UNIT 5, 6

UNIT - VPerfect Gas Laws – Equation of State, specific and Universal Gas constants – various Non-flow processes, properties, end states, Heat and Work Transfer, changes in Internal Energy – Throttling and Free Expansion Processes – Flow processes –Deviations from perfect Gas Model – Vander Waals Equation of State – Compressibility charts – variable specific Heats – Gas Tables.UNIT – VIMixtures of perfect Gases – Mole Fraction, Mass friction Gravimetric and volumetric Analysis – Dalton’s Law of partial pressure, Avogadro’s Laws of additive volumes –Mole fraction, Volume fraction and partial pressure, Equivalent Gas const. And Molecular Internal Energy, Enthalpy, specific Heats and Entropy of Mixture of perfect Gases and Vapour.

INTRODUCTION : An ‘ideal gas’ is defined as a gas having no forces of intermolecular attraction. The gases which follow the gas laws at all ranges of pressures and temperatures are considered as “ideal”. However, ‘real gases’ follow these laws at low pressures or high temperatures or both .This is because the forces of attraction between molecules tend to be very small at reduced pressuresand elevated temperatures. An ideal gas obeys the law pv = RT. The specific heat capacities are not constant but are functions of temperature. A perfect gas obeys the law pv = RT and has constant specific heatcapacities. A perfect gas is well suited to mathematical manipulation and is therefore a most usefulmodel to use for analysis of practical machinery which uses real gases as a working substance. In reality there is no ideal or perfect gas. At a very low pressure and at a very high temperature, real gases like hydrogen, oxygen, nitrogen, helium etc. behave nearly the same way as perfect gases. These gases are called semi-perfect or permanent gases. The term semi-perfect has the implication that the behaviour of the gases are nearly the same as that of a perfect gas. The permanent’ was used for these gases by earlier chemists who thought that these gases did not change their phase (i.e., did not condense to a liquid state). Hence they are called permanent gases. There is no gas which does not change phase, and there is no permanent gas in the real sense.However, these gases can be changed into a liquid phase only if they are subjected to a great decrease in temperature and increase in pressure.All gases behave in nearly in a similar way, especially at pressures considerably lower than the critical pressure, and at temperatures above the critical temperature. The relation between the independent properties, such as pressure, specific volume and temperature for a pure substance is known as the ‘equationof state’. For engineering calculations, the equation of state for perfect gases can be used for real gases so long as the pressures are well below their critical pressure and the temperatures are above the critical temperature.

EQUATIONS OF STATE OF A GASThe functional relationship among the properties, pressure p, molar or specificvolume v, and temperature T, is known as an equation of state, which may beexpressed in the form,

If two of these properties of a gas are known the third can be evaluated fromthe equation of state. The gas is the best-behaved thermometric substance because of the fact that the ratio .of pressure p of a gas at any temperature to pressure of the same gas at the triple point, as both p and approach zero, approaches a value independent of the nature of the gas. The ideal gas temperature T of the system at whose temperature the gas exerts pressure p defined as

The relation between pv and p of a gas may be expressed by means of a power series of the form .

……………1

where depend on the temperature and nature of the gas.

A fundamental property of gases is that, is independent of the natureof the gas and depends only on T. This is shown in Fig. 1, where the product pvis plotted against p for four different gases in the bulb (nitrogen, air, hydrogen,and oxygen) at the boiling point of sulphur, at steam point and at the triple pointof water. In each case, it is seen that as ,pv approaches the same value forall gases at the same temperature. From Eq. (1)

Therefore the constant A is a function of temperature only and independent ofthe nature of the gas.

The ideal gas temperature T, is thus

The term within bracket is called the universal gas constant and is denoted by

Thus …………..2

Fig1

………….3THE EQUATION OF STATE FOR A PERFECT GASBoyle’s law. It states that volume of a given mass of a perfect gas varies inversely as theabsolute pressure when temperature is constant.

If p is the absolute pressure of the gas and V is the volume occupied by the gas, then

V ∝ 1/pFig shows the graphical representation of Boyle’s law. The curves are rectangularhyperbolas asymptotic to the p-v axis. Each curve corresponds to a different temperature. For anytwo points on the curve,

Charle’s law. It states that if any gas is heated at constant pressure, its volume changesdirectly as its absolute temperature.In other words, V ∝ T Or V/T = Constant, so long as pressure is constant

If a gas changes its volume from V1 to V2 and absolute temperature from T without any

change of pressure, then

To derive the equation of state for a perfect gas let us consider a unit mass of a perfect gasto change its state in the following two successive processes (Fig. )(i) Process 1-2' at constant pressure, and(ii) Process 2'-2 at constant temperature.

The magnitude of this constant depends upon the particular gas and it is denoted by R,where R is called the specific gas constant. Then

The equation of the state for a perfect gas is thus given by the equation

or for m kg, occupying

If the mass is chosen to be numerically equal to the molecular weight of the gas then 1 mole of the gas has been considered, i.e., 1 kg mole of oxygen is 32 kg oxygen, or 1 kg mole of hydrogen is 2 kg hydrogen.The equation may be written as

where V0 = Molar volume, andM = Molecular weight of the gas.Avogadro discovered that V0 is the same for all gases at the same pressure and temperatureand therefore it may be seen that MR = a constant ; R0 and thus

R0 is called the molar or universal gas constant and its value is 8.3143 kJ/kg mol K. If there are n moles present then the ideal gas equation may be written as

where V is the volume occupied by n moles at pressure p and temperature T.

IDEAL GAS

VAN DER WAALS EQUATION OF STATE:

The ideal gas equation of state can be established from the positulatesof the kinetic theory of gases developed by Clerk Maxwell, with two importantassumptions that there is little or no attraction between the molecules of the gasand that the volume occupied by the molecules themselves is negligibly smallcompared to the volume of the gas. When pressure is very small or temperaturevery large, the intermolecular attraction and the volume of the moleculescompared to the total volume of the gas are not of much significance, and the real gas obeys very closely the ideal gas equation. But as pressure increases, theintermolecular forces of attraction and repulsion increase, and also the volume of the molecules becomes appreciable compared to the gas volume. So then the real gases deviate considerably from the ideal gas equation. van der Waals, byapplying the laws of mechanics to individual molecules, introduced twocorrection terms in the equation of ideal gas, and his equation is given below.

The coefficient a was introduced to account. for the existence of mutualattraction between the molecules. The term alv2 is called theforce of cohesion.The coefficient b was introduced to account for the volumes of the molecules, and is known as co-volume. Other equations of states are

COMPRESSIBILTY FACTOR :

The ratio is called the compressibility factor, Z. For an ideal gas Z = 1.The magnitude of Z for a certain gas at a particular pressure and temperaturegives an indication of the extent of deviation of the gas from the ideal gas

behaviour.As the pressure of anu gas is reduced its compressibility factor approaches unity, since the gasses act as perfect gas at lower pressure. The value of Z depends upon the pressure and the temperature of the gas.For each substance, there is a compressibility factor chart .It would be veryconvenient if one chart could be used for all substances. The general shapes of-thevapour dome and of the constant temperature lines on the p-v plane are sim-ilar for all substances, aJthough the scaJes may be different. This.similarity can be exploited by using dimensionless properties called reduced proper-ties.Thereduced pressure is the ratio of the existing pressure to the critical pres-sure of the substance, and similarly for reduced temperature and reduced volume. Then

where subscript r denotes the reduced property, and subscript c denotes theproperty ai the critical state.At the same pressure and temperature the specific or molal volumes of differentgases are different. However, it is found from experimental data that at the same reduced pressure and reduced temperature, the reduced volumes of differ-ent gases are approximately the same. Therefore, for all substances

This is called the critical compressibility factor. Thereforefrom above Eqs

COMPRESSIBILITY CHARTThe compressibility factor (Z) of any gas is a function of only two properties, usually temperature and pressure, so that Z = f(Tr, p r) except near the critical point. The value of Z for any real gas may be less or more than unity, depending on pressure and temperature conditions of the gas.The general compressibility chart is plotted with Z versus pr for various values of T.This is constructed by plotting the known data of one or more gases and can be used for any gas. Sucha chart is shown in Fig. This chart gives best results for the regions well removed from the critical state for all gases.

Fig Generalized Compressibility Chart

A change in the state of a system is called a thermodynamic process. All thermodynamics process may be classified into two types

1.non flow process

2.flow process

Non flow – process:

Occurs when the system is confined with in a closed boundary can change, but the mass of the system, remains constant throughout the process how ever the energy cross the boundary.

Non-flow process are analysed

1.constant volume process

2.constant pressure process

3. constant temperature process

4. Adiabatic process

5.polytropic process

6. throttling process

7.free expansion

1. Constant Volume process (dv=0)

The constant volume process is illustrated in this process volume remains constant through out the process since the volume of the system cannot change during the process the work done by the gas is equal to zero.

W = ∫ p dv= 0 [ ∵dv=0¿

The non flow energy equation for constant volume process may be written as Q= u2−u1=m cv (T2−T 1)

Where Q = heat added to the system

The initial and final states may be related by the equation

p1

T1

=p2

T2

It is also known as isochoric process or isometric process.

2) constant pressure process dp=0

As heat energy is supplied the gas will expand from state 1 to state 2 at the constant pressure thus the work is done by the system the amount of the work done

=R(T 2−T1)

= m (T 2−T1)

Adiabatic process

A process In which no heat is transferred to or from the gas (system) is called Adiabatic process. A reversible adiabatic process can be released in a cylinder with perfect heat insulated walls and is never perfect . however if an expansion or compression takes place very rapidly approximately adiabatic conditions may be achieved during the process heat transfer across the boundary is Zero. i.e., dQ=0 and therefore change in entropy is 0.

The friction less adiabatic process is often referred as isentropic process and it can be represented as

γ = adiabatic index =

The work done by an ideal gas in a reversible adiabatic process is given by

Reversible Isothermal process

When an ideal gas of mass m undergoes a reversible isothermal process fromstate 1 to state 2, the work done is given by

Polytropic Process An equation of the form pvn = constant, where n is a constant can be usedapproximately to describe many processes which occur in practice. Such aprocess is called a polytropic process. It is not adiabatic, but it can be reversible.It may be noted that r is a property of the gas, whereas n is not. The value of ndepends upon the process. It is possible to find the value of n which more or lessfits the experimental results. For two states on the process,

Expression for work done

Consider m kg of gas at conditions P1,v1, T1 changes to the condition P2,v2,T2

According to the law pvn = constant.

Consider a small change in the volme dv at pressure in dw =pdv work done between states 1 & state 2

W= ∫ pdv for a polytropic process

pvn = constant

∴w=∫v1

v 2 p1 v1n

vn dv

=p1 v1n∫

v1

v21vn dv

= p1 v1n¿]

W =mR (T 2 –T 1)

1−n

Polytropic index

For any polytropic process the value of n can be obtained by parameters p,v and T of any two points on the curve representing the polytropic process

taking logarithm on both sides ,

Some relations :

1.

2.

3.

The polytropic processes for various values of n are shown in Fig. on the

diagrams.

Internal Energy, Enthalpy and Specific Heats of Gas Mixtures

When gases at equal pressures and temperatures are mixed adiabatically without work, as by inter-diffusion in a constant volume container, the first law requires that the internal energy of the gaseous system remains constant, and ex-periments show that the temperature remains constant. Hence the internal en-ergy of a mixture of gases is equal to the sum of the internal energies of the in-dividual components, each taken at the temperature and volume of the mixture (i.e. sum of the 'partial' internal energies). This is also true for any of the ther-

modynamic properties like is known as Gibbs theorem. Therefore, on a mass basis

which is the average specific internal energy of the mixture.Similarly, the total enthalpy of a gas mixture is the sum of the 'partial'enthalpies

Many times the working substances encounter to be a mixture of gasses a particular example is that of combustion products expanding in the turbine of an open cycle gas turbine plant. The combustion products could be a mixture of

several gasses such as , , , in varying proportions.

To carry out a thermodynamic analysis of systems comprising gas mixtures it is necessary to know how the required thermodynamic properties of these mixtures could be obtained the mixture behaves like an ideal gas it self.

The mixture is homogeneous and non reactive.

Properties of Mixtures of Gases-Dalton's Law of Partial Pressures

Let us imagine a homogeneous mixture of inert ideal gases at a temperature T,

a pressure p, and a volume V. Let us suppose there are n1 moles of gas

moles of gas …….and upto n. moles of gas A (Fig).since there is no chemical reaction, the mixture is in a state of equilibrium with the equation of state

This is known as Dalton's law ofpartial pressures which states that the totalpressure of a mixture of ideal gases is equal to the sum of the partial pressures.

In a mixture of gases, if all but one mole fraction is determined, the last can be

Calculated from the above equation. Again, in terms of masses

from above eqs…

The gas constant of the mixture is thus the weighted mean, on a mass basis, of the gas constants of the components.The total mass of gas mixture m is

If µ denotes the equivalent molecular weight of the mixture having n totalnumber of moles.

A quantity called the partial volume of a component of a mixture is the volumethat the component alone would occupy at the pressure and temperature of theMixture. Designating the partial volumes by

The total volume is thus equal to the sum of the partial volumes.The specific volume a/the mixture, V, is given by

denote specific volumes of the components, each componentoccupying the total volume.Therefore, the density of the mixture