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The kinetic theory of gases

The kinetic theory of gases describes a gas as a large number of small particles (atoms or

molecules), all of which are in constant, random  motion. The rapidly moving particles

constantly collide with each other and with the walls of the container. Kinetic theory explains

macroscopic properties of gases, such as pressure, temperature, or volume, by considering

their molecular composition and motion. These particles have the same mass. The number of

molecules is so large that statistical treatment can be applied. These molecules are in

constant, random and rapid motion.

The rapidly moving particles constantly collide among themselves and with the walls of the

container. All these collisions are perfectly elastic. This means, the molecules are considered

to be perfectly spherical in shape, and elastic in nature.

xcept during collisions, the interactions among molecules are negligible. (That is, they exert

no forces on one another.)

The theory for ideal gases makes the following assumptions!

• The gas consists of very small particles. This smallness of their si"e is such that the

total volume of the individual gas molecules added up is negligible compared to the

volume of the container. This is e#uivalent to stating that the average distance

separating the gas particles is large compared to their  si"e. 

• These particles have the same mass.

• The number of molecules is so large that statistical treatment can be applied.

• These molecules are in constant, random and rapid motion.

• The rapidly moving particles constantly collide among themselves and with the walls

of the container. All these collisions are perfectly elastic. This means, the moleculesare considered to be perfectly spherical in shape, and elastic in nature.

• xcept during collisions, the interactions among molecules are negligible. (That is,

they exert no forces on one another.)

This implies!

$. %elativistic effects are negligible.

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&. 'uantummechanical effects are negligible. This means that the interparticle

distance is much larger than the thermal de roglie wavelength and the molecules

are treated as classical ob*ects.

+. ecause of the above two, their dynamics can be treated classically. This

means, the e#uations of motion of the molecules are timereversible.

• The average kinetic energy of the gas particles depends only on the temperature of the

system.

• The time during collision of molecule with the containers wall is negligible as

compared to the time between successive collisions.

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The thermodynamics of dry clean Air

&.- The rigin of the Atmosphere

The origin of the earths atmosphere is still a matter of speculation. /owever, most scientists

 believe, based on their knowledge of the gases in the universe, the earths first atmosphere

was composed of helium, hydrogen, ammonia and methane. thers believe that the first

atmosphere was probably released gases from volcanoes venting the earths hot inner core,

including 0&, 1&, /&, 1, and /&-. The earths second atmosphere can be traced to the

 planets heating and differentiation. 2t probably consisted of the same gases that are released

from volcanoes today! carbon dioxide, nitrogen, water vapour, and hydrogen and other trace

gasses. 3lanetary differentiation caused the lighter elements to rise to the outer layers of the

earth and initiated the escape of the lighter gases from the planets interior. The lighter gases

eventually formed the atmosphere and the oceans. 4or more information consult 5allace and

/obbs ($677).

2.1 Chemical Composition of the Atmosphere

The atmosphere is a mixture of solids, li#uids and gases. The gases in the atmosphere are

classified as either permanent (concentration remains constant) or variable (concentration

varies with time). The permanent gases include oxygen, nitrogen, neon, argon, helium and

hydrogen. The most abundant of these permanent gases are nitrogen (789) and oxygen

(&$9). The remainder of the permanent gasses and the variable gases exist in small

concentrations in the atmosphere. They are referred to as trace gasses. The atmosphere also

includes sulphur, chlorofluorocarbons, and dust and ice particles.

 0ote that the air is defined as a mixture of gases and that most weather is contained within

the troposphere. 2n this chapter we will deal with so called dry air, i.e. a gas not containing

water vapour. /owever, meteorologists often consider unsaturated air to be :dry;.

2.2 The equation of state of a perfect gas

<efinition

5hen a gas or vapour is so rarefied that the proportion of space occupied by the molecules

and the attractive forces between the latter are negligible, the gas is referred to as a perfect

gas. 2n real practice no gas can be exactly perfect, but under natural conditions, the mixture of

gases (dry air) is sufficient close to perfect for most meteorological purposes. The e#uation of

state for a perfect gas, involves the three variables p, T where

•  p denotes pressure in 3ascal (3a)

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• T denotes temperature in Kelvin

= denotes specific volume (volume>mass) or inverse of the density (mass>volume). 0ote that

the temperature T will always be assumed to be in Kelvin (K ? temperature in 1 @ &7+),

unless otherwise specified, and that 2 units are used in all calculations. The derivation of the

e#uation of state is by the combination of two experimental laws. The e#uation of state is

therefore not a law, but a relationship based on experiment.

Bets first use oyles law, pressure (p) specific volume ( ) ? constant$ which demands that

the temperature T be held constant, while pressure and specific volume (or density) is

allowed to change as follows!

3 ? constant, or 

 ? constant.

The second law is 1harles law, Temperature (T) specific volume ( ) ? constant& which

demands that the pressure remains constant while temperature and specific volume (ordensity) is allowed to change as follows!

This can be expressed as!

? 1onstant, or 

T ? constant.

The rest of the derivation is #uite easy if one remembers that we apply oyles law for the

change of state p$,  $, T$  (the initial state) to p&, $  (the intermediate state), with

temperature held constant and then apply 1harless law for the change p&, , T$ to

 p&, &, T& (the final state) with pressure held constant.

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xercise $

5rite down oyles law and 1harless law for the two indicated processes and eliminate the

intermediate specific volume . 5rite your resulting e#uation so that only initial state

variables appear on the left hand side and only final state variables appear on the right hand

side. Then, motivate that the e#uation of state can be written as follows!

 p ? %T or p ? %T!

The constant % is known as the specific gas constant and is measured at a volume occupied

 by a unit mass of the gas and selected pressure and temperature. The gas constant % for dryair e#uals &87 Ckg$ K $, by Dordon et al ($668). The e#uation of state is fundamental to the

understanding of thermodynamics of air (gases). This is not the only valid derivation.

3hysicists derive it using statistical physics. This e#uation expresses the three way

relationship between the state variables temperature, density and pressure of a gas. 2t may be

used to explain atmospheric behaviour if one of the variables is considered constant or it can

 be used to eliminate one of the state variables from a formula.

xercise &5hat is the density of a sample of dry air at the E-- h3a level if the temperature is &-F G

5hat is the density at the $--- h3a level if the temperature is +-F G (Answers! -.H6 and $.$E

kg m+)

2.3 The universal gas constant 

To derive the e#uation of state, we have used 1harless and oyles laws. Avogadros Baw

can be used for calculating the Iniversal gas constant. 2t shows that molar volume of a gas atthe same pressure and temperature will be the same for all permanent gases. Jolar volume

can be defined as a volume occupied by a mass of gas e#ual to unit mass multiplied by the

molecular weight of the gas e.g. $ gram molecule is m grams where, m is the molecular

weight. 1onsider a volume of gas with mass e#ual to m kg   and m a number e#ual to the

molecular weight of the gas. The volume occupied by this m kg  of gas is called the molar

volume. The molar volume changes with pressure and temperature.

1onsider the e#uation of state for dry air, with gas constant % d!

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 p ? % dT so that p? % dT ? %LT

where m is the mass of the molar volume and m numerically e#uivalent to the molecular 

weight. ut, noting that the units of is M N ? volume>mass, we may write m ? , the

molar volume and thus

 p ? %LT

%emembering that at the same temperature and pressure the molar volume is the same for all

gases (Avogadro), we can see that %L must be a universal gas constant and is given in

5allace and /obbs ($677) as 8.+$O+ $-+ C K $ kmol$.

2n general pm ?m%T ? %LT,

3roviding the relationship m% ? %L holds.

ubstitution in the e#uation of state results in!

 p ? ( )

2.4 Mixture of gases

y <altons law, a gas (e.g. air) occupying a volume , can be separated into its different

components, each occupying the same volume , each having its own pressure (partial

 pressure) and every component obeying its individual e#uation of state.

That is, if the partial pressures are p$, p&... ps, then

 pk  ? Jk % k TP k ? $, &,.......,s

5here the cubic meters of air contains the mass Jk  of each gas constituent. The sum

of the partial pressures gives the pressure of the mixture! pk  ? p and summing the above gaslaws above we get!

 ?

2f J denotes the total mass of the mixture, and % is chosen such that

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J% ? , or %?

then

 p ? J%T or p ? %T

5ith ? >J the specific volume of the mixture. #uation (&.8) is the e#uation of state

for a mixture of perfect gases and conforms to the e#uation of state of one perfect gas,

with gas constant % given by (&.7), pertaining to the mixture.

2.5 Molecular eight of !r" air 

The gas constants for the different gases in the atmosphere can be found in Dordon etal,

$668, and from section &.E the specific gas constant for dry air may be obtained by

considering the molecular weights and specific gas constants of the constituent gases. The

molecular weight of dry air can be defined using m% ? %L, where m denotes the molecular 

weight of dry air (md). 2t follows that

md  ? ? &8.67

2.# The first la of thermo!"namics

The first law of thermodynamics is expressed as follows!

d' ? dI @ d5 ($)

where

d' ? 1hange of heat (energy) of the system (d'). This is the energy added or taken from

  the system. Typical energy added by conduction, convection, radiation.

dI ? 1hange in internal energy of the system (dI). 2t can be related to the temperature of 

  the system and to the molecular motion of the substance.

d5 ? 5ork done on the system by the external forces (d5). 5hen a gas expands  it does work against the external pressure forces. 2t uses energy to perform this work.

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2f no energy is added or subtracted from the system (d# ? -) we call the process

A<2AAT21.

#uation ($), expressed per unit mass, becomes

d# ? du @ dw (&)

and using the e#uation for work expressed per unit mass dw ? pd>m ? pd P the familiar 

expression of the first law,

d# ? du @ pd

2.$ %pecific heats of gases

AJ (&---) defines heat capacity (also called thermal capacity) as the ratio of the energy or

enthalpy absorbed (or released) by a system to the corresponding temperature rise (or fall).

/eat capacities are defined for particular processes.

4or a constant volume process,

1v ?

5here I is the internal energy of a system and T is its temperature. 4or a constant pressure

 process,

1p ?

where / is the system enthalpy,

/ ? I @ p

a thermodynamic state function with I internal energy, p pressure, and volume.

AJ (&---) defines specific heat capacity (or specific heat) as the heat capacity of a system

divided by its mass. 2t is a property solely of the substance of which the system is composed.As with heat capacities, specific heats are commonly defined for processes

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occurring at either constant volume (cv) or constant pressure (cp). 4or an ideal gas, both are

constant with temperature and related by cp ? cv @ % with % the gas constant. 4or 

dry air at &7+ K,

1 p ? $--E.7 &.EC K $kg$

  1 ? 7$6 &.EC$kg$

and thus

% ? 1 p  1 ? &8H.7 &.EC$ K $kg$

4or moist air, the specific heat capacities of the dry air and water vapour must be combined in

 proportion to their respective mass fractions.

xercises +

$. Assuming cv ? 1v>m and 1v ? prove that

cv v

&. Assuming c p ? 1 p>m and 1 p ? , and the e#uation of state p ? %T, prove

that

1 p ? cv @ % 

2n an isosteric process, no expansion takes place, no work is done and pd ? -, meaning

that all the energy is used to increase the internal energy (and the temperature) of the gas.

The first law takes the simple form d# ? du ? cvdT! This provides a relationship between the

internal energy and the temperature of a gas for isosteric processes. 2n general

then, the first law is written as

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d# ? cvdT @ pd

%emarks

$. 2t is important to remember that the first law (&) expresses ($) per unit mass.

&. The specific heat at constant volume

cv  ( )v

  5hich gives the amount of energy (Coules) re#uired to heat $ kg of the gas by $ Kelvin

at constant volume and a value of 7$7 CK $kg$ is given for dry air.

+. The specific heat at constant pressure

c p  p

  5hich gives the amount of energy (Coules) re#uired to heat $ kg of the gas by $ Kelvin

  at constant pressure and a value of $--O CK $kg$ is given for dry air.

O. 0ote that c p Qcv since heating that takes place at constant pressure causes work to be

done due to accompanied expansion (d ).

xercises O$. ince the term pd= is difficult to deal with, eliminate it between the e#uation of state

and the first law of thermodynamics to give

d# ? (cv @ %)dT dp!

/int! first differentiate e#uation of state

2.& A!ia'atic process

2t is important to define the so called adiabatic process. This process simply means that

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 by and specific entropy s as entropy per unit mass. They add that the entropy of an isolated

system cannot decrease in any real physical process, one statement of the second

law of thermodynamics. Dordon et al. ($668) uses the symbol for specific entropy and

writes

d ?

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