CHAPTER 4

EQUATION OF STATE AND

ENERGY ANALYSIS OF

CLOSED SYSTEMS

Lecture slides by

Dr. Fawzi Elfghi

Thermodynamics: An Engineering Approach 8th Edition in SI Units

Yunus A. Çengel, Michael A. Boles

McGraw-Hill, 2015

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Objectives • Describe the hypothetical substance “ideal gas” and the ideal-gas equation of state.

• Apply the ideal-gas equation of state in the solution of typical problems.

• Introduce the compressibility factor, which accounts for the deviation of real gases

from ideal-gas behavior.

• Present some of the best-known equations of state.

• Examine the moving boundary work or P dV work commonly encountered in

reciprocating devices such as automotive engines and compressors.

• Identify the first law of thermodynamics as simply a statement of the conservation

of energy principle for closed (fixed mass) systems.

• Develop the general energy balance applied to closed systems.

• Define the specific heat at constant volume and the specific heat at constant

pressure.

• Relate the specific heats to the calculation of the changes in internal energy and

enthalpy of ideal gases.

• Describe incompressible substances and determine the changes in their internal

energy and enthalpy.

• Solve energy balance problems for closed (fixed mass) systems that involve heat

and work interactions for general pure substances

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Gas Laws Perfect gases The air that we breathe is a mixture of approximately 78% nitrogen, 21% Oxygen and 1% mixture of other gases; it is reasonable. It is reasonable to assume that the air is 79% nitrogen and 21% Oxygen. - All gases when compressed and cooled sufficiently will become liquid. At

atmospheric pressure, nitrogen liquefies at -196oC and Oxygen at -183oC. - The air that we breathe is therefore a vaporized liquid at a temperature well

above its boiling temperature; it is in fact a superheated vapour. The air is an example of a real gas, i.e. a gas that can be compressed into a liquid.

- Many chemists had dreamed of having an equation that describes relation of a gas molecule to its environment such as pressure or temperature. However, they had encountered many difficulties because of the fact that there always are other affecting factors such as intermolecular forces. Despite this fact, chemists came up with a simple gas equation to study gas behaviour while putting a blind eye to minor factors.

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Introduction - It is possible to imagine a gas that will never liquefy no matter how much it is

cooled or compressed. Such a gas is called a perfect gas or ideal gas, a gas is made of molecules which move around with random motion.

- Ideal gases are essentially point masses moving in constant, random, straight-line motion.

- In a perfect gas, the molecules may colloid but they have no tendency at all to stick together or repel each other. In other word, a perfect gas is completely inviscid.

When dealing with gas, a famous equation was used to relate all of the factors needed in order to solve a gas problem. This equation is known as the Ideal Gas Equation. As we have always known, anything ideal does not exist. In this issue, two well-known assumptions should have been made beforehand:

the particles have no forces acting among them, and these particles do not take up any space, meaning their atomic volume is completely ignored.

An ideal gas is a hypothetical gas dreamed by chemists and students because it would be much easier if things like intermolecular forces do not exist to complicate the simple Ideal Gas Law. Ideal gases are essentially point masses moving in constant, random, straight-line motion.

This definition of an ideal gas contrasts with the Non-Ideal Gas definition, because this equation represents how gas actually behaves in reality. For now, let us focus on the Ideal Gas.

The Ideal Gas Equation Before we look at the Ideal Gas Equation, let us state the four gas variables and

one constant for a better understanding. The four gas variables are: pressure (P), volume (V), number of mole of gas (n), and temperature (T). Lastly, the constant in the equation shown below is R, known as the the gas

constant, which will be discussed in depth further later:

PV=nRT

Another way to describe an ideal gas is to describe it in mathematically. Consider the following equation:

PVnRT=1

An ideal gas will always equal 1 when plugged into this equation. The greater it deviates from the number 1, the more it will behave like a real gas rather than an

ideal. A few things should always be kept in mind when working with this equation, as you may find it extremely helpful when checking your answer after

working out a gas problem.

Pressure is directly proportional to number of molecule and temperature. (Since P is on the opposite side of the equation to n and T)

Pressure, however, is indirectly proportional to volume. (Since P is on the same side of the equation with V)

Simple Gas Laws

• The Ideal Gas Law is simply the combination of all Simple Gas Laws (Boyle's Law, Charles' Law, and Avogadro's Law), and so learning this one means that you have learned them all. The Simple Gas Laws can always be derived from the Ideal Gas equation.

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•

• Avogadro's Law

Volume of a gas is directly proportional to the amount of gas at a constant temperature and pressure.

V∝n

Equation:

V1n1=V2n2

Avogadro's Law can apply well to problems using Standard Temperature and Pressure (see below), because of a set amount of pressure and temperature.

• Amontons's Law

Given a constant number of mole of a gas and an unchanged volume, pressure is directly proportional to temperature.

P∝T

Equation:

P1T1=P2T2

• Boyle's Law, Charles' Law, and Avogradro's Law and Amontons's Law are given under certain conditions so directly combining them will not work. Through advanced mathematics (provided in outside link if you are interested), the properties of the three simple gas laws will give you the Ideal Gas Equation.

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THE IDEAL-GAS EQUATION OF STATE

• Equation of state: Any equation that relates the pressure, temperature,

and specific volume of a substance.

• The simplest and best-known equation of state for substances in the gas

phase is the ideal-gas equation of state. This equation predicts the P-v-T

behavior of a gas quite accurately within some properly selected region.

R: gas constant

M: molar mass (kg/kmol)

Ru: universal gas constant

Ideal gas equation

of state

Different substances have different

gas constants.

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Equations of State

The relationship among the state variables, temperature, pressure, and specific

volume is called the equation of state. We now consider the equation of state for the

vapor or gaseous phase of simple compressible substances.

Ideal Gas

Based on our experience in chemistry and physics we recall that the combination of

Boyle’s and Charles’ laws for gases at low pressure result in the equation of state for

the ideal gas as

where R is the constant of proportionality and is called the gas constant and takes

on a different value for each gas. If a gas obeys this relation, it is called an ideal gas.

We often write this equation as

Pv RT

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The gas constant for ideal gases is related to the universal gas constant valid for all

substances through the molar mass (or molecular weight). Let Ru be the universal

gas constant. Then,

RR

M

u

The mass, m, is related to the moles, N, of substance through the molecular weight

or molar mass, M, see Table A-1. The molar mass is the ratio of mass to moles and

has the same value regardless of the system of units.

Mg

gmol

kg

kmol

lbm

lbmolair 28 97 28 97 28 97. . .

Since 1 kmol = 1000 gmol or 1000 gram-mole and 1 kg = 1000 g, 1 kmol of air has a

mass of 28.97 kg or 28,970 grams.

m N MThe ideal gas equation of state may be written several ways.

Pv RT

VP RT

m

PV mRT

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Here

P = absolute pressure in MPa, or kPa

= molar specific volume in m3/kmol

T = absolute temperature in K

Ru = 8.314 kJ/(kmolK)

v

Some values of the universal gas constant are

Universal Gas Constant, Ru

8.314 kJ/(kmolK)

8.314 kPam3/(kmolK)

1.986 Btu/(lbmolR)

1545 ftlbf/(lbmolR)

10.73 psiaft3/(lbmolR)

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Mass = Molar mass Mole number

Various expressions

of ideal gas equation

Ideal gas

equation at two

states for a fixed

mass

Real gases behave as an ideal

gas at low densities (i.e., low

pressure, high temperature).

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The ideal gas equation of state can be derived from basic principles if one assumes

1. Intermolecular forces are small.

2. Volume occupied by the particles is small.

Example 2-5

Determine the particular gas constant for air and hydrogen.

RR

M

R

kJ

kmol Kkg

kmol

kJ

kg K

u

air

8 314

28 97

0 287

.

.

.

R

kJ

kmol Kkg

kmol

kJ

kg Khydrogen

8 314

2 016

4124

.

.

.

How to calculate the molar mass of a

compound

element atomic mass # in formula contribution

C 12.01 g/mol x 6 = 72.06 g/mol

H 1.008 g/mol x 12 = 12.10 g/mol

O 16.00 g/mol x 6 = 96.00 g/mol

total = 180.16 g/mol

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Molar mass defined as the mass of one mole (also called gmole, abbreviated gram-

mole) of a substance in grams, or the mass of 1 kmol (also called kilogram-mole)

H2O , is (1.008 x 2) +(1x16.00) = its molar mass is 18 g/mol.

For more complex formulas it is convenient to use a table. The molar mass of

glucose, C6H12O6, is:

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Is Water Vapor an Ideal Gas? • At pressures below 10 kPa, water

vapor can be treated as an ideal gas, regardless of its temperature, with negligible error (less than 0.1 percent).

• At higher pressures, however, the ideal gas assumption yields unacceptable errors, particularly in the vicinity of the critical point and the saturated vapor line.

• In air-conditioning applications, the water vapor in the air can be treated as an ideal gas. Why?

• In steam power plant applications, however, the pressures involved are usually very high; therefore, ideal-gas relations should not be used.

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We see that the region for which water behaves as an ideal gas is in the superheated

region and depends on both T and P. We must be cautioned that in this course,

when water is the working fluid, the ideal gas assumption may not be used to solve

problems. We must use the real gas relations, i.e., the property tables.

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COMPRESSIBILITY FACTOR—A MEASURE

OF DEVIATION FROM IDEAL-GAS BEHAVIOR Compressibility factor Z

A factor that accounts for

the deviation of real gases

from ideal-gas behavior at

a given temperature and

pressure.

The farther away Z is from unity, the more the

gas deviates from ideal-gas behavior.

Gases behave as an ideal gas at low densities

(i.e., low pressure, high temperature).

Question: What is the criteria for low pressure

and high temperature?

Answer: The pressure or temperature of a gas

is high or low relative to its critical temperature

or pressure.

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The ideal gas equation of state is used when :

(1) the pressure is small compared to the critical pressure or

(2) when the temperature is twice the critical temperature and the pressure is less

than 10 times the critical pressure.

The critical point is that state where there is an instantaneous change from the liquid

phase to the vapor phase for a substance.

Critical point data are given in Table A-1.

Compressibility Factor

To understand the above criteria and to determine how much the ideal gas equation

of state deviates from the actual gas behavior, we introduce the compressibility factor

Z as follows.

Pv Z R Tuor

ZPv

R Tu

20 Comparison of Z factors for various gases.

Reduced

temperature

Reduced

pressure

Pseudo-reduced

specific volume Z can also be determined from

a knowledge of PR and vR.

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When either P or T is unknown, Z can be determined from the compressibility chart

with the help of the pseudo-reduced specific volume, defined as

vv

RT

P

Ractual

cr

cr

Figure A-15 presents the generalized compressibility chart based on data for a large

number of gases.

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For an ideal gas Z = 1, and the deviation of Z from unity measures the

deviation of the actual P-V-T relation from the ideal gas equation of

state.

The compressibility factor is expressed as a function of the reduced

pressure and the reduced temperature.

The Z factor is approximately the same for all gases at the same

reduced temperature and reduced pressure, which are defined as

TT

TP

P

PR

cr

R

cr

and

where Pcr and Tcr are the critical pressure and temperature, respectively.

The critical constant data for various substances are given in Table A-1.

This is known as the principle of corresponding states.

Figure 3-51 gives a comparison of Z factors for various gases and supports the

principle of corresponding states.

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These charts show the conditions for which Z = 1 and the gas behaves as an

ideal gas:

1.PR < 10 and TR > 2 or P < 10Pcr and T > 2Tcr

2.PR << 1 or P << Pcr

Note: When PR is small, we must make sure that the state is not in the

compressed liquid region for the given temperature. A compressed liquid state is

certainly not an ideal gas state.

For instance the critical pressure and temperature for oxygen are 5.08 MPa and

154.8 K, respectively. For temperatures greater than 300 K and pressures less than

50 MPa (1 atmosphere pressure is 0.10135 MPa) oxygen is considered to be an ideal

gas.

Example 2-6

Calculate the specific volume of nitrogen at 300 K and 8.0 MPa and compare the

result with the value given in a nitrogen table as v = 0.011133 m3/kg.

From Table A.1 for nitrogen

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Tcr = 126.2 K, Pcr = 3.39 MPa R = 0.2968 kJ/kg-K

TT

T

K

K

PP

P

MPa

MPa

R

cr

R

cr

300

126 22 38

8 0

3392 36

..

.

..

Since T > 2Tcr and P < 10Pcr, we use the ideal gas equation of state

Pv RT

vRT

P

kJ

kg KK

MPa

m MPa

kJ

m

kg

0 2968 300

8 0 10

0 01113

3

3

3

. ( )

.

.

Nitrogen is clearly an ideal gas at this state.

If the system pressure is low enough and the temperature high enough (P and T are

compared to the critical values), gases will behave as ideal gases. Consider the T-v

diagram for water. The figure below shows the percentage of error for the volume

([|vtable – videal|/vtable]x100) for assuming water (superheated steam) to be an ideal

gas.

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OTHER EQUATIONS OF STATE

Several equations have been proposed to

represent the P-v-T behavior of substances

accurately over a larger region with no

limitations.

Van der Waals

Equation of State Critical isotherm

of a pure

substance has

an inflection

point at the

critical state.

This model includes two effects not considered

in the ideal-gas model: the intermolecular

attraction forces and the volume occupied by the

molecules themselves. The accuracy of the van

der Waals equation of state is often inadequate.

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Beattie-Bridgeman Equation of State

The constants are given in

Table 3–4 for various

substances. It is known to be

reasonably accurate for

densities up to about 0.8cr.

Benedict-Webb-Rubin Equation of State

The constants are given in Table 3–4. This equation can handle substances

at densities up to about 2.5 cr.

Virial Equation of State

The coefficients a(T), b(T), c(T), and so on, that are

functions of temperature alone are called virial coefficients.

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MOVING BOUNDARY WORK

Moving boundary work (P dV work):

The expansion and compression work

in a piston-cylinder device.

Quasi-equilibrium process:

A process during which the system

remains nearly in equilibrium at all

times.

Wb is positive for expansion

Wb is negative for compression

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The boundary

work done

during a process

depends on the

path followed as

well as the end

states.

The area under the process curve on a P-V

diagram is equal, in magnitude, to the work

done during a quasi-equilibrium expansion or

compression process of a closed system.

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5 kPa

400 kPa

v, m3/kg

P,

kPa

400

Boundary Work for a constant-pressure

Boundary Work for a constant-pressure

• Example 4-2

A frictionless piston-cylinder device contains 5 kg of steam at 400 kpa and 200oC. Heat is now transferred to the steam until the temperasture reaches 250oC. If the piston is not attached to a shaft and its mass is constant . Determine the work done by the steam during this process.

• Solution

steam in a piston cylinder device is heated and the temperature rises at constant pressure. The boundary work done is to be determined.

Analysis:

A sketch of the system and p-v diagram of the process are shown Fig 4-7 (Refer to the book).

Assumption:

Even though it isn’t explicitly stated, the pressure of the steam within the cylinder remains constant during this process since both the atmospheric pressure and the weight of the piston remain constant. Therefore, this is a constant-pressure process and, from equation 4-2 (Refer to the book).

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•

34

35

What is the boundary

work for a constant-

volume process?

Boundary Work for a constant-Volume

•

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Boundary Work for an Isothermal Compression Process

• Example 4-3

• A piston-cylinder device initially contains 0.4 m3 of air at 100 kPa and 80oC. The air is now compressed to 0.1 m3 in such a way that the temperature inside the cylinder remains constant. Determine the work done during this process.

• Solution

air in a piston-cylinder device is compressed isothermally. The boundary work done is to be determined.

Analysis a sketch of the system and the P-V diagram of the process shown in Fig 4-8.

Assumption: at specified conditions, air can be considered to be an ideal gas since it is a high temperature and low pressure relative to its critical-point values To,

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Boundary Work for an Isothermal Compression Process

Apply the equation of boundary work related to isothermal compression of ideal gas

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SPECIFIC HEATS Specific heat at constant volume, cv: The

energy required to raise the temperature of

the unit mass of a substance by one degree

as the volume is maintained constant.

Specific heat at constant pressure, cp: The

energy required to raise the temperature of

the unit mass of a substance by one degree

as the pressure is maintained constant.

Constant-

volume and

constant-

pressure specific

heats cv and cp

(values are for

helium gas).

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• The equations are valid for any

substance undergoing any process.

• cv and cp are properties.

• cv is related to the changes in internal

energy and cp to the changes in

enthalpy.

• A common unit for specific heats is

kJ/kg·°C or kJ/kg·K. Are these units

identical?

True or False?

cp is always greater than cv

Formal definitions of cv and cp.

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Summary • The ideal gas equation of state

Is water vapor an ideal gas?

• Compressibility factor

• Other equations of state

van der Waals Equation of State, Beattie-Bridgeman Equation of State

Benedict-Webb-Rubin Equation of State, Virial Equation of State

• Moving boundary work

Wb for an isothermal process

Wb for a constant-pressure process

Wb for a polytropic process

• Specific heats

Constant-pressure specific heat, cp

Constant-volume specific heat, cv