**Isochoric process and
isobaric process for constant Thermodynamics
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**

Jegadeesan Singaram^{1} and P.
Sangeetha^{2}

^{1}
Department of Mathematics,
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Thiagarajar College of Education, Affiliated to Periyar
University, Salem, Tamil Nadu, India.

^{2}*Department of Mathematics, Jairam College
of Arts and Science, Affiliated to Periyar University, Salem,
Tamil Nadu, India.*

**Abstract**

There are several specific types of thermodynamic processes that happen frequently enough (and in practical situations) that they are commonly treated in the study of thermodynamics. Each has a unique trait that identifies it, and which is useful in analyzing the energy and work changes related to the process. The laws of thermodynamics are applicable to any system, regardless of its complexity. They are sometimes helpful to model the system by considering only some of their aspects, and thus substantially simplifying the analysis. Here we introduce the concept of a simple system, which has only one mode of quasistatic work. In many applications this simplified concept describes fairly accurately real substances.

**1. Introduction: Basic
Processes in Simple Systems**

In this part we apply the first law of thermodynamics to some basic processes in simple systems passing through equilibrium states.

The energy of a simple system is the internal energy and therefore the first law of thermodynamics, <?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" /> or (1)

It expresses the concept of conservation of energy in systems undergoing interactions.

Equation 1, is rewritten as

(2)

or in differential form

(3)

**1.1. The
Constant-Volume Process**

A process in which the volume of the system remains constant during a change of state is called a constant-volume or isochoric process.

Consider a simple compressible system inside a
rigid container undergoing a constant-volume change from states 1
to 2, shown by the vertical straight line on the *p-V*
diagram in Figure 1a. The state of the system may be changed by
several different processes.

*Case a.*
The change of state is accomplished by a heat
interaction as shown in Figure 1b. In this case *W* = 0 and the first law of thermodynamics, Equation 2,
yields

(4)

where Δ*U* = *U*_{2} − *U*_{1} is
the change in internal energy between states 1 and 2.

*Figure 1
Constant-volume process.*

*Case b.*
The change of state is accomplished by a work
interaction of a stirrer as shown in Figure 1b. In this case
*Q* = 0 and the first law of thermodynamics,
Equation 2, yields

(5)

We can also bring about the same change of state in the system by a process that involves both work and heat interactions in suitable amounts. For this case Equation 1 reads

Although the processes connecting points 1and 2 differ from each other by the interactions at the system boundary, they all result in the same changes in the internal energy. Heat and work interactions, however, are not the same for the processes between points 1 and 2, described earlier, because they are not properties.

The rate of variation of the internal energy
with temperature is important in thermodynamics; hence, a new
extensive property *C _{v}* is defined
as

(6)

This property is the *heat
capacity at constant volume.* The name is rather misleading,
because it has nothing to do with the capacity of the system to
store heat. Heat is not a property and cannot be stored. What is
actually stored is the internal energy, and the change in the
internal energy of the system can be achieved by either heat or
work interactions or by a combination of the two. The term heat
capacity is a remnant from the era of the caloric theory when
heat was considered to be a property. The heat capacity per unit
mass

(7)

is the specific heat capacity at constant volume
or, in short, *specific heat at constant
volume*. The units of heat capacity and specific heat are
kJ/K and kJ/kg K, respectively, or in British units and
Btu/lb°F.

The specific energy of a simple system is, in
general, a function of two independent properties, say,
*u*(*T*, *v*). It is
obvious from the below Equation 8 that the specific heat at
constant volume is also a function of the same independent
properties, *c _{v}*(

*T*,

*v*).

(8)

Now the internal energy, *U*,
is an extensive property. The corresponding *intensive* property, the specific internal energy,
*u*, is the internal energy per unit mass. An
example of a functional relationship which holds for a simple
system is

(9)

Differentiation of Equation. (9) results in

(10)

We now substitute Equation 8 into Equation 10 and obtain an alternative expression for the change in internal energy, which upon integration between states 1 and 2 yields

(11)

In the case of an isochoric process d*v* = 0, and Equation 12 simplifies to

(12)

or in terms of extensive properties

(13)

For the special case of *c _{v}* = constant, we obtain

(14)

**Example
1**

A rigid vessel contains 0.3 kg of carbon
monoxide (CO, *c _{v}* = 0.7423 kJ/kg K) at 25°C. The vessel is heated to 150°C. Find the
heat interaction of gas.

*Solution*

We use Equations 4 and 14 to calculate the heat interaction for this constant-volume process

**1.2 The
Constant-Pressure Process**

A process in which the pressure of the system remains constant during a change of state is called a constant-pressure or isobaric process.

Consider a simple compressible system inside a
piston-cylinder assembly. Properties of the system, such as
energy, temperature, and volume, may be changed by several
different processes. The pressure of the system, however, is kept
constant and equal to the equivalent external pressure imposed by
the piston. Figure 2a depicts the constant-pressure process on a
*p-V* diagram.

*Figure 2
Constant-pressure process.*

The constant-pressure path from states 1 to 2 is
shown by the horizontal line on the *p-V* diagram in
Figure 2a. The state of the system may be changed by several
different processes.

*Case a.*
The change of state is accomplished by a heat
interaction as shown in Figure 2b. This constant-pressure
interaction results in the movement of the piston and a change of
volume. The heat interaction is, therefore, accompanied by a work
interaction on the piston. As the external equivalent pressure is
equal to the pressure of the system *p*
throughout this process, the work interaction is given by the
product of the pressure and the change of volume.
Thus

(15)

The first law of thermodynamics requires that

(16)

or

(17)

As the pressure is constant, Equation 17 may be rewritten as

(18)

*Case b.*
As with the isochoric process, the same change
of state may be obtained by the nonquasistatic work of the
stirrer *W*s (Figure 2b), instead of
the heat interaction. The first law of thermodynamics requires
now that

(19)

or

(20)

Note that two kinds of works are involved in this process, the work associated with the change of volume, which is quasistatic, and the work of the stirrer, which is not.

We can also bring about the same change of state in the system by a process that involves both stirrer work and heat interactions in suitable amounts such that

(21)

The combination of properties (*U* + *pV*) describes a new extensive
property, with units of energy, which often appears in
thermodynamics. This property is called *enthalpy* and is denoted by *H*.

(22)

The corresponding intensive property is the
*specific enthalpy*, which is the enthalpy per
unit mass.

(23)

Equation 21 may now be rewritten for this constant-pressure process as

(24)

The variation of enthalpy with temperature at
constant pressure is also an extensive property called *heat
capacity at constant pressure* and denoted by *Cp*.

(25)

The corresponding intensive property is the
specific heat capacity at constant pressure called for short
*specific heat at constant pressure*.

(26)

The specific enthalpy is a function of two
independent properties, say, *T* and *p*.

(27)

Differentiating Equation 27 and using Equation 26 yields

(28)

which for an isobaric process, d*p* = 0, simplifies to

(29)

Hence, the change of enthalpy between two states at constant pressure is given by

(30)

The ratio of the specific heat at constant
pressure to specific heat at constant volume is an intensive
property denoted by *k* and generally it is a
function of two independent properties.

(31)

**Example
2**

A coal-oil mixture (COM) is a fuel that can replace heating oil for certain applications. To prevent settling of the coal particles, mixing is required.

An open tank contains 5000 kg of COM at 22°C. The COM was stirred
for 40 min by a 20 kW paddle mixer. During this operation the
temperature of the mixture rose to 27.6°C. The specific heat of
the COM is assumed to be constant at *cp *= 1.4
kJ/kg K.

1. Find the heat interaction of the COM with its surroundings.

2. What will be the temperature rise of the COM if the tank is perfectly insulated?

*Solution*

1. We use Equation 24 to calculate the heat interaction for this constant-pressure process.

∆*H* is calculated from Equation 30:

The work is

The work is negative because it is done on the system. The heat interaction is

The minus sign indicates that heat is removed from the system.

2. When perfectly insulated or in adiabatic process *Q* = 0; hence, from Equation 24

The process is
isobaric (d*p* = 0) and the specific heat is
constant. For this case Equation 31 becomes

Therefore

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