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Chapter 4

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

of energy.

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

energy?

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

classical thermodynamics.

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

atom, etc.

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

states.

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.

4.2 Enthalpy

Enthalpy, denoted by H, is deﬁned as follows:

H≡U+PV (4.1)

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-

thalpy has

h=u+Pv (4.2)

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

intensive properties.

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.

4.3 Summary

•Internal energy is a property of a system that represents the sum of all

microscopic modes of energy in the system.

•Enthalpy, H,isdeﬁnedas

H≡U+PV (4.1)

and the speciﬁc (or molar) enthalpy is therefore given by

h=u+Pv (4.2)

•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

molar) volume.

•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

phase.