Термодинамика, кинетика и расчеты металлургических процессов

 

 

Национальный исследовательский технологический университет 

«МИСиС» 

 

Кафедра металлургии стали и ферросплавов 

Кафедра высокотемпературных процессов, материалов и алмазов 

 

С.Н. Падерин                                                                              H. Jalkanen 

Д.И. Рыжонков                                                                           L. Holappa 

Г.В. Серов                                                                                   E. Heikinheimo 

 

THERMODYNAMICS, KINETICS AND CALCULATIONS  

on  METALLURGICAL PROCESSES 

ТЕРМОДИНАМИКА, КИНЕТИКА И РАСЧЕТЫ 

МЕТАЛЛУРГИЧЕСКИХ ПРОЦЕССОВ 

 

Допущено учебно-методическим объединением по образованию в области 
металлургии в качестве учебного пособия для студентов высших учебных 
заведений, обучающихся по направлению 150100 – Металлургия 
 

 

 

 

Москва Издательский Дом МИСиС 

2010 

УДК 669.04.997 
 
Т35 

Р е ц е н з е н т  
д-р техн. наук, проф. Г.Н. Еланский 

Издание на английском языке 

 
Термодинамика, кинетика и расчеты металлургических процессов /  
Т35 Падерин С.Н., Рыжонков Д.И., Серов Г.В., Jalkanen H., Holappa L., 
Heikinheimo E. – М.: Изд. Дом МИСиС, 2010. – 235 с. 
ISBN 978-5-87623-312-7 

 
Учебное издание предназначено для студентов, обучающихся по направлениям «Металлургия» 
и «Физическое материаловедение», магистрантов и аспирантов. 

Great advances have taken place in metallurgical processes during the past several decades. Modern, 
energy saving iron- and steel making or direct reduction technologies as well as copper and other nonferrous metals production are some examples of the recent progress. 
Energy and environmental issues have been generally considered as the main driving forces for this 
development. It is, however, clear that the key factor for the progress has been better knowledge of the 
basic phenomena in the processes. It means both thermodynamic bases of the prevailing phases and the 
reactions as well as chemical kinetics and transport phenomena in the reaction system, i.e. transport of 
heat, mass and momentum. All these are needed to create a representative model or a series of submodels 
to describe and to simulate the process. 
This book is mostly directed to thermochemical and thermodynamic properties of phases and reactions. 
However, in some parts also structure of phases e.g. slags are discussed as well kinetic and mass transport 
phenomena. 
This 
book 
was 
written 
as 
a 
joint 
project 
of 
professors 
at 
National 
University 
of Science and Technology «MISIS» and at Helsinki University of Technology (HUT). It is purpose is to 
be a general text book for undergraduate or postgraduate students who have metallurgy as their main 
subject. It is also suitable for self-studying as it has in each chapter, first a fairly thorough theoretical 
description of the problem followed by calculation examples and plenty of working examples and control 
questions. 
 

ISBN 978-5-87623-312-7 
© Падерин С.Н., Рыжонков Д.И., Серов Г.В., 
Jalkanen H., Holappa L., Heikinheimo E., 2010 

CONTENTS 
 
 
 
ПРЕДИСЛОВИЕ.........................................................................................................5 
INTRODUCTION ........................................................................................................6 
 
1.THERMOCHEMISTRY. SHORT BASIS OF CALCULATION OF ENERGY 
BALANCES .................................................................................................................7 
 
2. CHEMICAL THERMODYNAMICS OF PYROMETALLURGICAL 
SYSTEMS AND PROCESSES .................................................................................27 
2.1 ON THE DRIVING FORCE AND EQUILIBRIUM CONDITIONS FOR 
CHEMICAL REACTIONS AND SYSTEMS ...........................................................27 
2.1.1 Gibbs energy change in an isothermal chemical reaction .................................27 
2.1.2 Refractory-steel interaction as an example for chemical system  undergoing 
chemical reaction ........................................................................................................30 
2.1.3 Manual calculation of ( complex) reaction equilibria in a gaseous mixture......33 
2.1.4 Substitutional and competing chemical reactions..............................................39 
2.2 GAS AND SIMPLE GAS-SOLID SYSTEMS ....................................................46 
2.2.1 Composition and properties of gas systems most common in metallurgical  
processes ....................................................................................................................46 
2.3 METALLIC SOLUTIONS ..................................................................................49 
2.3.1 Standard states and composition co-ordinates for mixtures and solution 
phases..........................................................................................................................49 
2.4 INTERACTIONS IN METAL-GAS-SYSTEMS ................................................54 
2.4.1 Copper-sulphur and copper-sulphur-oxygen systems .......................................54 
2.4.2 Vacuum refining of blister copper ....................................................................58 
2.4.3 Decarburisation of chromium hot metal ...........................................................62 
2.5 OXID SOLUTIONS  AND  METALLURGICAL SLAGS ................................67 
2.5.1 Model of perfect ionic solution ( Temkin′s model) ..........................................67 
2.5.2 Using the model of a perfect ionic solution in calculation of metallurgical 
ssag component activities ...........................................................................................75 
2.5.3 Polymeric models of silica melts ( Masson′s model ) ......................................84 
2.5.4 The model of regular ionic solution  ( Kozheurov′s model)..............................99 
2.5.5 Thermodynamic model of slag as a phase, which has a shared electron  
System (Ponomarenko′s vjdel) ................................................................................124 
2.5.6 Optical  basicity ...............................................................................................129 
2.5.7 Electrochemical measurements and calculations in oxide solutions ..............134 
2.6 EXAMPLES OF REACTIONS BETWEEN METAL AND SLAG .................148 
2.6.1 Deoxidation of steel.........................................................................................148 
2.6.2 Desulphurisation .............................................................................................151 
2.6.3 Dephosphorisation of steel...............................................................................157 
2.7 DISTRIBUTION OF ELEMENTS BETWEEN MOLTEN COPPER OR 
COPPER MATTE AND SLAG................................................................................160 
2.7.1 Copper-slag distribution ..................................................................................160 
2.7.2 Distribution of elements between copper matte and slag ...............................164 
 
 

3. KINETICS OF HETEROGENEOUS METALLURGICAL PROCESSES ........171 
3.1 KINETIC EQUATIONS ....................................................................................171 
3.1.1 Temperature dependence of rate constant .......................................................171 
3.1.2 Kinetic equations for multistage reactions ......................................................174 
3.1.3 Conditions of the steady state. The limiting stage ..........................................179 
3.1.4 Mass transfer and chemical reaction................................................................180 
3.1.5 Simularity criteria and cakculations of the mass transfer coefficients ............182 
3.2 KINETICS OF FeO REDUCTION FROM SLAG MELT BY SOLID 
CARBON .................................................................................................................191 
3.3 KINETIC REGULARITIES OF ALLOYED STEEL  
DECARBURISATION ............................................................................................196 
3.3.1 Kinetic models of steel decarburization process and steel oxidizing  
refining process.........................................................................................................197 
3.4 KINETICS OF NITROGEN ABSORPTION BY MELTS FROM LOW 
TEMPERATURE PLASMA ...................................................................................210 
4.NANOMATERIALS.............................................................................................218 
NOMENCLATURE ................................................................................................226 
DICTIONARY..........................................................................................................227 

ПРЕДИСЛОВИЕ 

Книга 
"Термодинамика, 
кинетика 
и 
расчеты 
металлургических 

процессов" является результатом совместной работы русских и финских ученых 
и объединяет опыт теоретической подготовки инженеров-металлургов и 
инженеров-исследователей металлургических процессов в Национальном 
исследовательском 
технологическом 
университете 
«МИСиС» 
и 
в  

Технологическом университете г. Хельсинки (Финляндия), издание отражает 
большой прогресс в металлургии за последние два десятилетия. 

Термодинамика, термохимия и кинетика высокотемпературных процессов 

составляют теоретическую основу процессов получения и рафинирования 
металлов. На этой основе книга позволяет освоить проведение теоретического 
анализа сложных металлургических систем и процессов и их моделирование, 
формулировать и решать задачи, имеющие практическое значение. В книгу 
включены теория, модели и расчеты металлургических систем и процессов. 

В главе "Термохимия" приведены методы расчета энергетических 

равновесий при высоких температурах.  

В главе "Химическая термодинамика пирометаллургических систем и 

процессов" 
изложены 
основные 
теоретические 
положения 
равновесий 

химических 
реакций 
в 
металлургических 
системах, 
включающих 

металлическую, шлаковую и газовую фазы, и решения практических задач. 
Термодинамика металлических растворов и взаимодействие в системе металл–
газ даны на примерах систем медь–сера и медь–сера–кислород, вакуумного 
рафинирования меди и обезуглероживания хромсодержащего расплава на 
основе железа. На основе традиционных и новых термодинамических моделей 
оксидных расплавов применительно к металлургическим шлакам (ионные и 
полимерные ионные модели силикатных растворов) приведены расчеты 
активности компонентов основных и кислых шлаков. Представлены методика и 
результаты электрохимических измерений в оксидных расплавах и шлаках, 
расчеты активности оксида железа и парциальных мольных величин 
компонента по результатам электрохимических измерений. Рассмотрены 
примеры расчетов реакций между металлом и шлаком: раскисление, 
десульфурация и дефосфорация стали, а также распределение элементов между 
медью и шлаком, медным штейном и шлаком.  

В главе "Кинетика гетерогенных металлургических процессов» показано 

использование теоретических представлений о скоростях сложных процессов 
взаимодействия в системах металл–шлак–газ, теории и критериев подобия для 
оценки коэффициентов массопереноса. Рассмотрены кинетические модели 
процессов обезуглероживания и окислительного рафинирования стали и задачи 
с использованием модельных уравнений. Изложена кинетическая модель 
абсорбции азота расплавами на основе железа из низкотемпературной плазмы и 
выполнены расчеты по уравнениям модели. 
В главе «Наноматериалы» изложены основные закономерности получения 
наноматериалов. Приведены примеры расчетов образования зародышей при 
получении наноразмерных частиц, а также процессов взаимодействия с их 
участием. 
Дополнительно к изучению теории и овладению методами физикохимических 
расчетов 
книга 
позволит 
освоить 
физико-химическую 
и 
металлургическую терминологию на английском языке, на котором издается 
большая часть периодической литературы, журналов и книг по металлургии. В 
конце книги имеется англо-русский словарь терминов. 

INTRODUCTION 
 
The most important applications of thermochemistry and thermodynamics in 
metallurgy are determination of energy balances and equilibrium properties of 
whole reaction systems or their crucial parts in order to derive energy and material 
requirements and optimal thermomechanical and chemical conditions for 
processes. 
Real metallurgical (chemical) processes are quite rarely isothermic, i.e. take 
place at constant temperature. The most common and convenient simplification for 
thermochemical processes is, however, an isothermal one. Another useful and 
common approach is isobaric as most metallurgical processes take place at constant 
pressure, which often is atmospheric. The overwhelmingly most common 
thermodynamic abstraction for metallurgical processes is to treat them as a single 
isobaric-isothermic process or as a set of isobaric-isothermic processes. In some cases 
isochoric i.e. constant volume approach is the most proper one as for processes taking 
place in closed pressurized chambers (autoclaves) or extremely rapid, explosionlike 
reactions. 
Industry processes are either 1) continuous, steady-state processes which means 
that physicochemical and thermodynamic properties of the reaction system will vary 
along the length of a reactor but not with time. 2) nonsteady-state, batch processes in 
which the charge is treated in one stage and its composition and temperature are 
continuously changing. 
The final properties of the reaction system are often decisive and it is often 
enough to carry out the thermochemical and thermodynamic analyses only for the 
final (and initial if necessary) state of the process. All isobaric or isochoric continuous 
or batch processes can also be presented as a series of separate nonisothermal 
thermomechanic and isothermal chemical changes. So, isothermal thermodynamic 
analysis can be employed almost without restriction to the nonisothermic 
metallurgical processes. 
There 
are 
available 
several 
computer 
programs 
for 
computing 
thermomechanical effects and equilibrium compositions of high temperature reaction 
systems. However, as the main task of this book is to help students to understand and 
apply physical chemistry as a tool for solving problems in metallurgy and material 
technology the exercises in this book are aimed mainly for manual computing. 
Theoretical analyses of many metallurgical processes include thermodynamic 
calculations of interactions in slag-metal liquid systems. Activities of slag 
components should be calculated previously.  
Liquid slag is considered as an ionic solution. By comparing it with real liquid 
slags one can find out the reasons for deviations and create more complex models: 
polymeric models of silica melts, models of regular ionic solutions for basic and acid 
slags. These models are discussed below. There are some complicated models of 
subregular oxide solutions (Lumsden and Shiro-Banya models), which are not given 
in this book. The study of thermodynamic models allows to approach critically one or 
another model for calculation of the component activities of a slag, to determine and 
to specify model parameters on the base of phase diagrams and electrochemical 
measurements in slags. 

Kinetic calculations of metallurgical processes help to reveal mechanics and 
durations of real processes. It is of great importance to choose correct kinetic 
equations. In some complicated cases it is convenient to use the criteria of similarity 
for calculations of mass transfer coefficients. There are examples of kinetic 
calculations: ferrous oxide reduction from slag melts by solid carbon, kinetic models 
of steel decarburization process and steel oxidizing refining process and nitrogen 
absorption by melts from low temperature plasma. 
 

1. THERMOCHEMISTRY 
 

1.1 SHORT BASIS OF CALCULATION OF ENERGY BALANCES 
 
Thermochemical analysis and calculations are employed for determining energy 
balances of processes as whole or of their particular parts or zones in order to derive 
energy consumption or heat evolution and exchange with the environment (energy 
loss), etc. Energy balances give basis also for computation of reaction temperatures 
and temperature distribution in reaction environments. 
A material system can exchange energy with its environment in the form of heat 
or work. Energy exchange take place with the expense of kinetic energy of atoms, 
molecules or lattice, phase transformations (solid state phase transition, melting, 
evaporation) and chemical reactions (including dissolution, solution formation and 
chemisorbtion). Energy balances are computed either for isobaric as usual or isochoric 
conditions. 
The thermodynamic function employed in thermochemical analysis of chemical 
reaction systems undergoing isobaric changes is enthalpy, H. For an isochoric process 
energy exchange equals to the changes in internal energy, U (or E). 
In Fig. 1.1.1 the principles of an isobaric energy (heat) balance are presented 
schematically. The basic principle for an energy balance is: energy and matter 
transferred into the system equals to the sum of the energy and matter 
transferred from the system in different forms and accumulated inside the 
system. 
There are different ways to establish an energy or heat balance even when the 
basis for all them is basically the same. The method presented here is based on the 
computation of enthalpy of input and output substances and phases relative to pure 
elements at some reference temperature for which frequently room temperature is 
chosen. Another method is to compute the physical heat contents of input and output 
matter, and the extents of expected chemical reactions and the heats 
liberated/absorbed in them. Other methods are between these two extremes. As 
enthalpy and internal energy are state properties there are no other restrictions for 
construction the heat balance but the same reference is employed for both input, 
output matter. 
The enthalpy of a phase relative to pure elements at the reference temperature 
can be divided in two categories. The isobaric chemical heat content consists of heats 
of formation of all individual compounds present in the phase as well as heat of 
formation of solution (heat of mixing). The isobaric physical heat content consists of 
all heat absorbed in heating up the individual substances to the temperature, 
concerned including heats absorbed in phase transformations (solid-solid 
transformation, melting and evaporation). 

The quantities readily available for the most common substances are isobaric 
heat capacities, Cp(T), standard enthalpies of phase transformations 
and formation of chemical compounds. The corresponding isochoric quantities the 
isochoric heat capacity function, Cv(T) and the corresponding internal  
energy changes (ΔU or ΔE) can be readily computed from the isobaric ones. Several 
data compilations and computer databases give directly the "absolute enthalpies" of 
elements and compounds i.e. the sum of physical and chemical heat contents which 
are tabulated relative to elements at room temperature. Data for thermal effects 
involved in solution formation is very limited except for dilute molten or solid alloys 
of some common metals. 
For an isobaric process, in which the heats absorbed or evolved equal to the 
enthalpy changes we have, accordingly 
 
 
 I 
II 
ΔHph = Σ(HT - HT°)i =  Σni⌡⌠CpidT + ΣniΔtrHi 
(1.1.1) 

 
in which (HT -HT°)i, Cpi and ΔtrHi are the molar heat content, heat capacity and heat 
of phase transformation of an element, compound or solution, present in the input or 
output. 
Accordingly, the first term (I) is the sum of heats absorbed by homogeneous 
phases in heating, the second term (II) is the sum of isothermal heats absorbed in 
phase transformations which take place in substances between the reference 
temperature and the input or output temperatures. 
Chemical heat is the sum of heats of formation of compounds and solutions they 
form. 
 
I 
II 
ΔHch = Σni · ΔfH°i + ΣHmix 
(1.1.2) 
 
The first term on the right side of equation is the sum of heats of formation of all 
compounds from elements and the second term is the sum of heats of mixing 
(formation of solution from its components) of all solution phases. 
Pure energy (Qp) can be transferred into the reactor or reaction zone by 
conduction and radiation of heat or by direct transformation of other forms of energy 
into heat inside the reactor - e.g. resistive, inductive, arc, plasma, electron beam 
heating and other methods to transform electric energy into heat. The accumulation of 
heat in non steady state processes may take place by direct increase of heat content of 
substances or with the expense of accumulation of substance in the reactor or reaction 
zone. 
The most serious problem in calculation of energy balances for high 
temperature processes is often the lack of thermochemical data for solutions, heat 
capacities, heats of mixing, solidification, devitrification etc. Accurate values for 
heats of mixing for metallurgical solutions are not commonly available. In some cases 
they are given within the total heat content of the given molten phase measured 
relative to room temperature or some other reference temperature. As heats of mixing 
are small relative to heats of reactions they can often be neglected without causing 
any serious error in heat balance.  

The procedure presented in the following examples for computing energy 
balances of isobaric processes is valid for isochoric ones when enthalpy (H) and 
isobaric heat capacity (Cp) functions are substituted by internal energy (U or E) and 
isochoric heat capacity (Cv) functions. Relations between the isobaric and isochoric 
thermochemical functions are as follows: 
 
For chemical reactions 
 
ΔrU°T  ≈ ΔrH°T - Δng · RT 
(1.1.3) 

For heat capacities of condensed substances 
 
Cv(T) ≈ Cp(T) 
(1.1.4) 
 
and of perfect (ideal) gases 
 
Cv(T) = Cp(T) - R 
(1.1.5) 
 
Δng is the change in number of moles of gaseous reaction components in reaction and 
R is the common gas constant = 8.314 J·mol-1·K-1 (1.987 cal·mol-1·K-1) 
The first step in construction of energy balance of a chemical process is to 
establish a stoichiometric material balance, which gives the amounts, stoichiometric 
forms of substances and phases in the feed and product of process concerned. There 
are several ways to construct and compute an energy balance the main differences 
depending on the choice of reference state for substances and on the forms of 
available enthalpy data. It is not always reasonable to choose room temperature for 
the reference temperature and pure elements for the standard state of substances. If the 
material system includes exceptionally stable compounds, which do not undergo any 
chemical changes in processes concerned there is no need to compute the standard 
heats of formation at room temperature as they are equal on input and output side of 
the energy balance and cancel, accordingly, each other. 
Methods to establish energy balances for high temperature processes are better 
visualised in the following examples. 

Worked example 1 
 

Adiabatic reaction temperature of titanium carbide synthesis 
 
In "combustion synthesis" or "self propagating high-temperature synthesis" (SHS), 
ceramic materials are synthesised by applying the heat of reaction to generate high 
temperatures necessary to complete solid-solid reactions. For synthesis the specimens 
pressed from a mixture of reagent powders are placed into a reaction chamber, which 
is evacuated or spooled with an inert gas, the specimen is electrically ignited and the 
combustion reaction goes through the specimen very rapidly. 
We want to know, what is the theoretical maximum temperature (adiabatic 
reaction temperature) which can be achieved when titanium carbide is synthesised by 
SHS from a stoichiometric mixture of titanium powder and carbon black? Synthesis 
reaction is: 
 
Ti(s) + C(s) = TiC(s.l) 
1 
 
Adiabatic reaction temperature corresponds to the conditions when the reaction rate is 
very high relative to the rate of heat transfer from the reaction zone to the 
environment and all heat evolved is absorbed by the reaction product. If combustion 
takes place in a mechanically open reactor in which the total pressure equals the 
atmospheric one this heat equals to the enthalpy of formation of one mole of TiC, 
ΔfH°298(TiC) The principle of calculation of adiabatic reaction temperature is 
schematically presented in Fig. 1.1.2 and table 1.1.1 in the form of simple heat 
balance 0.5 kg of TiC synthesised from a stoichiometric mixture of titanium powder 
and carbon black. 
 
Table 1.1.1. Material & energy (heat) balance. 

IN: (reagents at room temperature) 

subst. 
amount 
T 
"phys.heat" 
"chem.heat" 

 
kg 
mol 
 
ni(HT-H298)i 
niΔfH°298 

Ti(s) 
0.400 
8.34 
298 
8.34·0 
8.34·0 

C(s) 
0.100 
8.34 
" 
8.34·0 
8.34·0 

Σ = 
 
 
 
0 
0 

OUT: (products at final temperature Tad) 

TiC 
0.500 
8.34 
298 
8.34(HTad
H298) 

8.34·ΔfH°298 

Σ = 
 
 
 
8.34((HTad-H298) + ΔfH°298)) 

 
 
From the material and heat balance we get the following simple equation: 
 

IN 
OUT 
0 = 8.34((HT - H°298)TiC + ΔfH°298(TiC)) 

0 = (HT - H°298)TiC + ΔfH°298(TiC) 
(1.1.6) 

From Eq. 1.1.6 we get further 
 

ΔfH° + ⌡⌠
298

Tm
Cp
(T)
soldT + r·ΔmH°TiC + ⌡⌠
Tm

Tad
Cp
(T)
ligdT = 0 
(1.1.6') 

 
in which Tm and Tad are the melting temperature of TiC and the adiabatic (maximum) 
temperature of reaction, r is the mole ratio of molten to total amount of reaction 
product (TiC). If r = 0, Tad ≤Tm; if 0  ≥  r  ≤   1 Tad = Tm and if r = 1,  Tad ≥ Tm. 
 
By introducing data compiled in the end of example into the Eq. 1.1.6' we get 
 

-184,100 + ⌡⌠
298

3290
(49.95 + 0.979·10-3·T - 14.77·105·T-2 + 1.59·10-6·T2)dT + 

  
+ r·71,100  + ⌡⌠
3290

Tad
62.76 dT = 0 

 

-184,100 + 
3290
⎜
298
(49.95T + 0.979·10-3/2·T2 + 14.77·105·T-1 + 1.59·10-6/3·T3) + 

  
+ r·71,000 + 
298
⎜
Tad
62.76T = 0 

 
We first compute the energy needed to heat one mole of TiC up to the melting 
temperature in order to check the magnitude of r. Solution of Eq. 0.6’ gives for r a 
value of 0.16. I.e. the adiabatic reaction temperature is equal to the melting 
temperature of TiC and 16 percent of TiC is melted. 11.5 kJ of energy released per 
mole of TiC in the reaction 1 is left in excess when one mole of TiC is heated up to its 
melting temperature whereas 71.1 kJ is needed for complete melting of TiC. 
 
 Tad = 3290K = 3017°C; r = 0.16 
 
Thermochemical data for calculations 
 
TiC1: ΔfH°298 = 4.184·(-44) J/mol = -184,100 J/mol 
ΔmH3290 = 4.184·17,000 = 71,100 J/mol 
Cp(T)sol = 4.184·(11.939 + 0.234·10-3T - 3.531·105T-2 + 0.451·10-6T2) J/mol; 
Cp(T)liq = 4.184·15 J/mol 

Worked example 2 
 

Adiabatic reaction temperature of combustion synthesis of titanium nitride 
 
Titanium nitride is synthesised from titanium powder and gaseous nitrogen under high 
pressure by SHS-method. A 0.2 kg specimen loosely pressed from titanium powder is 
placed into a pressure chamber of 2 dm3. Reaction chamber is filled by pressurised 
nitrogen to 18 MPa at room temperature and the charge is electrically ignited. What 
would be the theoretical maximum temperature in the reaction zone and the 
corresponding pressure in the reactor under adiabatic conditions ? 
 
Relation between the real process and its thermochemical model used for 
computing the maximum adiabatic temperature is as follows: 
 
Heat released in the chemical reaction 
 
 
Ti + 1/2N2(g) = TiN  
 
 
 
 
 
2 
 
is assumed to be completely absorbed by the reaction product TiN and remaining gas 
(N2). As reaction takes place in a closed vessel the reaction system does not perform 
any work against the external pressure and thermochemical conditions of the process 
are isochoric. Heat of reaction which equals to the change of the internal energy of the 
titanium nitride formation (ΔfU°) heats up the reaction product (TiN) and the 
remaining nitrogen gas. 
 
The change of the mole number of gaseous species, Δng, is according to the 
reaction equation 2 equal to - 0.5. If we consider nitrogen to behave as a perfect gas 
even at 18 MPa the amount of nitrogen in the reaction chamber is according to the 
ideal gas law (PV = nRT) equal to nN2 = 1.8·107·0.002·8.314-1·298-1 = 14.53 mol 
Table 1.1.2. Heat & Material balance 
 
IN: (internal energy of reactants at 298K) 

subst. 
amount 
T 
"phys.heat" 
"chem.energy" 

 
kg/m3 
mol 
 
ni(UT-U298)i 
ni·ΔfU°298 

Ti 
0.200 
kg 
4.184.
18.. 
298 
4.18·0 
4.18·0 

N2 
0.002 

m2 

14.53 
" 
14.53·0 
14.53·0 

Σ = 
 
 
 
  0               + 
  0 

OUT: (internal energy of products at Tad) 

TiN 
0.258 
kg 
4.18 
Tad 
4.18(UTad-U298)TiN 
4.18·ΔfU°29
8 

N2 
 
12.44 
" 
12.44(UTad-U298)N2 
5.99·0 

Σ = 
4.18(UTad-U298)TiN    +   12.44(UTad-U298)N2      +     

4.18·ΔfU°298