INTERNAL ENERGY & ENTHALPY
There the eye goes not, speech goes not, nor the mind.
We know not, we understand not
How one would teach it.
−Maitri Upanishad, 6.17
This chapter gives a brief introduction to the properties internal energy
and enthalpy, that are extensively used in the ﬁrst law analyses of engines.
Internal energy is a property used in the ﬁrst law analyses of closed systems,
and enthalpy is a property used in the ﬁrst law analyses of open systems.
30 Chapter 4
4.1 Internal Energy
Internal energy Urepresents the total of all the microscopic modes of
energy associated with the random motion and the internal structure of
molecules. By microscopic modes of energy, we mean, for example, the
energy associated with the translational, rotational and vibrational motion
of individual molecules and the energy associated with the intermolecular
forces. The energy associated with the position, motion and spin of the
electrons, with the nucleus-electrons interactions in an atom, and with the
nucleus-nucleus interactions also contribute towards the microscopic modes
These diﬀerent microscopic modes of energies are strongly related to
macroscopic properties such as the temperature T, pressure P,volumeV,
electrical charge, magnetic dipole-moment, surface tension, etc. Internal
energy U, which represents the total of all the microscopic modes of energy
in a system, can therefore be taken as a function of all the macroscopic
properties, and can be expressed as
U=U(P, T, V, electrical charge,magnetic dipole-moment,etc.)
When dealing with simple compressible systems, as we do here, there is
no need for us to consider properties, such as electrical charge and magnetic
dipole-moment. The properties that are relevant to simple compressible
systems, apart from the internal energy, are pressure, temperature, volume,
enthalpy, and entropy.
The state postulate says that an equilibrium state of a simple com-
pressible system can be determined by specifying two independent, inten-
sive properties alone (see Section 2.8). Let us choose these two intensive
properties as speciﬁc (or molar) volume vand temperature T, then the
speciﬁc (or molar) internal energy is described as
u=u(v, T )at an equilibrium state of a simple compressible system.
Student: Teacher, I have a question. Property is any characteristic of a system
which can be measured or calculated. We can measure the temperature
of a system with a thermometer and the pressure by a barometer or by a
pressure gauge. What is the meter that we use for measuring the internal
Internal Energy & Enthalpy 31
Teacher: Internal energy cannot be measured, dear Student. For that matter,
no energy can ever be measured by any meter. You know very well what
kinetic energy is. Is there a meter to measure kinetic energy?
Student: Let me think about it. Hm..... No teacher, I have never used any
meter to measure kinetic energy. Well..., there is no meter to measure
potential energy, either. I have just calculated them from formulas de-
scribing these energies. Well...then there must be a formula to calculate
the internal energy. What is that formula?
Teacher: Oh.. dear Student, we cannot give a formula for internal energy in
Student: What is classical thermodynamics? Why cannot we give a for-
mula for internal energy in classical thermodynamics?
Teacher: Classical thermodynamics deals only with macroscopic proper-
ties such as T,Pand V. I say temperature is a macroscopic property
since a thermometer inserted in a gas does not record the kinetic en-
ergy of each molecule, but the average kinetic energy of the molecules.
Pressure, for example, results from the bombardment of molecules. Even
though each molecule would bombard diﬀerently from the other, pressure
probe measures only the local average value of all those bombardments.
Therefore, pressure is a macroscopic property. Classical thermodynamics,
which deals only with macroscopic properties, cannot provide a formula
for internal energy that represents the total of all microscopic forms of
energy. To have a formula for internal energy, one shall study statisti-
cal thermodynamics which deals with microscopic properties, such as
the kinetic energy of an individual molecule, the spin of an electron in an
Student: Teacher, are you going to teach me statistical thermodynamics?
Teacher: No, I am not going to teach you statistical thermodynamics, because
the ﬁrst law analyses of most engineering systems can satisfactorily be
carried out with what we will learn in classical thermodynamics.
Student: Teacher, there is no meter to measure the internal energy. There is
no formula to calculate the internal energy in classical thermodynamics.
How are we then going to know the value of internal energy at a given
state of a system?
Teacher: Ah.. we are lucky there. We need not know the value of internal
energy at a given state for the successful application of the ﬁrst law of
32 Chapter 4
thermodynamics. Take a look at the ﬁrst law applied to a closed system,
which is Qin +Win =ΔU, where ΔUis simply the internal energy
diﬀerence between two given states of a system. Chapters 5 and 6 will
show you how to evaluate the internal energy diﬀerence ΔUbetween two
given states, without knowing the absolute value of Uat either of the two
Internal energy Utakes the unit of energy which, in SI units, is the joule,
abbreviated J. We could also use multiples of this unit, such as kilojoule and
megajoule, abbreviated kJ and MJ, respectively. Internal energy per unit
mass is known as speciﬁc internal energy, and it is usually given the
unit J/kg. Internal energy content of a substance divided by the amount of
substance is known as molar internal energy, and it is usually given the
unit J/mol. Note that, internal energy is an extensive property, whereas
speciﬁc and molar internal energies are intensive properties.
In this textbook, we use the unit kJ for internal energy, kJ/kg for spe-
ciﬁc internal energy, and kJ/kmol for molar internal energy. We use Uto
denote the internal energy, and the notation uto denote both the speciﬁc
and molar internal energies.
Enthalpy, denoted by H, is deﬁned as follows:
Since U,Pand Vare properties, His also a property. The unit of enthalpy
is the same as that of internal energy, which in general is the joules. The
unit of PV, pressure multiplied by volume, is also joules provided Pis in
pascals (which is equivalent to N/m2)andVis in m3.
Dividing (4.1) by the mass of the substance, we get the speciﬁc en-
where uis the speciﬁc internal energy and vis the speciﬁc volume. Common
unit of speciﬁc enthalpy is J/kg.
Internal Energy & Enthalpy 33
Had we divided (4.1) by the amount of substance, we would have got
the molar enthalpy.Thenuwould have been the molar internal energy
and vthe molar volume. Common unit of molar enthalpy is J/mol. Note
that enthalpy is an extensive property. Speciﬁc and molar enthalpies are
In this textbook, we use the unit kJ for enthalpy, kJ/kg for speciﬁc
enthalpy, and kJ/kmol for molar enthalpy. We use Hto denote enthalpy,
and the notation hto denote both the speciﬁc and molar enthalpies.
Student: Teacher, is enthalpy the same as heat?
Teacher: No. Enthalpy is very diﬀerent from heat, and you will learn about
the diﬀerence between enthalpy and heat in detail in Chapter 8. Now, let
me tell you that the enthalpy of a system may be increased by supplying
heat to the system.
Student: Teacher, forgive me for asking. What is enthalpy?
Teacher: My dear Student, there is only one answer for that question. Enthalpy
is a property deﬁned by H=U+PV.
Student: Teacher, I want the physical meaning of enthalpy.
Teacher: Forgive me, dear Student, I am unable to give you a physical meaning
of enthalpy. All what I can tell you is we frequently encounter the combi-
nation of properties U+PV, particularly when applying the ﬁrst law to
open systems. For the sake of convenience, U+PV is deﬁned as a new
property enthalpy. However, let me assure you that you will appreciate the
signiﬁcance of the property enthalpy when dealing with ﬂuids in motion,
as we will do with open systems in Chapters 9 and 10.
We know that a state of a simple compressible system can be described
completely by two independent, intensive properties (see Section 2.8). If the
properties are chosen as the speciﬁc (or molar) volume vand temperature
T, then the speciﬁc (or molar) enthalpy can be written as
h=h(v, T )at an equilibrium state of a simple compressible system.
34 Chapter 4
Student: Teacher, you have chosen vand Tas the independent, intensive
properties, and expressed uand has functions of vand T. Why don’t we
choose, for example, Pand Tas the independent, intensive properties,
and express uand has functions of Pand T.
Teacher: Yes, we could also choose the independent, intensive properties to
be Pand T, which are in fact easier to measure and control. However, it
is important to note that Pand Tcan be independent of each other only
when the substance is in a single phase. When a substance exists in two
phases, like boiling water in a closed vessel, Tcannot be changed without
aﬀecting P,orPwithout aﬀecting T. Therefore, we can write u=
u(P, T )or h=h(P, T )for an equilibrium state of a simple compressible
system only when the system remains in a single phase.
•Internal energy is a property of a system that represents the sum of all
microscopic modes of energy in the system.
and the speciﬁc (or molar) enthalpy is therefore given by
•Speciﬁc (or molar) internal energy for a simple compressible system at an
equilibrium state can be written as a function of temperature and speciﬁc
(or molar) volume.
•Speciﬁc (or molar) enthalpy for a simple compressible system at an equi-
librium state can be written as a function of temperature and speciﬁc (or
•For a simple compressible system at an equilibrium state uand hcan also
be written as functions of Tand Pprovided the system remains in a single