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Chapter 3
THE FIRST LAW
APPLIED TO
CLOSED SYSTEMS
One only ever understands what one tames. People no longer have
the time to understand anything.
−AntoineDeSaintExup´ery (The Little Prince)
Chapter 1 stated that the ﬁrst law of thermodynamics is simply the
law of conservation of energy, and Chapter 2 introduced us to the ther
modynamic meaning of terms such as system, property, and others. This
chapter demonstrates how to apply the ﬁrst law of thermodynamics to
closed systems.
22 Chapter 3
3.1 Energy Balance for a Closed System
The ﬁrst law of thermodynamics states that energy is conserved, which
means energy cannot be created or destroyed, but it may change its form.
Applying the ﬁrst law to a closed system involves accounting for energy,
that is simply writing an energy balance over the closed system as follows:
change in total energy content of the system =
energy that entered the system
−energy that left the system.
Let us choose the closed system comprising a gas trapped within the
pistoncylinder device shown in Figure 3.1. The dashed line of Figure 3.1
shows the crosssection of the boundary that separates the system from
its surroundings, which includes the piston, the walls of the cylinder, and
everything outside the piston and the cylinder. One may ignore the parts
of the surroundings that have no inﬂuence on the system, or are not inﬂu
enced by the system.
Figure 3.1 A closed system.
cylinder
boundary
gas

piston
Initially, let us say that the closed system, that is the gas, contains
an amount of energy Eo. The gas is compressed by pushing the piston,
and Win amount of energy is provided to the gas as work. The walls of
the cylinder allow heat to pass through, and Qout amount of energy is lost
to the surroundings. Because of these work and heat energy interactions
The First Law applied to Closed Systems 23
between the system and its surroundings, the energy content of the system
changes to a new value, say Ef. Energy balance for the closed system then
gives the following:
Ef−Eo=Win −Qout
Using Qout =−Qin, the energy balance could be rewritten as
Ef−Eo=Win +Qin
which could be rearranged to give
Qin +Win =ΔE(3.1)
where the notation ΔEis used to represent the change in the total energy
content of the system (Ef−Eo). Note that the value of ΔEdepends on
the initial and ﬁnal states alone, and E, the energy content of the system,
is a property of the system.
Qin +Win =ΔE
is the First Law of Thermodynamics
applicable to closed system.
``euu
The total energy content of the system, denoted by E,ismadeup
of internal energy, gravitational potential energy, kinetic energy, electrical
energy, magnetic energy, surface energy, etc. Chapter 4 attempts to give
an insight into the internal energy, denoted by U.Forthetimebeing,itis
adequate to know that changes in the thermodynamic properties, such as
temperature, pressure, and volume, are accounted for by the change in in
ternal energy, denoted by ΔU. Changes in electrical, magnetic, and surface
24 Chapter 3
energy are not considered in this text since these changes are insigniﬁcant
for most engineering systems.
Consequently, we write
ΔE=ΔU+Δ(mgz)+Δ
mc2
2(3.2)
where the ﬁrst term on the right hand side of (3.2) represents the change
in internal energy U. The second term on the right hand side represents
the change in gravitational potential energy, where mis the mass, gis
the acceleration due to gravity, and zis the elevation of the system above
a reference level. The last term of (3.2) represents the change in kinetic
energy of the system owing to the motion of the system, where cis the
speed at which the system is moving as a whole.
A closed system which does not experience any change in its gravita
tional potential energy or kinetic energy during a process is known as a
stationary system. For a stationary system experiencing no change in
electrical, magnetic or surface energy, (3.2) reduces to
ΔE=ΔU(3.3)
which, in turn, transforms the ﬁrst law of thermodynamics applied to a
closed system, given by (3.1), to
Qin +Win =ΔU(3.4)
This is the ﬁrst law of thermodynamics applied to closed stationary
systems for which changes in electrical, magnetic and surface energy are
insigniﬁcant. Such a system is known as the simple compressible system
(see Section 2.7), and for which internal energy Uis the sole representative
of the total energy content of the system E.
The ﬁrst law given by (3.4) can be presented in diﬀerential form as
dQin +dWin =dU (3.5)
Since the internal energy Uis a property of state, integration of dU
from an initial state (o) to a ﬁnal state (f) gives
Uf
Uo
dU =Uf−Uo=ΔU(3.6)
which is the total change in the internal energy of the system.
The First Law applied to Closed Systems 25
Since Qin and Win are not properties of state, they depend on the path
of the process. Therefore, integration of dQin and dWin gives
f
o
dQin =Qin (not ΔQin)(3.7)
and f
o
dWin =Win (not ΔWin)(3.8)
where Qin and Win are the quantities of energy transfers between the
system and the surroundings in the form of heat and work, respectively,
when the system goes from state (o) to state (f) executing a process.
Thus, we see that integration of (3.5) yields (3.4).
ij
tt Aa...Ca...
Xv...
Zk...
Cb... Wy...
℘
0
ΦΞ=ΔΥ
♥
♣
∃ℵ
¬
qqq
q
q
q
Qin +Win
=ΔU
26 Chapter 3
3.2 A Word of Caution
In this textbook, we write the ﬁrst law applied to closed simple com
pressible systems in the form given by (3.4). Since Win =−Wout, (3.4)
can also be written as
Qin −Wout =ΔU(3.9)
If the subscripts in (3.9) are omitted, we get
Q−W=ΔU(3.10)
which is how some textbooks present the ﬁrst law of thermodynamics ap
plied to closed simple compressible systems. The apparent confusion in
using diﬀerent forms of the ﬁrst law of thermodynamics applied to closed
simple compressible systems vanishes if we learn to look at the equation
concerned with proper subscripts, as shown in Figure 3.2.
Q+W=ΔUis the same as Qin +Win =ΔU
Q−W=ΔUis the same as Qin −Wout =ΔU
Figure 3.2 Diﬀerent forms of the ﬁrst law applied to closed systems.
One can choose whichever the form that one prefers from Figure 3.2
to represent the ﬁrst law of thermodynamics applied to closed simple com
pressible systems. There is absolutely no problem with it as far as one uses
consistent set of signs for work and heat transfers in and out of the systems
through the boundary.
The First Law applied to Closed Systems 27
It is important to keep in mind that for anyone of the four equations
of Figure 3.2 to be valid, internal energy Ualone must solely represent the
total energy content Eof the closed system considered. For such systems,
changes in gravitational potential energy, kinetic energy, and electrical,
magnetic and surface energy are all either completely absent or negligibly
small.
3.3 Worked Example
Example 3.1
It is reported that a closed system undergoes
a 3step process in such a manner that the internal energy of the system
at its initial state is identically the same as the internal energy at its ﬁnal
state. In the ﬁrst step, the system absorbs 300 kJ of heat and performs 200
kJ of work on the surroundings. In the second step, the system loses 100
kJ of heat while it neither receives nor does any work. In the third step,
there is 100 kJ of work done on the system and it looses 50 kJ of heat.
Can you say whether this report is correct or not? Explain your answer.
Solution to Example 3.1
The system is said to undergo a 3step process in such a manner that the
internal energy of the system at its initial state is identically the same as the
internal energy at the ﬁnal state. Since internal energy is a property of a system,
we have
ΔU=Uat ﬁnal state −Uat initial state =0
The ﬁrst law of thermodynamics applied to the closed system undergoing
the given 3step process therefore yields
Qin +Win =0
Let us check this out by calculating Qin +Win for the 3step process from
the data provided as follows:
28 Chapter 3
Qin = 300 kJ −100 kJ −50 kJ = 150 kJ
Win =−200 kJ + 0 + 100 kJ = −100 kJ
Therefore, Qin +Win = 150 kJ −100 kJ = 50 kJ =0
There is something wrong here. One possibility is that there are errors in the
experimental observations. The other possibility is, of course, that there may
have been changes in some other modes of the energy of the system although
the net change in the internal energy Uof the system is zero.
Remember that the most general form of the ﬁrst law of thermodynamics
applied to a closed system is Qin +Win =ΔE, where ΔE, the total energy
change of the system, is given not only by ΔUbut also by changes in the
kinetic and potential energy, as given by (3.2). It is therefore possible that, even
though there is no change in the internal energy of the system, the potential
or the kinetic energy of the system as a whole could have undergone a change
giving rise to ΔE= 50 kJ, and thereby explaining why Qin +Win =50kJfor
the given process even though ΔU=0.
That is, the given system may not be a stationary system.
3.4 Summary
First law applied to closed, simple compressible
systems:
Qin +Win =ΔU(3.4)
First law in diﬀerential form:
dQin +dWin =dU (3.5)
pz