and its surroundings, and there are many
similarities between them:
1. Both are recognized at the boundaries of a system as they cross the boundaries. That is,
both heat and work are boundary phenomena.
2. Systems possess energy, but not heat or work.
3. Both are associated with a process, not a state. Unlike properties, heat or work has no
meaning at a state.
4. Both are path functions (i.e., their magnitudes depend on the path followed during a
process as well as the end states).
Path functions have inexact differentials designated by the symbol d. Therefore, a differential
amount of heat or work is represented by δQ or δW, respectively, instead of dQ or dW.
Properties, however, are point functions (i.e., they depend on the state only, and not on how a
system reaches that state), and they have exact differentials designated by the symbol d.
A small change in volume, for example, is represented by dV, and the total volume change
during a process between states 1 and 2 is
That is, the volume change during process 1–2 is always the volume at state 2 minus the volume
at state 1, regardless of the path followed. The total work done during process 1–2, however, is
That is, the total work is obtained by following the process path and adding the differential
amounts of work (δW) done along the way. The integral of δW is not W2 _ W1 (i.e., the work at
state 2 minus work at state 1), which is meaningless since work is not a property and systems do
not possess work at a state.
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Reversibility and Irreversibility:
Thermodynamic processes may have the change of state occurring in two ways. One is the
change of state occurring so that if the system is to restore its original state, it can be had by
reversing the factors responsible for occurrence of the process. Other change of state may occur
such that the above restoration of original state is not possible. Thermodynamic system that is
capable of restoring its original state by reversing the factors responsible for occurrence of the
process is called reversible system and the thermodynamic process involved is called reversible
process. Thus, upon reversal of a process there shall be no trace of the process being occurred,
i.e. state changes during the forward direction of occurrence of a process are exactly similar to
the states passed through by the system during the reversed direction of the process. It is quite
obvious that the reversibility can be realized only if the system maintains its thermodynamic
equilibrium throughout the occurrence of process.
The irreversibility is the characteristics of the system which forbids system from retracing the
same path upon reversal of the factors causing the state change. Thus, irreversible systems are
those which do not maintain equilibrium during the occurrence of a process. Various factors
responsible for the nonattainment of equilibrium are generally the reasons responsible for
irreversibility. Presence of friction, dissipative effects etc. have been identified as a few of the
prominent reasons for irreversibility. The reversible and irreversible processes are shown on p-v
diagram in figure 1.9. by „1–2 and 2–1‟ and „3–4 and 4–3‟ respectively.
Figure 1.9: Reversible and Irreversible Process
Causes for irreversibility:
Generic types of irreversibilities are due to;
(i) Friction,
(ii) Electrical resistance,
(iii) Inelastic solid deformations,
(iv) Free expansion
(v) Heat transfer through a finite temperature difference,
(vi) Non equilibrium during the process, etc.
(i) Friction: Friction is invariably present in real systems. It causes irreversibility in the process
as work done does not show equivalent rise in kinetic or potential energy of the system. Fraction
of energy wasted due to frictional effects leads to deviation from reversible states.
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(ii) Electrical resistance: Electrical resistance in the system also leads to presence of dissipation
effects and thus irreversibilities. Due to electric resistance dissipation of electrical work into
internal energy or heat takes place. The reverse transformation from heat or internal energy to
electrical work is not possible, therefore leads to irreversibility.
(iii) Inelastic solid deformation: Deformation of solids, when of inelastic type is also
irreversible and thus causes irreversibility in the process. If deformation occurs within elastic
limits then it does not lead to irreversibility as it is of reversible type.
(iv) Free expansion: Free expansion as discussed earlier in chapter 3, refers to the expansion of
unresisted type such as expansion in vacuum. During this unresisted expansion the work
interaction is zero and without expense of any work it is not possible to restore initial states.
Thus, free expansion is irreversible.
(v) Heat transfer through a finite temperature difference: Heat transfer occurs only when there
exist temperature difference between bodies undergoing heat transfer. During heat transfer if
heat addition is carried out in finite number of steps then after every step the new state shall be a
non-equilibrium state. In order to have equilibrium states in between, the heat transfer process
may be carried out in infinite number of steps. Thus, infinitesimal heat transfer every time causes
infinitesimal temperature variation. These infinitesimal state changes shall require infinite time
and process shall be of quasi-static type, therefore reversible. Heat transfer through a finite
temperature difference which practically occurs is accompanied by irreversible state changes and
thus makes processes irreversible.
(vi)Non equilibrium during the process: Irreversibilities are introduced due to lack of
thermodynamic equilibrium during the process. Non equilibrium may be due to mechanical in
equilibrium, chemical in equilibrium, thermal in equilibrium, electrical in equilibrium etc. and
irreversibility are called mechanical irreversibility, chemical irreversibility, thermal
irreversibility, electrical irreversibility respectively. Factors discussed above are also causing non
equilibrium during the process and therefore make process irreversible.
First Law of Thermodynamics:
The first law of thermodynamics, also known as the conservation of energy principle, provides a
sound basis for studying the relationships among the various forms of energy and energy
interactions. Based on experimental observations, the first law of thermodynamics states that
energy can be neither created nor destroyed during a process; it can only change forms.
Therefore, every bit of energy should be accounted for during a process.
Thus first law of thermodynamics states that “in a closed system undergoing a cyclic process, the
net work done is proportional to the net heat taken from the surroundings” or “for any cycle of a
closed system the net heat transfer equals the net work”.
For non-cyclic process: Let us now take up a system undergoing a non-cyclic process where
transfer of heat and work take place and there is some change in the state of system i.e., initial
and final states are different.
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If we assume the system to have the heat interaction ΔQ and work interaction ΔW, then from the
basic principles it can be said that: Energy lost = Energy gained; as the energy can neither be
created nor destroyed.
Therefore, between states 1–2 one can write energy balance as,
Q1–2 – W1–2 = U1–2
Where Q1–2, W1–2 and U1–2 are the heat, work and stored energy values. This stored energy is
called as internal energy for a system having negligible electrical, magnetic, solid distortion and
surface tension effects.
For elemental interactions, dQ = dU + dW
Thus, the first law of thermodynamics for non-cyclic processes can be given by
∫ dQ = ∫ dU + ∫ dW
The first law of thermodynamics for cyclic process can be employed,
Corollaries:
First law of thermodynamics based on law of energy conservation has proved to be a powerful
tool for thermodynamic analysis. But over the period of time when it was applied to some real
systems, it was observed that theoretically first law stands valid for the processes which are not
realizable practically. It was then thought that there exist certain flaws in first law of
thermodynamics and it should be used with certain limitations. Say for example let us take a
bicycle wheel and paddle it to rotate. Now apply brake to it. As a result of braking wheel comes
to rest upon coming in contact with brake shoe. Stopping of wheel is accompanied by heating of
brake shoe.
Examining the situation from first law of thermodynamics point of view it is quite satisfying that
rotational energy in wheel has been transformed into heat energy with shoe, thus causing rise in
its temperature: Now, if we wish to introduce the same quantity of heat into brake shoe and wish
to restore wheel motion then it is not possible simply, whereas theoretically first law permits the
conversion from heat to work (rotation of wheel in this case) as well.
Therefore, it is obvious that first law of thermodynamics has certain limitations as given below:
(i) First law of thermodynamics does not differentiate between heat and work and assures full
convertibility of one into other whereas full conversion of work into heat is possible but the viceversa is not possible.
(ii) First law of thermodynamics does not explain the direction of a process. Such as theoretically
it shall permit even heat transfer from low temperature body to high temperature body which is
not practically feasible. Spontaneity of the process is not taken care of by the first law of
thermodynamics.
Perpetual motion machine of the first kind (PMM-I) is a hypothetical device conceived, based on
violation of First law of thermodynamics. Let us think of a system which can create energy as
shown below.
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Figure 1.10: PMM-I, based on violation of First law of thermodynamics
Here a device which is continuously producing work without any other form of energy supplied
to it has been shown in figure 1.10 (a), which is not feasible. Similarly a device which is
continuously emitting heat without any other form of energy supplied to it has been shown in (b),
which is again not feasible. Above two imaginary machines are called Perpetual Motion
Machines of 1st kind.
First law applied to a Process and Flow System:
Let us consider an open system as shown in figure 1.11having inlet at section 1–1 and outlet at
section 2–2. The cross-section area, pressure, specific volume, mass flow rate, energy at section
1–1 and 2–2 are
Section 1–1 = A1, p1, v1, m1, e1
Section 2–2 = A2, p2, v2, m2, e2
Open system is also having heat and work interactions Q, W as shown in figure above.
Applying the energy balance at the two sections, it can be given as,
Energy added to the system + Stored energy of the fluid at inlet = Stored energy of the fluid at
outlet
Figure 1.11: I Law Applied to Open System
Steady Flow Energy Equation:
Steady flow refers to the flow in which its properties at any point remain constant with respect to
time. Steady system is the system whose properties are independent of time, i.e. any property at a
point in system shall not change with time.
Let us take an open system having steady flow. Figure 1.12 shows steady flow system having
inlet at section 1–1, outlet at section 2–2, heat addition Q and work done by the system W.
Figure 1.11: Steady Flow System
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UNIT II
Limitations of the First Law
The first law of thermodynamics is a law of conservation of energy. It does not specify the direction of
the process. All spontaneous processes processed in one direction only. The first law of thermodynamics
does not deny the feasibility of a process reversing itself. The first law of thermodynamics does not
provide answers to the following questions.
Is a particular process / reaction feasible?
To what extent do the process / reaction proceed?
Is complete conversion of internal energy into work possible?
There exists a law which determines the direction in which a spontaneous process proceeds. The law,
known as the second law of thermodynamics, is a principle of wide generality and provides answer to
the above questions.
Thermal reservoir
It is a large body from which a finite quantity of energy can be extracted or to which a finite quantity of
energy can be added as heat without changing its temperature. A source is a thermal reservoir at high
temperature from which a heat engine receives the energy as heat. A sink is a low temperature thermal
reservoir to which a heat engine rejects energy as heat.
Heat Engine
A heat engine is a device which converts the energy it receives at heat, into work. It is a cyclically
operating device. It receives energy as heat form a high temperature body, converts part of it into work
and rejects the rest to a low temperature body. A thermal power plant is an example of a heat engine.
Figure 2.1: Heat Engine
In the boiler, the working fluid receives a certain amount of heat (Q1) from the hot combustion
products. The superheated steam enters a turbine where it undergoes expansion performing the shaft
work (WT). The low pressure steam enters a condenser where it exchange energy as heat at constant
pressure with the cooling water and emerges as the condensate. The condensate rejects a certain
amount of heat (Q2) to the cooling water. The low pressure condensate from the condenser enters the
pump. Work (WP) is done on the pump to elevate the condensate to the boiler pressure and return it to
the boiler.
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Heat pump
Heat Pump is cyclically operating device which absorbs energy form a low temperature reservoir and
reject energy as heat to a high temperature reservoir when work is performed on the device. Its
objective is to reject energy as heat to a high temperature body (space heating in winter). The
atmosphere acts as the low temperature reservoir.
Refrigerator
A refrigerator is a cyclically operating device which absorbs energy as heat from a low temperature body
and rejects energy as heat to a high temperature body when work is performed on the device. The
objective of this device is to refrigerate a body at low temperature. Usually it uses atmosphere as the
high temperature reservoir.
Figure 2.2 : Heat pump/Refrigerator
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Parameters of performance
Let QH and QL represents the amount of energy absorbed as heat from the low temperature reservoir
and the energy rejected as heat to the high temperature reservoir respectively, Let W be the work done
on the device to accomplish the task.
QH and QL =W
Second Law of Thermodynamics: Kelvin-Planck and Clausius Statements
KELVIN PLANCK STATEMENT: It is impossible to construct a cyclically operating device such that it
produces no other effect than the absorption of energy as heat from a single thermal reservoir and
performs an equivalent amount of work.
Figure 2.3: Kelvin Planck Statement
CLAUSIUS STATEMENT: Heat always flows from a body at higher temperature to a body at a lower
temperature. The reverse process never occurs spontaneously. Clausius statement of the second law
gives: It is impossible to construct a device which, operating in a cycle, will produce no effect other than
the transfer of heat from a low-temperature body to a high temperature body.
Figure 2.4: Clauisus Statement
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Equivalence / Corollaries
Let us suppose that the Kelven Planck statement is incorrect.
Figure 2.5: Equivalence of Kelvin Planck and Clausius Statement
It is possible to construct a device I which, working cyclically, absorbs energy a heat from a source at
temperature and performs an equivalent amount of work. Next consider a device II which absorbs
amount of energy from a low temperature body at and delivers energy as heat to a high temperature
reservoir at. To accomplish this, work is done on the device. The device II does not violate the Clausius
statement. For device II, we can write. Now combine I and II. The work delivered by device I is used by
device, II.
This combined device (which is no more aided by any external agency) working cyclically, is not
producing any effect other than the transfer of energy as heat from the low temperature reservoir to
the high temperature reservoir. This is in violation of the Clausius statement.
To prove that violation of the Clausius statement leads to violation of Kelvin Planck statement, let us
assume that the Clausius statement is incorrect. That is, it is possible to constructs a device I such that it
transfers energy as heat from a body at lower temperature to a body at higher temperature unaided by
any external agency. Consider another device II which receives energy as heat from a body at higher
temperature, delivers work and rejects energy as heat to the body at a low temperature. Device II does
not violent Kelvin Planck statement.
Figure 2.6: Equivalence of Clausius Kelvin and Planck Statement
PMM of Second kind
A perpetual motion machine of the second kind, or PMM2 is one which converts all the heat input into
work while working in a cycle.
Carnot’s principle:
The most efficient heat engine cycle is the Carnot cycle, consisting of two isothermal processes and two
adiabatic processes. The Carnot cycle can be thought of as the most efficient heat engine cycle allowed
by physical laws. When the second law of thermodynamics states that not all the supplied heat in a heat
engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat
which can be so used.
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In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be
reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no
real engine processes are reversible and all real physical processes involve some increase in entropy.
Carnot cycle and its specialties
The Carnot cycle when acting as a heat engine consists of the following steps:
1. Reversible isothermal expansion of the gas at the "hot" temperature, T1 (isothermal heat addition or
absorption). During this step the gas is allowed to expand and it does work on the surroundings. The
temperature of the gas does not change during the process, and thus the expansion is isothermal. The
gas expansion is propelled by absorption of heat energy Q1 and of entropy ΔS= Q1/T1 from the high
temperature reservoir.
2. Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step the
mechanisms of the engine are assumed to be thermally insulated, thus they neither gain nor lose heat.
The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of
internal energy. The gas expansion causes it to cool to the "cold" temperature, T2. The entropy remains
unchanged.
3. Reversible isothermal compression of the gas at the "cold" temperature, T2. (isothermal heat
rejection). Now the surroundings do work on the gas, causing an amount of heat energy Q2 and of
entropy ΔS= Q2/T2 to flow out of the gas to the low temperature reservoir. (This is the same amount of
entropy absorbed in step 1, as can be seen from the Clausius inequality.)
4. Isentropic compression of the gas (isentropic work input). Once again the mechanisms of the engine
are assumed to be thermally insulated. During this step, the surroundings do work on the gas, increasing
its internal energy and compressing it, causing the temperature to rise to T1. The entropy remains
unchanged. At this point the gas is in the same state as at the start of step 1.
Figure 2.7: Carnot Cycle PV diagram
Figure 2.8: Carnot Cycle TS diagram
Thermodynamic scale of Temperature
A temperature scale, which does not depend on the thermodynamic property of the substance can be
established by making use of the fact that the efficiency of a reversible heat engine does not depend on
the nature of the nature of the working fluid but depends only upon the temperature of the reservoirs
between which it operates. The establishment of thermodynamic temperature scale is also a
consequence of the second law of thermodynamics. It can be understood by the following figure.
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Figure 2.9: Thermodynamic scale of Temperature
Clausius Inequality
Consider a system undergoing a reversible cycle. The given cycle may be sub-divided by drawing a family
of reversible adiabatic lines. Every two adjacent adiabatic lines may be joined by two reversible
isotherms.
Figure 2.10: Clausius Inequality
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The equality holds good for a reversible cycle and the inequality holds good for an irreversible cycles.
The complete expression is known as Clausius Inequality.
Entropy
Entropy is a measure of the "disorder" of a system. The disorder refers to is really the number of
different microscopic states a system can be in, given that the system has a particular fixed composition,
volume, energy, pressure, and temperature.
Clausius inequality can be used to analyze the cyclic process in a quantitative manner. The second law
became a law of wider applicability when Clausius introduced the property called entropy. By evaluating
the entropy change, one can explain as to why spontaneous processes occur only in one direction.
Consider that, it is possible to connect the states 1 and 2 by several reversible paths and see that
has the same value irrespective of the path as long as the paths are reversible. Therefore
an exact differential of some function which we identify as entropy. Hence it can be said that there
exists a function S, called entropy, the change in entropy is expressed as
is
The following facts should be kept in mind while calculating the change in entropy for a process
1.
For a reversible process
2. Entropy is a state function.
3.
The entropy change of a system is determined by its initial and final states only, irrespective of
how the system has changed its state. In analyzing irreversible processes, it is not necessary to
make a direct analysis of the actual process. One can substitute the actual process by a
reversible process connecting the final state to the initial state, and the entropy change for the
imaginary reversible process can be evaluated.
Principle of Entropy Increase
Now let us apply the above results to evaluate the entropy of the universe when a system interacts with
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the surroundings.
Figure 2.11: Principle of Entropy Increase
Finally entropy increase means
 The energy of the universe in constant (first law).
 The entropy of the universe tends towards a maximum (second law)
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Energy Equation, Availability and Irreversibility
Available Energy: In order to determine that part of the energy which can be converted into work by an
engine, we require consider a thermal reservoir at constant temperature T from which a quantity of
energy Q is being absorbed as heat.
Consider the Carnot cycle as shown in a T-S diagram below:
Figure 2.12: Carnot TS Diagram
The area 1234 represents the portion of energy delivered as work by the reversible engine and is called
available energy. The area 43BA represents the portion of the energy which is rejected to the
atmosphere. The portion of energy is not available for conversion into work is called unavailable energy.
Therefore, the available energy represents the portion of the energy supplied as heat which can be
converted into work by means of a reversible engine. It is also known as EXERGY.
The availability of a given system is defined as the maximum useful work that can be obtained in a
process in which the system comes to equilibrium with the surroundings or attains a dead state. Clearly,
the availability of a system depends on the condition of the system as well as those of the surroundings.
Thermodynamic Potentials
Eight properties of a system, namely pressure (p), volume (v), temperature (T), internal energy (u),
enthalpy (h), entropy (s), Helmholtz function (f) and Gibbs function (g). h, f and g are sometimes referred
to as thermodynamic potentials.
g = h – T.s is known as Gibb’s function or free energy function
f = u – Ts is known as Helmholtz function. This gives maximum possible output when the heat Q is
transferred at constant temperature and is the case with a very large source.
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Gibbs and Helmholtz Functions
The work done in a non-flow reversible system (per unit mass) is given by :
W = Q – (u0 – u1)
= T.ds – (u0 – u1)
= T (s0 – s1) – (u0 – u1)
i.e., W = (u1 – Ts1) – (u0 – Ts0)
The term (u – Ts) is known as Helmholtz function. This gives maximum possible output when the heat Q
is transferred at constant temperature and is the case with a very large source.
If work against atmosphere is equal to p0 (v0 – v1), then the maximum work available,
Wmax = W – work against atmosphere
= W – p0 (v0 – v1)
= (u1 – Ts1) – (u0 – Ts0) – p0 (v0 – v1)
= (u1 + p0v1 – Ts1) – (u0 + p0v0 – Ts0)
= (h1 – Ts1) – (h0 – Ts0)
i.e., Wmax = g1 – g0
where g = h – T.s is known as Gibb’s function or free energy function.
The maximum possible available work when system changes from 1 to 2 is given by
Wmax = (g1 – g0) – (g2 – g0) = g1 – g2
Maxwell Relations
If dx is called an exact differential then
dx = Mdy + Ndz
where
M=(∂x/∂y)z
and
N=(∂x/∂z)y
We know from the first law of thermodynamics,
dQ = du + pdv
Tds = du + pdv
du = Tds – pdv
….eq(1)
We know enthalpy as h=u+pv,
dh = du + pdv + vdp
dh = Tds + vdp
….eq(2)
Helmholtz free energy function f=u-Ts,
df = du – Tds – sdT
df = – pdv – sdT
….eq(3)
Gibb’s free energy function g= h-Ts,
dg = dh – Tds – sdT
dg = vdp – sdT
….eq(4)
Therefore the Maxwell’s equations can be rewritten as follows when above equations are considered as
exact differentials.
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Elementary Treatment of the Third Law of Thermodynamics
‘Third law of thermodynamics’, an independent principle uncovered by ‘Nernst’ and formulated by
‘Planck’, states that the “Entropy of a pure substance approaches zero at absolute zero temperature.”
This fact can also be corroborated by the definition of entropy which says it is a measure of molecular
disorderness. At absolute zero temperature substance molecules get frozen and do not have any
activity, therefore it may be assigned zero entropy value at crystalline state. Although the attainment of
absolute zero temperature is impossible practically, however theoretically it can be used for defining
absolute entropy value with respect to zero entropy at absolute zero temperature. Second law of
thermodynamics also shows that absolute zero temperature can’t be achieved. Third law of
thermodynamics is of high theoretical significance for the sake of absolute property definitions and has
found great utility in thermodynamics.
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UNIT – III
Perfect Gas Laws
An equation of state of a particular substance is an empirical result, and it cannot be deduced from the
laws of thermodynamics. Nevertheless the general form of the equation may be predicted from
hypotheses about the microscopic structure of matter. This type of prediction has been developed to a
high degree of precision for gases, and to a lesser extent for liquids and solids. The simplest postulates
about the molecular structure of gases lead to the concept of the perfect gas which has the equation of
state pv = RT. Experiments have shown that the behaviour of real gases at low pressure with high
temperature agrees well with this equation.
Boyle’s law: It states that volume of a given mass of a perfect gas varies inversely as the absolute
pressure when temperature is constant.
Charle’s law: It states that if any gas is heated at constant pressure, its volume changes directly as its
absolute temperature.
Equation of State
Let us imagine a series of experiments in which the volume of a substance is measured over a range of
temperatures while the pressure is maintained constant, this being repeated for various pressures. The
results might be represented graphically by a three-dimensional surface, or by a family of constant
pressure lines on a v-T diagram. It is useful if an equation can be found to express the relation between
p, v and T, and this can always be done over a limited range of states. No single equation will hold for all
phases of a substance, and usually more than one equation is required even in one phase if the accuracy
of the equation is to match that of the experimental results. Equations relating p, v and T are called
equations of state or characteristic equations.
Specific and Universal Gas constants
Avogadro discovered that V0 is the same for all gases at the same pressure and temperature and
therefore it may be seen that
MR = a constant ;
R0 and thus
pV0 = R0T
R0 is called the molar or universal gas constant and its value is 8.3143 kJ/kg mol K.
Various Non-flow processes, properties, end states, Heat and Work Transfer
A process occurs when the system undergoes a change in a state or an energy transfer at a steady state.
A process may be non-flow in which a fixed mass within the defined boundary is undergoing a change of
state. Example : A substance which is being heated in a closed cylinder undergoes a non-flow process.
Closed systems undergo non-flow processes. A process may be a flow process in which mass is entering
and leaving through the boundary of an open system. In a steady flow process mass is crossing the
boundary from surroundings at entry, and an equal mass is crossing the boundary at the exit so that the
total mass of the system remains constant. In an open system it is necessary to take account of the work
delivered from the surroundings to the system at entry to cause the mass to enter, and also of the work
delivered from the system at surroundings to cause the mass to leave, as well as any heat or work
crossing the boundary of the system.
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Throttling and Free Expansion Processes
The throttling process is as shown in figure below. The velocity increases at the restriction, with a
consequent decrease of enthalpy, but this increase of kinetic energy is dissipated by friction, as the
eddies die down after restriction. The steady-flow energy equation implies that the enthalpy of the fluid
is restored to its initial value if the flow is adiabatic and if the velocity before restriction is equal to that
downstream of it.
Figure 3.1: Throttling Process
Consider two vessels 1 and 2 interconnected by a short pipe with a valve A, and perfectly thermally
insulated as shown in figure. Initially let the vessel 1 be filled with a fluid at a certain pressure, and let 2
be completely evacuated. When the valve A is opened the fluid in 1 will expand rapidly to fill both
vessels 1 and 2. The pressure finally will be lower than the initial pressure in vessel 1. This is known as
free or unresisted expansion. The process is highly irreversible; since the fluid is eddying continuously
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during the process.
Figure 3.2: Free Expansion Process
Flow processes
A process may be a flow process in which mass is entering and leaving through the boundary of an open
system. In a steady flow process mass is crossing the boundary from surroundings at entry, and an equal
mass is crossing the boundary at the exit so that the total mass of the system remains constant. In an
open system it is necessary to take account of the work delivered from the surroundings to the system
at entry to cause the mass to enter, and also of the work delivered from the system at surroundings to
cause the mass to leave, as well as any heat or work crossing the boundary of the system.
Deviations from perfect Gas Model
The behavior of real gases usually agrees with the predictions of the ideal gas equation to within 5% at
normal temperatures and pressures. At low temperatures or high pressures, real gases deviate
significantly from ideal gas behavior. In 1873, while searching for a way to link the behavior of liquids
and gases, the Dutch physicist Johannes vander Waals developed an explanation for these deviations
and an equation that was able to fit the behavior of real gases over a much wider range of pressures.
Vander Waals realized that two of the assumptions of the kinetic molecular theory were questionable.
The kinetic theory assumes that gas particles occupy a negligible fraction of the total volume of the gas.
It also assumes that the force of attraction between gas molecules is zero.
The first assumption works at pressures close to 1 atm. But something happens to the validity of this
assumption as the gas is compressed. Imagine for the moment that the atoms or molecules in a gas
were all clustered in one corner of a cylinder, as shown in the figure below. At normal pressures, the
volume occupied by these particles is a negligibly small fraction of the total volume of the gas. But at
high pressures, this is no longer true. As a result, real gases are not as compressible at high pressures as
an ideal gas. The volume of a real gas is therefore larger than expected from the ideal gas equation at
high pressures.
The deviation from ideal behavior is large at high pressure.
The deviation varies from gas to gas.
At lower pressures (<10 atm) the deviation from ideal behavior is typically small, and the ideal gas law
can be used to predict behavior with little error.
Deviation from ideal behavior is also a function of temperature: As temperature increases the deviation
from ideal behavior decreases; as temperature decreases the deviation increases, with a maximum
deviation near the temperature at which the gas becomes a liquid.
Two of the characteristics of ideal gases included: The gas molecules themselves occupy no appreciable
volume; the gas molecules have no attraction or repulsion for each other.
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Vader Waals Equation of State
Vander Waals’ equation (for a real gas) may be written as:
(p+(a/v2))(v-b) = RT
The constants a and b are specific constants and depend upon the type of the fluid considered,
‘v’ represents the volume per unit mass and R is the gas constant.
where units of
R = 8314 Nm/kg mol K,
a = Nm4/(kg-mol)2
b =m3/kg.mol
Compressibility charts
Let us plot to show the variation of volume with pressure for various constant values of temperature.
In this case, the gradient of a curve at any state is (∂v/∂p)T .
When this gradient is divided by the volume at that state, we have a property known as the
compressibility K of the substance. Since this gradient is always negative, i.e., the volume of a substance
always decreases with increase of pressure when the temperature is constant, the compressibility is
usually made a positive quantity by defining it as
K= (-1/v)*( (∂v/∂p)T).
K can be regarded as a constant for many purposes for solids and liquids.
The following figure shows the compressibility.
Figure 3.3: Compressibility
The compressibility factor (Z) of any gas is a function of only two properties, usually temperature and
pressure, so that Z = f(Tr, pr) except near the critical point. The value of Z for any real gas may be less or
more than unity, depending on pressure and temperature conditions of the gas. The general
compressibility chart is plotted with Z versus pr for various values of Tr. This is constructed by plotting
the known data of one or more gases and can be used for any gas.
Figure 3.4: Compressibility charts
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Variable specific Heats
They are usually expressed in a form:
cp = a + KT + K1T2 + K2 T3
cv = b + KT + K1T2 + K2T3
where a, b, K, K1 and K2 are constants.
Phase Transformations: Triple point at critical state properties during change of phase
If the vapour pressure of a solid is measured at various temperatures until the triple point is reached
and then that of the liquid is measured until the critical point is reached, the result when plotted on a pT diagram appears as shown in figure below.
Figure 3.5: p-T diagram for a substance such as water
If the substance at the triple point is compressed until there is no vapour left and the pressure on the
resulting mixture of liquid and solid is increased, the temperature will have to be changed for
equilibrium to exist between the solid and the liquid.
Measurements of these pressures and temperatures give rise to a third curve on the p-T diagram,
starting at the triple point and continuing indefinitely.
The points representing the coexistence of
(i) solid and vapour lie on the sublimation curve,
(ii) liquid and vapour lie on the vapourisation curve,
(iii) liquid and solid lie on the fusion curve.
In the particular case of water, the sublimation curve is called the frost line, the vapourisation curve is
called the steam line, and the fusion curve is called the ice line. The slopes of sublimation and the
vapourisation curves for all substances are positive. The slope of the fusion curve, however may be
positive or negative. The fusion curve of most substances have a positive slope. Water is one of the
important exceptions.
Triple point: The triple point is merely the point of intersection of sublimation and vapourisation curves.
It must be understood that only on p-T diagram is the triple point represented by a point.
Dryness Fraction
The term dryness fraction is related with wet steam. It is defined as the ratio of the mass of actual dry
steam to the mass of steam containing it. It is usually expressed by the symbol ‘x’ or ‘q’.
If
ms = Mass of dry steam contained in steam considered, and
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mw = Weight of water particles in suspension in the steam considered,
Then,
x= ms/(ms+mw)
Thus if in 1 kg of wet steam 0.9 kg is the dry steam and 0.1 kg water particles then x = 0.9.
Clausius – Clapeyron Equation
Clausius-Claperyon equation is a relationship between the saturation pressure, temperature, the
enthalpy of evaporation, and the specific volume of the two phases involved. This equation provides a
basis for calculations of properties in a two-phase region. It gives the slope of a curve separating the two
phases in the p-T diagram.
Let us consider the change of state from saturated liquid to saturated vapour of a pure substance which
takes place at constant temperature. During the evaporation, the pressure and temperature are
independent of volume.
This is known as Clausius-Claperyon or Claperyon equation for evaporation of liquids.
Property tables/Gas Tables/Steam tables
In engineering problem, for any fluid which is used as working fluid, the six basic thermodynamic
properties required are : p (pressure), T (temperature), v (volume), u (internal energy), h (enthalpy) and
s (entropy). These properties must be known at different pressure for analysing the thermodynamic
cycles used for work producing devices. The values of these properties are determined theoretically or
experimentally and are tabulated in the form of tables which are known as ‘Steam Tables’. The
properties of wet steam are then computed from such tabulated data. Tabulated values are also
available for superheated steam. It may be noted that steam has only one saturation temperature at
each pressure.
Following are the thermodynamic properties of steam which are tabulated in the form of table:
p = Absolute pressure (bar or kPa) ;
ts = Saturation temperature (°C) ;
hf = Enthalpy of saturated liquid (kJ/kg) ;
hfg = Enthalpy or latent heat of vapourisation (kJ/kg) ;
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hg = Enthalpy of saturated vapour (steam) (kJ/kg) ;
sf = Entropy of saturated liquid (kJ/kg K) ;
sfg = Entropy of vapourisation (kJ/kg K) ;
sg = Entropy of saturated vapour (steam) (kJ/kg K) ;
vf = Specific volume of saturated liquid (m3/kg) ;
vg = Specific volume of saturated vapour (steam) (m3/kg).
Also, hfg = hg – hf ...... Change of enthalpy during evaporation
sfg = sg – sf ...... Change of entropy during evaporation
vfg = vg – vf ...... Change of volume during evaporation.
The above mentioned properties at different pressures are tabulated in the form of tables as under :
The internal energy of steam (u = h – pv) is also tabulated in some steam tables.
Mollier charts
Dr. Mollier, in 1904, conceived the idea of plotting total heat against entropy, and his diagram is more
widely used than any other entropy diagram, since the work done on vapour cycles can be scaled from
this diagram directly as a length; whereas on T-s diagram it is represented by an area.
Figure 3.6: Mollier Charts
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Various Thermodynamic processes and energy Transfer
Steam Calorimetry
Steam calorimeter, in which the mass of steam condensed on a body, is used to calculate the
amount of heat supplied.
The dryness fraction of steam can be measured by using the following calorimeters:
1. Throttling calorimeter
2. Separating and throttling calorimeter.
The dryness fraction of wet steam can be determined by using a throttling calorimeter which is
illustrated diagrammatically in figure below:
The steam to be sampled is taken from the pipe by means of suitable positioned and dimensioned
sampling tube. It passes into an insulated container and is throttled through an orifice to
atmospheric pressure. Here the temperature is taken and the steam ideally should have about 5.5
K of superheat.
The throttling process is shown on h-s diagram in shown below by the line 1-2. If steam initially
wet is throttled through a sufficiently large pressure drop, then the steam at state 2 will become
superheated. State 2 can then be defined by the measured pressure and temperature. The
enthalpy, h2 can then be found and hence
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Figure 3.7: Throttling calorimeter
Figure 3.8: Throttling process
Hence the dryness fraction is determined and state 1 is defined.
Separating and Throttling Calorimeter:
If the steam whose dryness fraction is to be determined is very wet then throttling to atmospheric
pressure may not be sufficient to ensure superheated steam at exit. In this case it is necessary to
dry the steam partially, before throttling. This is done by passing the steam sample from the main
through a separating calorimeter as shown in figure below. The steam is made to change
direction suddenly, and the water, being denser than the dry steam is separated out. The quantity
of water which is separated out (mw) is measured at the separator, the steam remaining, which
now has a higher dryness fraction, is passed through the throttling calorimeter. With the
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combined separating and throttling calorimeter it is necessary to condense the steam after
throttling and measure the amount of condensate (ms). If a throttling calorimeter only is
sufficient, there is no need to measure condensate, the pressure and temperature measurements at
exit being sufficient.
Figure 3.9: Separating and throttling calorimeter
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