# JPRS ID: 10432 TRANSLATION CHEMICAL THERMODYNAMICS OF COMBUSTION AND EXPLOSION BY B.N. KONDRIKOV

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JF'RS L/ 10432
2 April 1982
Translation
CHEMICAL THERMC)DYf~AMICS OF COMBUSTION
~1ND EXPLOSION
By
~ B.N. Kandrikav
FBIS FOREIGN BROADCAST INFORRJIATION SER~~ICE
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JPRS I;/10432
2 April 1982
Y
CHEI~ICAL THERMODYNAMIC~ OF COM~USTION
r AND EXPLOS~ON
MoG~ow KHIMTCHESr.AYA TE~lOD~NA'NlTK~, ~ORENZYA T VZRY'VA in Russian 1980
PP 1-79
jBook by B. N. Kondrikav, USSR Min~stry of Higher and Secondary
Special Education, Moscow Order o~ Lenin and Ordex of the Red Labor
Banner Institute of Chem3cal Technology imeni D. I. Mendeleye~:]
C ONTENTS _
_ -
Annotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Concentration. Stoichiometric Relationships . . . . . . . . . . . . . . . . 2
2. Functions of State. Some Thermodynamic Relatic.zships . . . . . . . . . 6
3. The Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1. An Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2. Real Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3. Gases at Ultrahigh Pressure and Condensed Substances 29
4. Chemical Equilibrium in the Products of Combustion and Explosion 46
4.1. Moderate Pressure . . . . . . . . . ~ . . . . . . . . . . . . . . . . . 46
; 4.2. High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5. Enthalpy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6. Calculation of the Composition and Thermodynamic Characteristics
of Combustion and Explosion Products . . . . . . . . . . . . . . . . . . . . 57
6.1. Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . 57
6.2. Composition and Z'hermodynamic Characteristics of Cc:~~+ustion
Products at Low Constant Pressure . . . . . . . . . . . . . . . . . . 60
6.3. Refined Calculation of the Thermodynamic Characteristics of
Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4. Thermodynamic Characteristics of the Products a~ a High
Pressure Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5. Expansion of the Detonation Products of Gondensed Substances 78
' APPendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
R,~co~nended L~terature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
lI - L?SSR - H FOUO]
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- ANNOTATION
This training .._.xnual w:~xamir.es, on the basi's of Soviet an~i foreign literature, the
_ fundamental problems associa~ed with the chemical thermodynamics of combustion and
explosion--energy release, the composition of combustion groducts and the thermo-
~ dynamic characteristics of fuels and the products of their chemi.:al conversion. ~t
provides simnle methods for calculating thermodynamic variables required for assess-
ment ~f the effectiveness of explosion processes, the danger of spontaneous com-
bustioii or explosion and the quantity of toxic products formed by combustion and
detonation. Special attention is devoted to the state e~iiation for the initial
substances and explosion products at high temperature (thousands of degrees--and
ultrahi.qh preseu~:.--:~undreds of thousands of atmospheres).
FOREWORD
The principal unique feat~xre of combustion processes--that which nredetermines their
technical applications and the danger of spentaneous fires and explosions--is the
release of a significant quantity of energy. Hence follows, in particular, the
- primary significance of the thermoc~:ynamic aspects in the study of these processes.
At present, however, there are no training manuals whi;;ii~night examine the theoreti-
cal problems of combustion and explosion thermodynamics and the mefihods of calcula-
ting the principal thermodynamic variables ~rom the standpoint of the requirements
1
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" of institutions of higher education special:x_ng in chemical technology. Some books
dealing with these problems, for example the rem~~rkable textbook written by Andreyev
and Belyayev (1), have long been a bibliographic rarity while others are too voluminous
and difficult or fail to satis=y our instruction programs.
The purpose of this training manual is to corre�:t this situation. Ir! addition to
examining the commonplace ti~.ermodynamic relationshi~s and simple calculation methods,
it devotes significant attention to one of the most important problems of the theory
of explosive transformation---the state equation for matter at high pressure--tens
and hundreds of thousands of atmospheres. In the final analysis the objective i~
to make it possible for every process engineer to calculate the composition and other
characteristics of. the products of chemical reactions occurring at high temperature
and ultrahigh pressure, something which is now within the means of only a few special-
ists having access to expensive computer technology (2). Some particular paths of
solving this problem are already known, and they are described in this book. We will
probably ilot have to wait long at all for someone to arrive at a~eneral solution
similar to that reached for combustion processes at moderate pressure (3,4). Perhaps
some of the students for whom this guide is intended will themselv~s be able to take
part in reaching ~his solution.
1. Concentration. Stoichiametric Relationships
In physics, the principal characteristic describing the concentration of a substance
in a given point in space is density p(kg/m3). The reciprocal, V= 1/p, is called
the specific volume. If the substance is an individual chemical compaund, the pro-
duct VZM~ (m3/mole) is its molar volume. Here, VZ = 1/pi is specific volume and MZ
is the molecular weight of substance i(kg/mole).
2
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In -~hemistry. concentration C is usually given as the number of particles (molecules,
atoms, radicals, ions) per unit volume (1/m3, moles/m3, moles/liter). There are
other ways of expressing concentration involving simple relationships of p, V and C:*
e~ n~
Proportion of siibstance in mixture = C[ M:,V =
by weight ~kg2/kg ) - - - ( I.I . )
Q~..~H
p P . ' ' (2.2.)
Molar ro ortion (moles~/mole) �~H; 2'Ct K?
C?~~ v__~_, ? ( I.3. )
Proportion by volume (m32/m ~ w~~ - 2`~ N~ ~9`
Number of moles of the substance -C ~ ~ ~(1.4.)
per kg of mixture (moles2/kg) n` ` i� N
ti .
Here, M= 1/E(g2/MZ)=p~EC2 is the mean m~lecuiar weight of the mixture (kg/mole).
_ Symbol E means summation in relation~to all of the components of the mixture, or
in relation to a certain group of the components, for example just the gaseous pro-
ducts. Thus when we compute the composition of combustion products we use
- E (1.5)
No gas Ni
where summation is performea only in relation to the gaseous components of the
mixture.
If when the components are mixed together they do not interact in such a way to
cause a change in volume, then
~ ~ ~ ~ J Y.'
V = i ~ ~ ~ J~~ ~ "`1+: "Ci~V~" ,`j ~I.o.)
*A convenient way to check the corbectness of formulas (1.1)-(1.4), or formulas
similar to tHem, is to consider the units of u~easurement pertaining to a given
component of a mixture of substances. Following rec~uctions, the units of ineasure-
_ ment must be in tciie t~ight combination. For example for formula (1.1) ,[g2] = kg2/kg =
_[ivjZG'ZjI] = kCJZ.molesi.m3/molesZ.m3.kg. It should be considered that in formula
ct.a.~
(1.4), ~ ~j. ~ It stands to reason that following s~ach verification, the
Z ` ' ' M 3
units of ineasu~emeiit sh~uld be reduced to their conventional form.
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Relationships (l. 6) are always satisfied for mixtures of ideal gases . It would not be
difficult to note that the sums Eg2, Eajy2 and Ea~Z in relation to all components of
the mixture are equal to unity. Obviously g and a may also be expressed as percents.
When we use the theory of combustion and explosion to solve thermochemical problems,
first we usually determine the elementary (eTemental) composition of the mixture.
In this case the concentration of element n3 is determined in moles/kg by the formula:
l9;rj= Mj J�~~~ (I.'1.}
where is the number of atoms of the given element in the molecule of compound ~j
= C, O, H, N etc., moles/mole). S~nmation is performed in relation to all compo-
nents of the mixture. 7.'he variable g~j/M~ = n~ represents the number of moles of the
given compound per kg of the mixture. On calculating the amount of all [symbol
omitted; possibly n3] elements of which the mixture consists, we get the "empirical
formula" for the mixture:
C~~Un~,~H~w
the molecular weight of which is 1 kg.
Most systems capable of combustion and explosion contain a fuel and oxygen, The
- ratio between these ingredients is an important characteristic of the system. Several
_ ways of expressing this ratio are co~nonly employed in engineering.
In the case of secondary explosives, the best suited variable is t.he oxygen balance
(K~S)? expressed as a percent by weight. It is the difference between the quantity of
.
oxygen ..ontained in the syst~~m and the quantity necessary for complete combustion--
that is, for transformation of all carbon into C02r all hydrogen into ~T20, all
aiuminum into .~.1,203 and so on. If halogens such as chlorine or florine are pr~sent
in the syszem, it is believed that they interact with hydrogen to form hydroc~en halides.
_ 4
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K~ _ ~ ~n(o)-12n~ t n(2-n� }z ,1`,n~~~~) ir.e)
Coefficient 1.6/M in this fonnula is obtained by dividing the molecular weight of
oxygen ~0.016 kg/mole) bi% the molecular weight of system M and multiplying it by
100 so that tne result would be a percentage.
The oxygen balance of a mixture is an additive function of the oxygen balances of
its components. We can easily persua3e ourselves of this by stunming the products
g~Xd and applying formula (1.7). The oxygen balance of the mixture would be less
than zero if there is not enough oxygen in it for complete oxidation, and grpater
than zero if exceas oxygen is present. For a system that is "balanced in relation
to oxygen," Kd =
The coefficient of excess oxidant, ap~is the ratio of the quantity of oxygen in the
system and the quantity needed for complete combustion. For the purposes of this
handbook, in our calculation of ap we will consider only the amount of oxygen re-
maining following combu~tion of aZuminum to A1203 and lt?g to MgO, and we will consic3er
only carbon and excess hydrogen (following its interaction with halogens--chlorine
and fluorine) as the combus~tible substances. The formula for calculating ap would
have the form:
r~0---1 ~`�n.A~+n,~y)
~ s ~ 2n~ + ~i~ H _ ~~,r~.a, ( I . 9. )
When the system is "balanced in relation to oxygen," ap = 1, when excess oxygen is
present a~>1, and when axygen .is lacking a0..~a. , c~.~, ~
Isentropic compressibility S S(see formula (2.19)) is used to calculate the volumetric
speed of sound
CO =~r~~)sa-VZl~~3V/ISs,/l~s~~ (2.z6)
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~ .
The speed of sound in an ideal gas is
~'P~ ~ d'~T -
~o � - (z.s~
_ The velocity oi longitudinal waves in an organic liquid, also determined by equation
(2.26), may be calculateci using Rao's rule: '
3
. `o � F ~ j d~ ~ ' m/~sec ~ (2.28~ .
- where pp--density of the liquid, kg/m3; M--its molecular weight, kg/mole; ZZ--number
of chemical bonds of the give.z form; BZ~�-contributions of these bonds:
C-H 95,�_ C-N 20.?~ N-H 90,7
C-C 4,25 C=C I29 C-1102 302,5
C~ 0 34.5 C= 0 I86 0- 1102 360
C- 0. A1+,S 0-~H 99 M- p02 330,
(ether)
The speed of sound (of longitudinal waves) in solid organic substances having the
density of crystals may also be estimated by formula (2.28) in the absence of experi-
mental data.*
When we use the speed of sound, we can supplement relationship (2.20) by yet another:
C~ = CP
! f ~T,(~ ~i
*It should be remembered in this case that waves prcpa.gating in isotropic solids may
be both longitudinal (compressive strain) and transverse (shear strain). The speed
of sound in a thin plate and in a thin rod differs from the speed of sound in a
boundless meiiium and from the velocity of surface waves. The calculation formulas
may be found in handbooks (see for example (7)). Estimates may be arrived at very
conveniently with the formula C= where ~--Young's modulus and p--density of
the solid. This is the speed of sound in a thin rod, close to the longitudinal speed
of sound in a boundless medium.
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Here, Cp in the denomi.nator is the specific heat (inasmuch as ~he sc~uare of the
speed of sound has the units of specific energy).
~
Table 3 gives the values of Cp, Cy, ap and Cp for a number of substances.
Table 3. Density, Specific Heat, Coefficient of Thermal Expansion and the Speed
of Sound for Some Organic Substances
_ po 3_ C~(3 C�~3. It 3 'y c(4j
~ Be~d~"~rDO ~ r/~~ ~ Kf.~ ac/ ~ K- IO Q 2:17 o~dre(6y
. Tpus~:. ( 293H) I,663 I,373 I,243 I~I05 0,32~ 2 95
(354K) I.470 I,604 0,983 I,63 I,OS ISI6 I600
Talpwa ~8~ I,?3 0,942 0,6I6 I,IS ~ 0'.32 2I9I
Peucnren (9) 1,,8I6 I,248 I,I35 I,IO 0,25 262T
= Tau (10) I,7?3 1,67 I,33 I,26 0,5 243I
Hxrpr.;~aNUepaa (11 I,60 I,67 I,22 I,3? 0~85 I7I4
Haspurar~uona (12) I,49 I,67x~ I,26 I,33' 0.85 I604 �
Yer~nadtpo= (1 3) I~2I 2,035 I,45~ I~40 1~3 I275
Iiexpove:ea (~4) j,i3 I,74P I,223 I,42 I,=: I293 ~6
~ Hespo6eaaon ~i5) it2p7 I,SII I,I68 I,293 Q,83 I540 I473
ioayoa (16) 0,867 I~704 I,235 T~JB I,I I330 I300
Yerencn (17) . 0,79I 2,5?I 2~I9? I,1? I,2 III2 II22
- 7Giopo~opw (~a) I~489 0.967 0,653 I~48 I.23 9~9 995
9eee+pe~acuupxcxyl~~ I,594 0~858 0,592 I,45 I,236 866 930
yrne oA ,
fipo~o~opa (2U) ~ 2,85 O~SIS 0.289 I,?8 I,27 94t 908
*The value of Cy for nitroglycol is assumed to be the same
as for nitroglycerin; formula (2.23) for an ideal gas
produces close values for these two substances.
Key:
1. Substance 11. Nitroglycerin
2. gm/cm3 12. Nitroglycol
3. kj/kg�OK 13. Methylnitrate
4. m/sec 14. Nitromethane
5. From formula (2.17) 15. Nitrobenzene
6. Experimental 16. Toluol
7. Troi:yl 17. Methanol
8. Tetryl 18. Chloroform
9. Hexogen 19. Carbon tetrachloride
10. PETN 20. Bromoform
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The system's entropy may be found by equation (2.1):
a~8� dE f ~v, ~~Tt~v~1v~,~ay=
T r~ T ~d~ T
- ~ _ ~ f,(~)~Jv = .-t . .
In addition to (2.1), we use equations (2.13) and (2.14) ~~o write these~ualities.
Integrating (2.30), we get:
r
.4_S,=~~~~.,rf~~T~Jv ~2.~c)
The dependence of specific heat Cy on temperat~:re is obtained experimentally, or
it is found from formula (2.23). 7.'he derivative
(~p/~r)~ - ~P= ~r/J4 Cs.~.)
The entropy of an ideal gas (or mixture of gases) may be calculated using equation
(2.2).
~ d.t� ~+-~P- ~'r~~ ~'PTT a~ p~.3~~
or f
s~~~~ ~~~'~,o~,P~iTr1Q~J ~~M/~ ~Z.~,
T T~ ~
'in an isobaria process,
. ,~T - sr� - rPPJ~ r tT.~ .
f. .
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in an isothermic proce:;s,
~ ` f� ~~n~~~_~-7 ~n p_
~ ~'.r ~'r ~ a ~ ~ ' ~ Pp ( Z .:i5)
~
�
In an adiabatic (isentropic) process,
N s e e p ~
f(~~,~~~i~ l k~1 ~~..~P~P~ ~ t�r' T~T." sr gl- (2.~6)
r, � .
It follows from relationships (2.34)-(2.36) that when adiabatic expansion of a gas
occurs, pressure may be calculated if we know the entropy of the gas at atu?ospheric
pressure and at a temperature corresponding to the given degree of isentropic ex-
pansion:
p e ~
Cr~ ( ~~o = ~�-~o ~Sr - s (2.37)
5~ is the entropy of an ideal gas at the given temperature, and pressure Pp = 105 Pa
is called the; standard entropy. It follows from (237) that change in entropy in
an isobaric process may serve as a measure of pressure change in an adiabatic
process.
At Np, Cp = const. in an adiabatic isentropic process,
H P~,~~/1?,~ A~ ~n(Tn'~) ~ar P/Pa =~~T/`1'o)~P/R
Considering that Cp = YR/(~y-1) and P= RT/V, we can write
Pm~ a ~T~o~d%~'d~i) ~V )r (2 ~311)
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- This is Poisson's adiabatic equation.
To complete our list of basic thermodynamic relationships, we need one more function
, of state that is often used in calculations associated with combustion processes--
the reduced thermodynamic potential.:
� ,
(~J " ~ ,!/i_` ~i
� T JT T . (2.39)
As with entropy, the reduced thermodynamic ~tential is characterized by the fact
that it converts to zero at absolute zero for most substances.
3. The Equation of State
3.1. An Ideal Gas
Before writi.ng the equation of state for an ideal gas--one of the fundamental laws
of nature*--we will examine partial gas laws from which it was obtained. These
laws were obtained as result of physical measurements made over a period of
about two centuries, and as we know; they =se of interest on their.own. We will
adhere not to the chronological but to the logical sequence of presentation and
to the commonly accepted modern terminology.
1. Dalton's law (John Dalton, 1766-1844): The pressure of a mixture of several
nonreacting gases is equal to the sum of the partial pressures of each of these
gases. ,
*It would be interesting to note that despite the fact that this law is so funda-
mental, the equation of state fdr an ideal gas is purely approximate. From this
standgoint this is a typically technical or technological law. Incidentally it
does en~oy broad use in engineering.
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2. Law of multiple proportio~s (Prout, 1~99; Dalton, 1808): Voltunes of gases
entering into a reaction relate to one another as small whole numbers.
3. Avogadro's law (Amadeo Avogadro, 1776~1856): Equal volumes of gases at the
same temperature and pressure contain the same number of molecules (at given P and T,
the molar volumes of all gases are equal). ~
4. The Boyle-Mariotte law (Boyle, 1662; Mariotte, 1676): At constant temperature,
the pressure of a certain mass of gas is inversely proportional to the volume of the
gas.
5. The Gay-Lussak law (Joseph Louis Gay-Lussak, 1778-1850): At constant pressure,
the volume of a gas is proportional to its absolute temperature. ,
6. Charles' law (1787): At constant volume, pressure is directly proportional to
the absolute temperature of a gas; (a different interpretation is: The pressure of
a certain mass of gas, when heat.ed 1�C at constant volume, increases by 1/273d of
the pressure at 0�C).
Let us write laws (4), (5) and (6) for 1 mole of a gas:
(4) at 'P z C[~;~1t f~= ~j / ~M
) at P x~ d%1.S ~ V~y s r.+( t T
_ (6) at V,y::L'D/lS~' � ~(q T"
where Vjy--molar volume, al, aZ - a3--constants.
Note that inasmuch as at given pressure and temperature, according to Avogadro's
law (3) the molar volume of all gases would be the same a nd constants al,
a2 and a3 would be exactly the same for all gases.
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Let us obtain the appropriate partial derivatives from relationships (4)-(6):
r~J%' t' , ?!~M y~_ r~~~ _
\ ~~H~= --?N' ,vT~,= 7. foT - r c3.t.) .
~ , ~ r
. T / M
We immediately see that these derivatives f:~rm a general thermodynamic identity:
~ ~ni% j 1
I, ~D?/l, 'T' ~i (~'t~~ .V/N .
f
But this means that the pressure, molar volume and temperature of ~ gas are asso-
ciated with each other by a single functional dependence common to all gases:
_ ~ ~ Vrl ~ r~ " ~f or - ( ~ , ~ l 3.?. . ) .
The form of this dependence is easily found by differentiating (3.2):
c, c1 VM -f/ ~1~-'~ c+ T,. t, o/' c~ l
C' ~~'D-v~)~ ` 7 i/`~M I
and integrating the resulting expression:
�Lil~ 6i~ V~ rfi~ ~~l~v~~S~ or f'`i~~~j= Li~;~st -~1"~, (3.i.1
~
Thus we arrive at the famous Clapeyron law--the equation of state of an ideal gas.
Constant Rp may be obtained experimentally; it is the same for all gases.
Rp= 8.315 Pa�m3/(mole�OK) = 8.315 j/(mole�OK) = 1.987 cal/mole�OK. Substituting the
molar volume by specific volume, we get
P~,_ R~r - c3.3,~~
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where R= Rp/M--specific gas constant (different for different gases); M--molecular
weight of the gas.
If VC is the volume of the vessel and M is the mass"of the gas within it, then
VM ~ V I~1~r>> ana ~l c-~~'~ k I .
( 3.3,~)
This is the Clapeyron-Mendeleev law.
Inasmuch as 1/VM = C is the concentration of a gas,
_ ['~u I ( 3.sC)
or for a mixture of gases,
f~~ - A','I (~.5~[~
where PZ is the partial pressure,of the gas.
In accordance with Dalton's law P~EPZ, since C= ECZ (the total number of molecules
in a given volume is equal to the sum of the numbers of molecules in the different
gases). Other relationships for partial pressure are:
1 ~fy~.~' ' ~n'' %i~~i ( s.a.>
"~P ~
Partial pressures enter directly into the expression for the equilibrium constants
of an ideal gas:
,
- ~r- IJ/~,' ' a '~i."
_ h;,_ ~ ~er = ~~,,.~J. -A, ~s.y.)
Here ]T--multiplication symbol; v2 and v~--stoichiometric coefficients of corres-
pondingly the reaction products (the right side of the reaction equation) and the
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initial substances; ~G~,and ~~g*--change in Gibbs' -:iergy and in the reduced thermo-
dynamic potential as a result of the reaction:
M
A ~r T ~ a~~~- as,-
3.2. Real Gases
3.2.1. Noble's and Abel's Equation
When the temperature of combustion products is high (1,000�K and above) the equation
for an ideal gas may be used with relatively small error (up to 2 percent) at
pressures up to several hundred atmospheres.
At higher pressure (up to several thousand atmospheres) we use a simplified
van der Waa]s equation*:
~
/ _ _ r
r.~ ( V-. ~c l' , orr R T ~R j y ~ 3.6. )
where V--specific volume of combustion products at pressure P and temperature T;
Vg--so-called covolume, which accounts for the volume of the molecules themselves
in gaseous products (Vg is about four times as large as the total volume of gas
molecules) and, in the case of gaseous suspensions, the volume of the condensed
substance. Vg is expressed in cubic meters per kilogram of explosion products.
If ~Z is the proportional weight of condensed product i and Vp is the specific
~ volume of gases under normal conditions, then
(1 /I~~~ ~o * ~ (i.7.) .
*We should note the fact that the first to discover and explai:~ the difference in
behavior of real and ideal gases was M. V. Lomonosov; this is sometimes called
Lomonosov's equation (8).
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The following terms are introduced for the internal ballistics of gunbarrel systems
(in relation to which we mainly employ equation (3.6)):
~
~l 7;, - n ~ I~?' = v
Ty--temperature generated by combustion of powder in an enclosed space in adiabatic
,
conditions. The variable fn, which is expressed in the units.of specific energy,*
is called the "powder power," and variable ~ is referred to as the "loading density."
Equation (3.6) takes the form:
a f~ (3.8.)
~J ~'r a
Formula (3.8), which was obtained experimentally at the end of the last century,
is called the Noble-Abel equation (sometimes Abel's equation).
If gas pressure is greater than several thousand atmospheres the dependence of
covolume on pressure must be considered. Assuming that covolume is a function of
just specific volume alone at high temperature, M. Cook obtained a single dependence,
VK(V), for the explosion products of 14 explosives in an interval of V values from
0.2 to 1.4 cm3/~cn (a specific volume of 0.2-0.4 cm3/gm--that is, a density of 5-2.5
gm/cm3--corresponds to the explosion products of lead azi~le, mercury fulminate and
mixtures of trotyl with a large quantity of lead nitrate). The dependence VK(V)
can be expressed approxi.mately by the following formula (Johanson and Person, 1970):
~/K ` K u e _ V / . ( 3 . 9. )
*At constant heat capacity fn =(Y-1)E, where y= Cp/Cy; E--specific intrinsic energy
of the gas.
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There are two constants in this formula that are expressed in units of specific~
volume:
VK~ = 1. 0 cm3/gm and VQ = 0. 4 cm3/gm
3.2.2. Equation of State With Virial Coefficients
If we write equation (3.6) in the form
, ~
_1 _ . . r . i ~
~c _ = ~ .r VL~ .r + . � 3 . i0. )
~i'T- l~ 1 I- 1'~c/~~ V f
(we use the expansion (1-x)-1 = l+x+... at x�1 to obtain the latter equality),
tti e
it is easy to see t,~at relationship between the Noble-Abel eqaation and an equation
of state for a gas containing virial coefficients:
;
T ! -r f ~ j . . ~ ~
- E Vi 'f� , (3.II.) .
obtained on the basis of Boltzman's virial theorem (the theorem on expansion of
~ the function of state into a series in relation to small powers of density).
Here, B1, B2.etc. are the second, third and subsequent virial coefficients depending
on temperature and not depending on pressure. The second virial coefficient in the
model of rigid spherical molecules is equal to the covolume (that is, it is �our
times larger than the volume of the molecules themselves), and the third and fourth
virial coefficients are proportional to the corresponding powers of the second:
2 3 4
: ~ 82 1 0,6~5 ~2+C~?d~~.~is~id-~{s.12.)
r . . .
The second virial coefficient in this equation, B2, is obtained by summing the con-
tributions made to its magnitude by gaseous explosion products:
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Table 4 gives values for coefficients B22.
Table 4. Second Virial Coefficients for Gaseous Substances, B22, cm3/gm, at 3,500�K
_ _ .
Mli~ NU CU~ _M. CU M. U N- U~. CII~ I~
15,~� 2I,2 3'1,0 54,0 3~~0 63.9 I4.U 3U,~ 3?~D '1,9
3.2.3. The Becker-Kistyakovskiy-Wilson Eqt~ation
Substituting B2/V = X in equation (3.11), we get (ignoring all but the first three
terms of the polynomial)
~ ~T = 1 -I~ X~7~-~'~ ~A') - l-r Xt'~/x ~S.I3.)
Here S'= B3/B2; the latter equality is obtained by using the expansion E~~X=1+s~X.
When we use the potential for interaction between molecules in the form
n, , n~;
~'r, : ~,,1,~ - !t ~ , r
the second virial coefficient may be expressed in the form
t, h, - ~
J
~ ~
where K is proportional to the covolume and represents the sum of the products of
- the corresponding values of each of the gaseous explosion products times their
molar proportions in the mixture. In order to keep pressure increasing to infinity
as the temperature tends toward zero and keep (aP/8T)y positive, constant T' had
to be added within the range of specific volumes of interest to us. The expression
for X is found to contain three constants, the selection of which must be made
with a consideration for experimental data (the fourth constant contained in
equation (3.13) is S
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~
~ _ ~ n k:
Y 1 Y 7'~''~' v ~r T~'~ ~ ( 3. I4. )
Equation (3.13) with parameter X, given i.n form (3.14), is called the Becker-
Kistyakovskiy-Wilson (BKW) equation. It is broadly employed in the USA (2) to
calculate the detonation characteristics of explosives. Good results were obtained
for trotyl and hexogen using the constants shown in Table 5. ~
Table 5. Constants in the BitW Equation of State Used to Calculate Detonation
Characteristics of Trotyl and Hexogen
- - _ _ . - -7~-- � - -
~ ~ ~ d i ~ i._.._.-- _
~
- - . _ .r N,c, . . cc~t _ co _
_~T 0~(~96 12 ~ 69_ ii ~ So q00 250 600 390 380
Hexogen p116 . 14s~I
3.3. Gases at Ultrahigh Pressure and Condensed Substances
- OnP sho?-tc-oming of all of the fo7ms of the equation of state given above is that
they do not contain the cold and elastic components of pressure and intrinsic
. energy. In all cases except (3.13), at T= 0 pressure becomes zero independently
of V~--the specific volume of the substance. But the theory of the structure of
matter indicates that at absolute zero, pressure and intrinsic energy are functions
of volume, and when the latter changes (especia.lly when it decreases) they change
a very great deal.
The general form of an equation of state taking account of this circumstance was
given earlier as relationships (216) and (217):
/~-P~~f~r ana ~`=EX~Er
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where PX and E~y--potential and elastic or "cold" components of pressure and intrin-
sic energy, associated by the relationship ~cl E,~;~V= ~D~'~~~~~~ ; pT and E~-
thermal (that is, depending on temperature) components of pressure and intrinsic
energy.
The dependence of the elastic energy of a solid on specific volume (Figure 3) has
qualitatively the same form as the curves describing the dependence of the energy
of interaction of two atoms in a molecule on the distance between them. The depen-
dence of elasti^. pressure on specific volume is also shown in Figure 3.
The value for specific volume at T= 0�K, VpoK, corresponds to mechanical equilibrium.
Cold pressure at this point would be equal to zero.* The forces of attraction and
repulsion balance each other out. Elastic energy is minimal in this case. ~
When a substance is heated, it undergoes thermal expansion: Thermal pressure,
which is always assiuned to be positive, arises. Elastic pressure becomes negative
in this case: It compensates for the action of the stretching forces that increase
specific volume. In terms of absolute value,
I~xl=1~'=~�
Negative pressure can be estimated from the heat of sublimation of the substance.
By definition, the area beneath the curve representing cold expansion of a solid
between zero volume and infinity is equal to the energy of sublimation:
j j~ . aV
E~ J Px ~ V" ' M kcux
VO~f
*We ignore external (atmospheric) pressure when examining condensed substances.
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Assuming that bonding forces diminish significantly as the interatomic distance
doubles (that is, when OV=10VpaK), we get
1 ~~x k~ur ( ~ i ~ ~U V~K .
For iron: ES = 94 kcal/mole = 7.0�106 j/kg
For alumintun: ES = 55 kcal/mole = 8. 5� 106 j/kg
This means that for iron and aluminum, ~PX~~ is, respectively, =6�109 and
=2.5�10-9 Pa (that is, 60�103 and 25�103 acm).
The forces of repulsion, which increase dramatically as atoms come closer together,
play the main role in relation to compression. Therefore when V ( /a,~~, (P, /n,~ (!',/n,~
~or a given type of oscillations (longitudinal for example) the ntunber of possible
wavelengths greater than ap would be equal to the number of positive whole niunbers
Z1, Z2r Z3 satisfying the condition:
t - ~ ~ . 2
P~ F, ?
i ~ ~ ti:
l i~~) ~lTs (~.i~ (.1~.~ ,
that is, it is equal to the quantity of integral points within an ellipsoid:
2= /
j fP,. ) ~ + ~ l .~a/ '
l l (
the coordinates Zi of which are positive whole numbers and the semiaxis of which
is bz = 2 az/ao. The volume of such an ellipsoid is 3
blb2b3. If we divide the
ellipsoid in Cartesian coordinates by planes xlx2, xlx3 and x2x3 we get eight
octants, the volume of each of which is (~r/6)blb2b3. One of these octants
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corresponds to positive values of b2 (the one located in space (Z1,Z2,Zg)~0).
The number of possible wavelengths greater than ap is
r?~r~t . y,f V
~ - ~ ~ - - ~ ?
where V is the volume of the box. Thus the number of stationary longitudinal
waves is directly proportional the volume of the box V= ala2ag. The velocity of
a wave in an isotropic medium does not depend on direction, and when ~p� (al,a2,a3),
it does not depend on wavelength either. The relationship between the velocity of
a longitudinal wave on one hand and frequency v and length ~ on the other has the
conventional form CZ =~v (a quantity v of waves with length a would fit within
CZ). The number of possible frequencies less than vp in this case would be
3~ � L'g 3 V,
Similarly if the velocity of all transverse waves is C~, then considering the
existence of two independent transverse oscillations, we find that the total number
of frequencies less than vp is
s � s � ~'j
. . ~1:_ f ~ ~ ' '~C a � L � f
~ ;r ~ ~ '~~~Y,
3 ~ 3
Q
~
where s~ 3( i c~- ~ ) --average velocity of waves for oscillations ~of all
c
types.
Differentiating expression (3.15), we get the number of simple oscillations with
frequencies in the interval between v and v+dv:
1?:~~~ (S.IE~.)
l.' ~ .
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Let us interpret each such oscillation as one of a harmonic oscillator, the zero
energy of which is adopted as the energy of its lowest quantum state. The oscilla-
tory statistical sum of such an oscillator would be
~ ~hd/Kr -r
~KVA. ~ ~ - ) ' ( S .1'l . )
The sum total in relation to the states of all oscillators, Z(T), is the derivative
of a function having the form (3.17), the powers of which are determined from (3.16):
w : . ~1~? ,)~1.1.
:J Y
~ ~ - ( ~ - ~ ~ s . 1 ~i . ~
Expressing (3.18) as a loga~rithm and correspondingly substituting the product by a
sum and the sum by an integral, we get
1?,1 V ~ t~ , d~'~Kf
.t~, ( ~I ) ' + .f J t~n - t' ~ s. [y. )
0
We integrate in (3.19) in relation to all stationary frequencies from 0 to v~ (in
compliance with Debye's theory we assume that there are no frequencies higher than
v~) . �
A real solid consisting of N material points may have not more than 3N oscillation
frequencies. Basing ourselves on this condition, we can estimate v~. We find from
(3.15)
4zv~~' a .~e_;
~ ~V" or = y ~
Z' ( 3.2tl. )
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This frequency is characteristic of the given elastic medium. It is called the
Debye frequency, and it depends on the number of particles per unit volume and on
the mean velocity of elastic waves. Computations show that at v< v~ the wavelength
corresponds to several thousand elementary cells, and consequently interpretation of
the medium as an elastic continuum is fully valid.
Let us in'troduce Debye's temperature:
(.~N 1 (3.2I.) .
R;D ~ K^ ~ K~ y ir'~ _
The oscillatory sum (3.19) takes the form:
H~, d~/T
~?~~T~~~-~~-Il~~~~f-~ ~~.~7.-y~(f;~l f~e~~~-Q~~~3.~~.,
v
where 'l " ~ / ~o ~n~l ~ = ti ~j
k' 1~= ~.~y~T are integration variables .
Statistical physics providesthis expression for free energy, F(T):
f~ (r)_ -K~'~, ~ (r) c~.z~.~
Ignoring the contribution made by free and bound electrons and the orientation of
nuclear spins, we get the following f~r the free energy of an elastic solid:
It~ d1~
i" = t~ ~ E ~~~~r f
p b~~r- e')~~ c~.z4.)
t�~�,,~�~x is the contribution made by the potential energy of atoms in zero oscilla-
' tion state at T= 0�K. Using (3.16), (3.20) and (3.21) we find the contribution made
by the energy of zero oscilla}ions:
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vn vr a� a~.~~aa. vva: w~ a. �
( :.G:1.) N~(5.~1.)'
~ . ~ ; ~ ti ~':,!,~~r `i~ . - ~ ~ K~~; c
C ~
,7
For many solids, energy EHK is small in comparison with crystal bond energy. It
plays a noticeable role in molecular crystals, and it is so great for helium that
at normal pressure, even at T= 0�, helium remains liquid.
It follows from the thermodynamic identity (2.4) that
,
' ~ `D ~r~~} ~D~ (3.:'_c.
/ - ~
~ - ~ ~ ~ ;--1,. = - T ~i--; Y ; = - rD J
\
Keeping in mind that En, EHK, vp, v and consequently n do not depend on tempera-
ture, we find from (3.24) that
~~~-fr1g u'~T ~~i f ~ , f ~7'a'7
~ ~ . 1 ~ . ' j ~ . fdl~' ~ ~ ^E~~ ~ /T ` ~n ~C'll~ ' yA'~' ~ ~ v; ~jr 1
` ~ ! 1
ox' _ t.{ ~ i i r.S4'TD/��!~ ~ (�~~A `K ~ (3.?'/
' A ~
`I'
where f, -,i .Q 1~.~ ~ da i 7;) ( 3.28;
is the component of intrinsic energy contributed by thermal oscillations of particles
at the junctions of the crystalline lattice.
D~'t~~ ~ _i(~ )i ~ ~j~tf.r~~l (3.29)
~ ~ � t f~,, ; , � :
is Debye's function (see (5)).
At low (UD/T�1) and high (OD/T�1) temperatures expression (3.29) is integrated
analytically. At low temperature Debye's function D=(T/OD)3. At high (OD/T�1),
D= 1. In the intervening temperature interval ( 3. 29) can be integrated numerically.
We calculate pressure using (3.24) in correspondence with the second expression
of (3.26) :
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, J` . ~'G~iy ~ ~r ~
� . . . _ . , ~ ~ . ; _ ~l'~', r. t> ; !~~l ~
� ~~r7) ~ . /
. 1! . ~ �j~,Qi ' ~~0 ~ r ~I{,Y%~i,J~ " -
.l{~~ .P ~ ..~q~;~~ ~ ,~iJ ir/~'i~~/ ~
Ib !_I _ ` /
A I ..1..__._.
Nj~ ~f~�r ~ ~~1 ~a~:~~./L~IZ~~ , r'~ ~ Q)~o rl- !
or �
? ~or`J~ ~ ~ ~ / ( iil)
_ f' - r s~' D i~l, l 7 l ~ I;,~ F~. t~' ~ f� ,
, e ~ ii�
where
~ ~ ~D s -J�-- (3.31)
~ is Gryunayzen's parameter, a dimensionless variable depending on volume and (in tlie
approximation under exami.nation here) not depending on temperature.
Formula (3.30) or the Mi-Gryunayzen equation, written using the symbols of (3.28),
(3.29) and (3.31), is the sought equation of state.
Pressure resulting from zero oscillations of a solid is
~My 9~e,r _ B R~ g~ ~
~
Under normal conditions this pressure attains several kilobars, and it increases
as the volume of the solid decreases. Correspondingly we can also obtain, for a
solid, the contribution made by zero oscillations to the modulus of ~ubic com-
pression:
k N~ ~-~~~VNK ~ ev Qe,r ~r+!- ~e v~
The characteristic Debye temperature--OD--is a significant element of the equation
of state for a crystalline substance. It may be determined from the dependence of
atomic heat capacity on temperaturQ:
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` ~f b~ {7~~ e ~~l~r~~ r~~e r,4fT~
~ ~i.. _ ~
~ r rp T . 9R $
+ ~ r 9.~ ` ri 1~e , 2 ~
9~ (Z i..z r~/r3~e t ~ 3R.~ t 3.32)
+ (t - ~ 'C T~
- where D1(OD/T) is another Debye function (see Appendix 4 in (5)). This function
also varies from zero to one as the temperature rises from zero to infinity. At
- low temperature (OD/T>10) it equals 78(T/OD)3, which corresponds to the relation-
ship CVcT3, obtained by Debye back in 1912. At high temperature CV~3Ro 25 j/mole�OK.
This is the well known Dulong-Petit law. Of course the latter was obtained for
specific heat at constant pressure Cp, but according to expression (2.20), in terms
of gram-atoms the difference between Cp and Cy is not very large, averaging just
~-1.6 j/mole�OK.
OD in (3.32) is found as a reducinq parameter from the experimental dependence of
heat capacity on temperature (see Appendix.5 in (5)).
Another way for calculating OD requires interpretation of a crystal as an elastic
isotropic medium. An isotropic solid has two coefficients of elasticity--the
- compressibility coefficient R T and Poisson's constant Qn. The speeds of soun3 CZ
and CT are expressed in terms of S T and ai-I as follows:
r: ..s~a-~;~.~ nz 3( f;2vn ~ 4IV k 1S ~ 2
~:'f~n),pr.;' ' 4~Z~~~ri+),prf 07C e~~' 9k fh
J(C~�f1{J ) (3.341
J
When we know .S T, Qn and p, we can find OD.
The ta.ble below compares OD values obtained by these two approaches for four metals.
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Characteristic Debye Temperatures
~ _ . _ _
' ~ r/cK3 ~t,IpIIy~.~1i C 3~A~, H I~4~e ~-H
~I~..~ .._..L~_.. ~ _ N3 C (T) n ~.34
~_-f~~. Y` 2,7I I,36 0,33? 398 402
~ i~:i ~ i~,96 0~74 0,334 3I5 33?
~ al~,~ ~ i0.53 Q~9~ O,.i79 2I5 2I4
I-- II,32 ~ 2,0 U,446 88 Y ~ ?3
Key:
1, gm/cm3 3. From
2. m2/Pa 4. According to
~ The closeness of the values is doubtlessly very good for a theory derived so approxi-
mately.
Crystals Consisting of Molecules
The formulas of Debye's theory may be applied directly only to crystals consisting
of atoms of the same sort. If a crystal consists of N molecules, each with s atoms,
then in addition to 3N oscillations distributed as per Debye's theory, to describe
the state of the crystal we would have to add 3N(8-1) oscillations associated with
the internal degrees of freedom of the molecules (it is usually asstuned that motion
within molecules and relative motion of molecules within the crystalline lattice
do not depend on each other). We will interpret internal degrees of freedom as
simple harmonic oscillations of the same frequency, and correspondingly we repre-
sent their contri.bution as Einstein's statistical sum:
3r -B~[~,y
n (~-E
The log~~~rithm of the statistical sum would be:
s ti, ~ t t; d; j'f ~
p ~ ( Y. - 9A~~ ~ ~ ~~r. r. )J ~ _ a ( -1 ~
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Free and intrins~cenergy would correspondingly equal
?_t�ijd-'~rz
3s -9~ '1'~.
ti-~-.+9Qr(-i~.,!~ ~ ~~~-e j~~i+Ri~4~(f-e ~ (3.35)
3S ~_7 � i,j -l1)
~'~t'3k,~(f9D/`r~+~r~~ ~~`ir j_-~'r~Ft ~[T ,
, ( 3.35)
where E:'~ is the energy contribution of thermal oscillations of atoms in the
molecules. Differentiating (3.36) with respect to T, we get the specific heat
- ; d.~
~Y _ ~r~~a e~`Z~= 3Q`r~~ (9~ /r) ~ ~s i El~ ; `r1 r
~e e, ir z c3.37)
Comparing (3.37) with (2.22) and (2.23), we find that in the approximation under
examination here, the expression for specific heat of a molecular crystal at
pD~T�1 is similar to the relationship for heat capacity of an ideal gas consisting
of nonlinear molecules. If we assume that the internal degrees of freedom of a
molecule depend weakly on pressure and volume, and if we ignore the influence of
V on OZ, then in accordance with (3.26) and (3.30) we get the following expression
for pressure:
~Q A~ ~ (
T
x l , px ~
- PX f ~ y ~ ~ r
or �
n _ ~~y. ~ ~ (L - ~ X - t~i.`2~ c ~i.3S)
The third term of the expression for free energy (3.35) does not contribute to
pressure.
Debye's theory is one of the fundamental theories of physics. Naturally, however,
in view of the structural complexity and diversity of condensed substances, its
conclusions n?ust be approximate in nature. This is especially true of the
40
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in-between range in which the frequency distribution function (3.16) obtained f`or
long-wave low-temperature oscillations applies to the entire interval of tempera-
tures, up to the extreme highest, at which point the solution once again becbcties
Z
statistically valid (10). Also important is the fact that the theory does not
account for the anisotropy typical of crystalline substances or the relatively
high mobility of particles in a fluid. As was noted earlier, the latter becomes
less significant at high pressure, which li.mits the mobility of particles so much
that the state of both a liquid and even a gas could be described by equations such
as (3.30) and [equation number illegible; possibly (3.38)]. We will henceforth need
only these two equations. A reader wishing to acquaint himself with the equation
for the state of crystals with ionic and complex lattices and consider the influence
of free and bound electrons would need to refer to more-detailed handbooks (9).
Gryunayzen's Parameter I'
Gryunayzen's parameter, which is fundamental to an equation of state for a condensed
substance, may be obtained by differentiating (3.38) with respect to energy at con-
stant volume:
~ ~ 1 ~ ?~.L) ~ : r~ ~A ~ ~ ~ ~rj ~L~~ f t' �r i f , ~ 1~~~ . {
~~rp i M ~0 ~ 1~ _ 1.~p~ ~~a~V ' rD i~, r~i~ a
_ ! - ~ ( + t :p E~'~~r/ , l~- f ~r~~ v
\ ) = ~ ' ~
or ~,~r,~ ~rE I a~~ ~~r~ _ ~r,~(r~ } ~ ( 3.3~)
~r ~CPrIi~,~
In the general case CV~1~ is determined by formula (3.32). At (OD/T) �1, we can
assume Cy~l~ = 3R. The values of ap, ST and p are known for many substances under
normal conditions. 'I"herefore there is usually no difficulty in calculating I'(Vo)�
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Slater's formula can be used to calculate Gryunayzen's parameter at high pressure.
_ This formula was derived with the assumption that all frequencies in an isotropic
solid are proportional to the speed of sound lCY =V (~pX/nV ) I~~J and inversely pro-
portional to the distance between particles 1' ~T3~:
l~a ,ypR I/2
( ~Y )
From whence we get
~D~i}~ , V '~2~r ~2
r 5 e'c~ V ?,~Px 1" ~~s ( 3.40)
- The following formula gives values closer to experimentally obtained values for
ionic crystals and metals: ~
= aM/~ ~ 2
~PrV /
~ ~ 2 /~(PwV 2w%:"yi,pv + - 2.1 (.3.41)
In thi.s case m= 1 for metals in most situations, while for ionic crystals m= 2.
In many cases the dependence of Gryunayzen's parameter on the specific volume of a
solid is close to a direct proportion:
- ~ = L'{)il~~ ( 3.W?)
v -
Potential Components of Energy and Pressure (11)
The energy of a crystalline lattice is determined by forces acting between its
elements. These include attraction forces--Coulomb, van der Waals and valent,
and forces of repulsion--Coulomb or quantum.
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~ Coulomb attraction dominates in ionic lattices. Its energy has the form -aq/r,
i
where q is the least charge of the ion, r is the least distance between ions and
a is Madelung's constant. Forces of repulsion are produced in ionic crystals by
overlapping of the electron shells of the ions. These are quantum forces which
- decrease exponentially with distance. The potential of an ionic crystal may be
written as: ~
' t n ' ~~a~ /Q+~t' uj ~ ,jQiV,(~f~~ ) (3.43).
\
where al, a2 and agare constants and r=V1/3
The potential part of pressure is
F~ - - �~6!' ~ q~ ( ~a~ [~i ' r - ai rJ ~ + 1 '
J~` t I', , (r, ~ s'~4~~)i
The potential part of the modulus of cubic compression is:
'~n - J~n _ ~rJ~`~j/'. ~?~ex~~?,~~' ~ Q,~(r)~(3.q5)
The forces of attraction in molecular crystals are van der Waals forces. Their
dependence on distance is -4'(r)/rs, where `Y(r) is some function of distance. As
in ionic crystals, forces of repulsion are produced by overlapping of electron
shells.
Potential energy has the form
En _ ..4~1f, /4,~~0 �,rf> I !1, Z \ ~j ~ ~ ` (3.46)
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Potential pressure is
9
~'p _ Q, ~ i ~~~e~[r~ ~?r ;~:~1- ~ ; ~ 3.k~1,
The potential part of the modulus of coa~ression is
9
kA ~ ~(?~)~(A2r ~1)er~~q:~l-r~]-.lA~~r~~ (3.4a)
In the case of inetals the forces of attraction are produced by Coulomb interaction
of free e]ectrons with positive ions as well as by the volume energy of free
electrons. In both cases energy is proportional to r-1. Coulomb forces of repul-
sion may be combined into a single term together with forces of attraction. Repul-
sion forces steauning from overlap of electron shells have the same form as in pre-
1 vious cases. Fermi kinetic energy of free electrons also causes repulsion. It is
proportio;nal to r-2.
The resultant potential is
r~ i /
t � _ n, ~~c,~ ~ ~1: i ~ - r~ ~J ~ ~?s - r~~( "r" ) c3.a~)
/
In alkali metals the forces of repulsion are produced mainly by Fermi kinetic energy
of conducting electrons: The first term in (3.49) may be ignored. In a number of
cases exponential repulsion plays the main role, and the second term of (3.49)
may be ignored~ In the latter case the components of pressure and the modulus of
cubic compression are determined by formulas (3.44) and (3..45).
- 44 '
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~
Many authors use exponential potential: ,
li b _
~ri~ ~ . r~ r~ (3.50)
where, as in the previous cases, the positive term represents forces of repulsion
and the negative term represents forces of attraction. Obviously as r decreases
the repulsion forces decrease faster than the attraction forces (n>m). In most
cases n= 9-14; for molecular crystals m= 6, and for ionic crystals m= l.
Leonard-Johns usecl potential (3.50) at m= 6 to describe the behavior of compressed
gases and (in the free volume theory) to arrive at an equation of state for fluids.
The theory of free volume gives us the potential for spherical apolar molecules
(3.50) in the form
~ n, c
E (r~- S~`,,~ ~6)~ f~r+>s (r~)~~ (3.5I)
where E m-maximum energy of interaction, and ro -effective collision radius, for
which E(r) = 0. At S= 12 this is the so-called Leonard-Johns potential (6-12).
For it,
E~ . 4 ~ ( / ~*~21 + ~ `~""12 /~~.1~ j
v ~ 1 ns ~
~n ' ~Y LaU ~ v)~- ~'~~:1 ( ~.52)
These expressions are much simpler than (3.46)-(3.48).
We can also calculate Gryunayzen's parameter:
. r~3~~~~Fr-3)-~
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. . .
The relationship between V*, VooK and Vo is diagramtned in Figure 3.
~ - - - _
P E~ pn
,
,.t
v~ v,~ ,
0 V'------V.
~ ~ ~
, ~ .
~
~ ; .
~ -
~
Figure 3. Dependence of Potential ComQonents of Iiitrinsi~c Energy and
Pressure on the Specific Volume of a Substance
4. Chemical Equilibriiun in the Products of Combustion and Explosion
4.1. Moderate Pressure
Table 6 shows the basic equilibrium reactions in the explosion and combustion pro-
ducts of commonly encountered explosive systems (C, H, N, 0, C1, P). The enthalpies
of these reactions and the expressions for the equilibritun constants are given.
Partial pressures of gaseous substances are expressed in atmospheres. The partial
pressures of condensed substances are given in dimensionless units. The formulas
for the depen~ence of equilibrium constants on temperature were obtained by plotti.ig
the tabulated data in coordinates 1/T-[symbol illegible]. For all reactions except
(1), straight lines are obtained with good accuracy within a broad interval of
temperatures (500-[number and units illegible]). The constants for the reaction
' were obtained in a 2,000-4,000�K interva~..
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- Table 6. Basic Equilibrium Reactions in Combustion and Explosion Products; 1{alue
of L~oefficients A and B in the formula lg Kp = A-(B/T)
IVo. ~action Equations A B
~r - _ _ .
I CU2 + H2 ~ CO t H20 I~23 II00
2 C02 t Cx ~ 2C0 8,?3 8480
- I
; }~ZO Oll + �2 H2 3 ~65 I4600 ~
CH~ t H20'.:= CO t 3H2 I3 ~06 II580
5 CH4 ~ Cx + 2H2 5,75 4?4Q
6 NN3 NZ t-~HZ b~00 ~7I0
'J ~112 t ~02 NU ~ 0~65 4750
b IICI ~ ~i2 + CI2 -C;35 4~0
9 flF ~-~li2 t~'~ -O~IO 142E0
10 M1 c 2N ' 7,20 499I0
II Uz 2(~l) 7,I0 2649U
~
12 � k12 .r 2fl 6,30 23470
_ I3 CI2 2C! b,40 I32i0
Ih F~ !F G,70 6~Op
I5 CG2 CO +~2 4, 30 14`;IU
I
Note: The eguili.brium constant for .the reaction (i. ) uA +d B~. ~fD I'
~ n~ / ~j..~~r~-d
has the form
~ t rt p \ ~o
For example, , _ nav ~~H ~ .~~~Q ~M~ , ~ 2
G -
k~ ?t w,o no ~ K" nlNy � lI N~O n~,~
Thermochemical calculations are now usually made using the equilibrium constants for
atomization reactions. These constants may be used on their own: It is presumed
t'~at the initial substance had broken down into atoms, which then group together
in c.orrespondence with the atomization reaction constants into the appropriate
~ groups--molecules or radicals. On the other hand these constants can be used to
derive the equili.brium constant for any equili.brium chemical process. For example
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for reaction (1) (Table 6),
CQZ t N1 CO t Ii20
the equilibrium constant is calculated as
~'~o Pq,a Ku i ~
xi' ^ Y~oj pe~ xw x r�o
where K.ia are the atomization constants of the appropriate substances: For example
p 1'~, k'c
x~p = I~GZ .
~e values of X.La are taken from handbook (3).
It follows from Table 6 that at moderate pressure the equilibrium of reactions
3, 7, 11 and 12 is shifted leftward to a temperature of about 3,000�K and the equil-
ibrium of reactions 8-10 is shifted leftward to a temperature of up to 5,000�K.
When the temperature is not too high (up to about 3,000�K) and pressure is high
(above 10~ Pa) there is little OH in the products, and almost no N, O and H at
all. The equili.brium of reaction 2, 4-6, 13 and 14 is shifted right at high temper-
ature. Methane and ammonium do not form at high temperature and moderate pressure
in systems of conventional composition. Higher temperature promotes formation of
these substances while simultaneously blocking dissociation as described by the
equations (reactions 3 and 10-14).
For practical purposes alumi.num and magnesium experience combustion reactions in
the entire range of combustion conditions. Z"hese reactions are not shown in Table 6.
It may be presumed that when oxygen is sufficient aluminum transforms quantitatively
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into A1203k, while Mg transforms into Mg9k. In the presence of chlorine and
fluorine Na, K and a.i transform into the appropriate halides, while in the absence
of these elements they transform into oxides, hydroxides or carbonates.
Finally the equili.brium of the reaction of water gas establishes itself in accord-
ance with the size of the equili.brium constant of reaction 1 at the given tempera-
ture and concentration of carbon, hydrogen and oxygen in the system.
4,2. High Pressure
At high pressure, in the range within which the equation of state for an ideal gas
becomes inapplicable, the expression for the equilibriiun constant becomes more
complex. In an isothermic process
~G= Vap x) _ (4.I)
At V= RT/P (Clap~eyron' s equation) we get
a 6= RTd P,n P and a G=. RTP~+~ P/Po~ ~4.2)
This is the source of the simple relationship (3.5).
At high pressure V~ RT/P, relationship (4.2) breaks down, and a dependence arises
between the equilibrium constant (3.5) and pressure. The effect of pressure on the
equilibrium constant may be determined by Lewis' method by substituting partial
pressure in (3.5) by fugacity:
`~{T)e R~' or f' _VRTt4.3)
~ - ti~~;p -So~l', at . =Cn~,S~ we get. ~a.I~
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Integrating, we get
~i+Cf/f.) = RT,~vdP
0
~ We introduce the coefficient of activity ~'u f~p rn =~~/po) t
_ P
Pal ( ~c1 /~u
) = P.n (Po /P) 1 (!l~T) I ~"~f~ t4.4)
r~
We represent volume as V= RT/P-~V (correspondingly, ~V =(RT/P) -T~ , and we account
for the fact that as Pp->0, Ya~-~1 and ln YQ~->0. Then ~
~"u~_ RT f e vdP c
o ~~:ti)
Usually the coefficient of activity is found by approximate integration of (4.5).
Having the isotherm V(P) for a real gas, we plot the curve ~V = RT/P-V = ~V (P) .
The area between this curve and the abscissa from 0 to P is prnportional to the
log,arithm of the activity coefficient at pressure P. For oxygen at 0�C and
carbon dioxide at 60�C, at a pressure of 2�10~ Pa ya = 0.87 and 0.45 respectively,
while for carbon monoxide at 1.2�108 Pa, ~ya = 2.22. If a simpl:e analytical ex-
pression is obtained for the isotherm of a real gas, the coefficient of activity
may be found by quadrature.
Thus expressing V(P) as in formula (3.6):
~ s R T~?,~ M a V~- V t~ fansf
P
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we get
r'ti'~ r'{ ..~i~~ P r.
P~~ J ~ + ~ R ro . PQ = +-~n -y~=
At VK/Vo = 1.10 3, P/Po = 5, 000 and To/T =1 � 10-1, we get ln ~y =+0. 5; ~y = 10 65 .
When we find the equation of state with virial coefficients (3.12) applicable,
V~ " - ~ c' ( x.) . .
where G'(Y) = 1 t x t O~iu5x2 + U~28'!x3 t O~I93;c~~
x _ ~,/v, ~Q f n~ BY~ ~ .
B2Z--second virial coefficient for 1 mole of the given reaction product (see Table 4),
C,~ e` - l ~~t
~ ~ r ' e?' ct ".~a:~ ~~o--- , ) ,
� ~ 2
Knowing the activity coefficient Yai and substituting partial pressure by fugacity
in the formula for the equili.brium constant (3.5), we get �
~l �~I f t~,/~~~ ~J~~~`: ~/JS ``/~~a~''~ n l~ ~~PL''
~ . �
Here the II symbolizes multiplication of fugacities (or partial pressures) of the
reagents to a~ower consistent with the reaction's stoichiometric coefficient vZ
(for substances in the right side of the reaction equation vi>0, and for substances
in the left side v2
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then we get the following simple formula for the coefficients of activity from (4.8):
t/f ? r _ ~ 1 ~ I. 3 `'~'~~y ~y �l1~~ ~u j`rrr)~~ -'t
1 p (`?�91
) D
(1 Q
- 5. Enthalpy o~~ Formation
Besides the equation of state , tY~e equilibrium constants and the dependence of
intrinsic energy or enthalpy on temperature, to calculate the thermo.dynamic relation-
shipsin combustion and explosion praducts we need to know the enthalpy of formation
of the initial substances and the reaction products. The enthalpies of formation
of organic compounds--the principal ingredients of combustible and explosive
systems--are obtained mainly by measuring the heats of combustion reactions in
a calorimetric bomb. Values obtained in this fashion are tabulated in handbooks
(3,6,12). Some of the values used most often are given in the Appendix in (5).
At the same time~in view of the tremendous diversity of organic compounds on one
hand and the complexity and high cost of reliably determining enthalpy of formation
on the other, methods of calculating this variable are under intensive development.
All existing methods of calculating enthalpy of formation , as well as a number of
other thermodynamic variables (entropy, Gibbs' energy, specific heat), are based on
the principal of additive contributions by groups and bonds. Table 1 in the
Appendix compares some methods in application to a number of organi.c substances,
as studied in a degree project conducted by I. Ye. Esterman (1977). It also gives
the r:sults of his calculations for group contributions obtained by Esterman and
V. M. Reykova from an analysis of formerly published experimental values of
~Fif. These contributions are brought together in tables 7 and 8. Table 1 in the
Ap~~eridix gives the differences (in kcal/mol~) b~tween thp ~Hf values obtained
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Table 7. Contributions of Functional Groups to Enthalpy of Formation of Aliphatic
Compounds (From Data Cited by I. Ye. Esterman and V. M. Raykova)
I'pC�~nite~ Pag ~1+A- TpYnna 'd3 ~1AKOCTb
~1J f2~2 KOCTb ~1~~ ~Z~ I3y
-Cti3 -I0~09 II,53 -Nr;~ -9,6 -I6,i.
;CH~ -4~93 -6,II ~CH(H02) -I3,8 -2U,3
`-.CN -I,6 -Z,0 -CH(~0~)~ -IC,6) -I9,2)
rrC~ I,0 I,5 ' -C(f~C2)~ -3,3 -II,3
K2C=CH- I5~0 I2,3 -UNC: -9,8 -I3,6
H2C=C~ I6,6 I~+,O -GSCZ -2i,7 -27,7
HC=C< � 20,3 I7,5 -C~ 0 -47,2 -5�,I
2
}iC~=CH-uxc ~(4 I8,2 I5,6 -~;}1-tCL -~,6
HC~=Cki-x~aac (5 I7,4 I5,0 iN-yG~ I9,o �,6
=Cii-(~1E~~ 24,7 23,2 rll-NO I�,6 (I2,4)
=Cl~-Cc1paHC 20~6 I9,0 -NF2 (-9,6) -I:,I)
~;=C~ (PS) 23 ~ -C:5 2'T,6 2I,8
-C:C- 55,5 SI,7 -H_C~ 52,C 46,3
HC~C- 54,5 SI,4 -H=N- 5~,t 50~2 ~
-OH -kI,7 49~I -CI -II;7 -I4,3
-G- -30~I 3I,0 -P -4'r -48,8) ~ .
--CUi,H -92,6 I04 ~(~)11 N H]1 N 4
~C~ 0 -29,5 33,5 Q" I4,4
N
-C
D (-79~5) b4~0 lT S,I (?,I7
o-
-C=0 93,5
~O~~N3 Cr -IS,~ -?I,5
-HH~ 3,I -0,63
-hH- I2,I ~G,3 ~ -~6,I -33~1~
(22) 2z~6
;~=U 3I,8 37,8 -Cd-yt{- -~~~~3
_ _ - . . . .
Note: Contributions obtained from the enthalpy
of formation of one compound are given in paren-
theses. Contributions are given in kcal/mole,
1 kcal = 4.184 kj .
Key:
1. Group 4. Cis
2. Gas 5. Trans
3. Liquid 6. Rings
experimentally (usually the averages of several sources) and values obtained by
calculation. Comparative calculations using six to eight methods were made for
100 compounds. From cne to three compounds were chosen by chance to represent
many of the examined series. Omissions of figures in the table mean that OHf~
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Table 8. Contributions by Functional Groups of Ziquid Aromatic Compounds
~t3W~CTNT CO CII80dN ~`.i8it8C?NTCIIB C CBAbHHY
8aH ^ - � -
- P38tlY0A8NCSH88Y ~ t~` B38~10A8HCTBjICY
. . . . r .-:fy1=-
-CH3 -II.53 -k~ ~-I0~4)
- 0 - -3I,'? ~ M `-IQ2 t23,2 ~
-OH -49,I -1$2. -6,3
C'~~ -33,y ) AH +4,4 I
~
>C = U -36,6 ) N- tI6,ts ~
~ -CUUIi -IUI -REI-kH ~iI8,4 j
~ ~Rti l-5a) C6Hy ~as 2I,64 ~
G 2 YeAa 1n,4I