Simcem is a computational thermodynamics package and database with aspirations of process simulation. It currently allows the calculation of thermodynamic properties from thermodynamic models (i.e., equations of state), and can calculate the equilibrium state of model thermodynamic systems. It mainly focuses on combustion and cement chemistry, but it is planned to evolve into a general chemical engineering toolkit.

Everything is available via the main menu (click the icon in the top left to open it). The console at the top provides information on the progress of calculations (and any errors).

The combustion calculator works and most of our cement database has been made available. A cement kiln and tubular furnace simulator should be available soon.

If you are a collaborator, you can access our full data set by logging in.

These are notes written while developing my own understanding of applied thermodynamics. They are brief and incomplete but they are provided to help others understand SimCem, to check my workings, and to implement their own Gibb's free energy minimisation as I struggled to find explanations of certain key aspects of the technique (such as eliminating redundant molar constraints, or a generalised Euler's solution, or even just what a complete thermodynamic models look like, and no its not $P\,V=Z\,N\,R\,T$).

This work wasn't possible without some of the excellent work already out in the literature. In particular, nothing was more useful to me than the excellent NASA CEA program, its database, and its highly educational report.

First, lets define the problem of thermodynamics and its
terminology. A thermodynamic *system* is the partitioning of
some quantity of mass and energy from its *surroundings*
through an enclosing *boundary*. The key idea is the division
of what we are interested in (the system) from the uninteresting
(the surroundings).

The boundary of the system may be physical (e.g., the walls of a vessel such as a balloon) or may be defined by some arbitrary division of space (e.g., a finite volume in a CFD simulation). If the boundary is physical, then it may or may not be included as part of the system. For example, water droplets in air have a surface tension which acts like the skin of a balloon and pull the drop into a spherical shape. This surface has an associated energy and it is at our discretion whether to include the energy as part of the system or as part of the surroundings (or neglect it entirely as an approximation).

Both physical and unphysical boundaries may be fixed or change shape
over time. In addition, if mass can pass through the boundaries then
the system is deemed *open* and *closed* if it
cannot.

The mass and energy contained inside a thermodynamic system may take
many forms, but only the *observable* properties at the
boundary of the system, such as volume, mass, surface area, and
pressure are visible to us. Thus thermodynamics focuses on these
variables and tries to find mathematical relationships to link these
variables together. Any other *internal* effect the mass and
energy of the system has cannot be seen unless it appears at the
boundary and so thermodynamics does not immediately concern itself
with them. E.g., a thermometer measures the temperature at its surface,
not in the bulk of a fluid. If the system is small enough, then the
observable properties should be approximately constant across the
system. For larger systems, we can always split it into smaller and
smaller sub-systems until the observable properties of a system are
approximately constant. Moving on, we will always assume each system
we consider is small enough (and thus homogeneous enough) to ignore
any changes in observable properties across its volume.

The sum total of the energy inside a system (the *internal
energy*) is not directly observable as most of it is inside,
away from the boundary; however, the conservation of energy tells us
it must exist. It should come as no surprise then that there are
other *internal* variables, such as the entropy, which can
only be discerned indirectly and yet are very interesting to
thermodynamics (but more on this later).

The key external and internal variables are the
so-called *state* variables. These are the variables that
change whenever the system itself has changed. This is a circular
argument that makes perfect sense as far as the external variables
are concerned as we only know if a system has changed if the
observable properties have changed. Counter examples of non-state
variables are the age of the system or the distance it has moved
(assuming there's no external field like gravity). If a complete set
of state variables is collected such that any change of the system
results in a change in one (or more) of the collected state
variables then we have what is called an *ensemble*. Which
variables are collected into an ensemble depending on what we are
observing and how we are observing it. For example, perhaps our
external variables are pressure and temperature for a balloon. I
haven't mentioned the balloon's volume but as we will see later
thermodynamics tells us volume is directly related to the pressure
and temperature, so there's no need to specify or observe it, or we
can swap the volume for either the pressure or temperature in our
ensemble (more later). Sometimes we need to add additional
variables. For example, a battery uses the same variables as a
baloon but must also include the electric potential across its
terminals. The key to understanding when an ensemble is incomplete
is that any process connected to the transfer of mass and/or energy
in the system must have its associated state variables.

Now that the initial terminology has been outlined, a governing equation for the changes in the energy of a thermodynamic system is derived.

The power of thermodynamics arises from its ability to find simple universal relationships between observable state variables. These relationships are a direct consequence of the laws of thermodynamics.

The *first law of thermodynamics* is an observation that
energy is neither created or destroyed but only transformed between
different forms. Every thermodynamic system may contain internally
some energy, $U_{sys.}$. The first law can then be stated as as a
conservation of this *internal* energy between a system and
its surroundings:
where the ${\rm d}X$ indicates an infinitesimal change in $X$ and
that this is an *exact* differential ($U_{sys.}$ cannot
change without $U_{surr.}$ changing).

From further observation of real systems, two types of energy
transfer are identified: *heat transfer* and *work*,
where $\partial Q_{surr.\to {sys.}}$, is heat transferred to the
system due to temperature differences and $\partial W_{sys.\to
surr.}$ represents all forms of work carried out by the system (the
negative sign on the work term is a conventional choice). The work
term represents many forms of energy transfer, so why is heat
transfer singled out as a separate term? Nature appears to maximise
heat transfer over work whenever possible, and this is discussed
later when reversibility is introduced.

You should note that a $\partial$ symbol is used for the
work/heat-transfer terms to indicate *inexact differential*
relationships. A thermodynamic system may transfer arbitrarily large
amounts of heat, and perform arbitrarily large amounts of work, but
only the remainder $(\partial Q_{\to {sys.}}-\partial W_{sys.\to surr.})$
will actually cause a change in the energy $U_{sys.}$.
The internal energy is a *state* variable as it describes the
state of the system; however, work and heat transfer are not.

Physical examples of this include engines, which are thermodynamic
systems that can perform arbitrary amounts of work provided
sufficient heat/energy is supplied but they return to their initial
state at the end of every cycle. An inexact differential implies
there is no unique relationship between the variables (we cannot
integrate this equation). Interestingly, inexact differentials can
often be transformed into simpler *exact* differentials
through the use of constraints. For example, if the engine is seized
and no work can be carried out ($\partial W_{sys.\to surr.}=0$),
then only heat transfer can change the energy of the system and we
now have an exact differential relationship, ${\rm d}U_{sys.}={\rm
d}Q_{\to {sys.}}$. This constraint is far too restrictive in general
and another constraint, known as reversibility, must be invoked to
generate exact differential equations we can integrate.

In the next two subsections, the concept of reversibility is introduced through consideration of cycles and is used to find exact differential descriptions of work and heat.

A thermodynamic *cycle* is a *process* applied to a
thermodynamic system which causes its state (and its state
variables) to change but eventually return to its initial state (and
so it also returns to the initial values of its state
variables). For example, the combustion chamber inside an engine
will compress and expand during its operation but it returns to its
starting volume after each cycle. This leads to the following
identity where the sum/integral of the changes over a cycle are
zero, i.e., $\oint_{\rm cycle} {\rm d} V=0$, and similar identites
must also apply for every state variable.

In 1855, Clausius observed that the integral of the heat transfer
rate over the temperature is always negative when measured over a
cycle,
This is known as the Clausius inequality. It was found that this
inequality approaches zero in the limit that the cycle is performed
slowly. This limiting result indicates that the kernel of the
integral actually contains a state variable, i.e.,
where $S_{sys.}$ is the state variable known as *entropy* of
the system. Interestingly, the entropy (like the internal energy) is
not directly observable and its existence is only revealed by this
inequality.

As the inequality is generally negative over a cycle, it indicates
that entropy always increases and must be removed from a system to
allow it to return to its initial state (except in the limit of slow
changes). This has led to the terminology of
the *irreversible* cycle, $\oint_{\rm cycle}{\rm d}S>0$, and
the idealised *reversible* cycle, $\oint_{\rm cycle}{\rm
d}S=0$, which can be returned to its starting state without removing
entropy.

Further careful reasoning which is omitted here results in the
statement of the second law of thermodynamics: *the total entropy
of a isolated system can only increase over time*. Assuming the
universe is an isolated system (at least over the timescales we're
interested in), our thermodynamic system and its surroundings must
always together have a positive (or zero) entropy change.

One last thing to note, thermodynamic processes which are not cycles may also be reversible or irreversible. For a general process to be reversible the total entropy change of the system and its surroundings together must remain zero. This allows the entropy to increase or decrease in the system, but only if the surroundings have a compensating opposite change. Irreversibility is further explored later but for now our understanding is sufficient to introduce the various forms of work.

The work term $\partial W_{sys.\to surr.}$ represents all methods of transferring of energy other than as heat. Reversible paths reduce total entropy changes to zero, which minimizes the heat transferred and actually maximizes the amount of work performed by the system for a given process. It also turns work into an exact differential!

As an illustrative example, consider the emptying of a balloon via popping it versus untying the neck and letting it go. In the first case, no work is done as the air is immediately released into the surroundings: this is the quickest path of deflating the balloon thus it maximizes entropy. Untying the neck, the air jet leaving the balloon will perform work by propelling the balloon around the room (thus yielding kinetic energy). This slower release of air has allowed work to be extracted.

All work can be expressed as a generalized driving *force*,
$\vec{F}_{sys.}$, which is *displaced* by a change in the
corresponding generalized distance, $\vec{L}_{sys.}$. For the
balloon, the force is the pressure difference in the neck (and the
air resistance, which should be equal and opposite when the system
is reversible) and the distance is the travel of the balloon. The
reversible limit corresponds to infinitesimally slow/small changes
of the distance (i.e., ${\rm d}\vec{L}$) allowing all opposing
forces time to remain in balance resulting in the following general
expression for the work.

For the balloon, there are three forms of work taking place. First,
as the volume of the balloon is decreased work must be performed to
compress the volume against the pressure of the air within. The
reversible *pressure-volume* work is then as follows,
where the pressure $p$ is the generalised force and the volume $V$
is the generalised displacement. In addition, the balloon itself is
shrinking, releasing the tension within its elastic surface. This is
known as *surface* work,
where $\gamma_{sys.}$ is the surface tension and $\Sigma_{sys.}$ is the surface
area.

As air leaves the balloon through the neck it will carry away energy
with it. This is known as *chemical* work,
where the chemical potential, $\mu_{i,{sys.}}$, is the energy added
to the system if one mole of the component $i$ (from one of the
$N_C$ components of the system) is added to or removed from the
system by any process (e.g., flow through the boundaries or internal
reactions). The definition of a component, $i$, in a thermodynamic
system is flexible and may be used to represent a single type of
atom, molecule, or elementary particle (i.e., electrons), or even a
mixture of molecules (such as "air").

The term ${\rm d} N_{i,{sys.}}$ represents changes in the amounts of
a species $i$. This change may be due to mass flowing in or out of a
system, but it may also result from reactions within a system;
However, for *closed* system (a system which cannot exchange
mass with any other system), chemical work is impossible and thus
the conservation of energy requires that the following holds true
(even if ${\rm d} N_{i,{sys.}}\neq 0$ due to internal processes such
as reactions),
Closed systems are typical during process/unit-operation
calculations; however, as these closed systems are often composed of
multiple open sub-systems (i.e. multiple interacting phases within
a closed vessel) the chemical work term is always useful to retain.

In summary, under the constraint of a reversible system, the expression for entropy (Eq. \eqref{eq:entropydefinition}) and any relevant work terms (Eq. \eqref{eq:pressurevolumework}-\eqref{eq:materialwork}) can be substituted into the energy balance of Eq. \eqref{eq:initialebalance}, to yield the fundamental thermodynamic equation, where the subscripts have been dropped from every term for convenience. Other work terms, such as the surface or electrical work, can be added to this equation depending on the system studied; however, the pressure-volume and chemical work terms are the most important from a process engineering perspective.

As we have an exact differential in Eq.\eqref{eq:fundamentalThermoRelation} if the internal energy is taken as function of $U(S,\,V,\,\left\{N_i\right\})$, then the total derivative of the internal energy in these variables is as follows, where, for clarity, the variables which are held constant while a partial derivative is taken are written as subscripts on the parenthesis surrounding the derivative (this is needed for clarity as in thermodynamics we often change the set of independent and dependent variables).

Comparing the total derivative above to the fundamental thermodynamic relation of Eq.\eqref{eq:fundamentalThermoRelation} yields the following definitions of the partial derivatives, This is the first indication that thermodynamics is a powerful tool as it has already found a differential relationship between the internal energy and the intensive properties. Also, as the variables of $U(S,\,V,\,\left\{N_i\right\})$, are all extensive, Euler's solution for homogeneous functions applies.

Consider some function (in our case an extensive thermodynamic
property), $Z$. Assume that the property is only a function of the
extensive quantities $\left\{{sys.}_i\right\}$ (these may be the molar
amounts $\left\{N_i\right\}$, volume $V$, and entropy $S$ which
are all extensive). If all of the extensive properties
$\left\{A_i\right\}$ are scaled equally by some factor, $k$, the
extensive thermodynamic property must also scale. Thus,
where $k$ is some arbitrary scaling factor. $Z$ is therefore a
homogeneous
function of first order in $\left\{A_i\right\}$. Taking the
derivative of both sides with respect to $k$ (chain rule on the
LHS):
At this point it is selected that $k=1$ and expanding the dot
product as a sum,
This allows us to solve for $Z$ if the partial derivatives in
terms of each of its extensive parameters are known.

This allows the equation to be "solved" immediately as it is a
first-order homogeneous function of the extensive properties.
This is the remarkably simple solution for the internal energy which
is the first
*thermodynamic potential* we encounter.

When performing calculations in thermodynamics, we are free to
specify our system state using any of the state variables introduced
so far
$(U,\,T,\,S,\,p,\,V,\,\left\{\mu_i\right\}^{N_C},\,\left\{N_i\right\}^{N_C})$,
but how many are required and which ones are independent? Each term
of the fundamental thermodynamic equation consists of a
so-called *conjugate pairing* of an *intensive*
property such as $T$, $p$, or $\mu_i$ and a corresponding
conjugate *extensive* property $S$, $V$, or
$\left\{N_i\right\}$ respectively. Provided all the relevant work
terms have been included, it has been observed that a thermodynamic
state is fully specified if *at least* one variable is
specified for each of the conjugate pairs considered.

The *natural variables* for a particular function are
whichever choices result in an exact differential relationship for
that function. For example, the internal energy has the natural
variables $U(S,\,V,\,\left\{N_i\right\}, \ldots)$. This is apparent
from Eq.\ref{eq:fundamentalThermoRelation}, where these variables
are all exact differentials related to ${\rm d}U$. Unfortunately,
these variables are not particularly nice (and the internal energy
is not particularly interesting) as we cannot directly measure the
entropy or internal energy in experiments. There are other
thermodynamic potentials which have more convenient natural
variables and these can be derived by considering the consequences
of the second law of thermodynamics. These are all Legendre
transforms of the internal energy thus their natural variables will
always correspond to one variable from each conjugate pair.

The second law of thermodynamics has already been introduced via the
Clausius inequality and is formally written as follows,
i.e., the total entropy of the universe (our system and its
surroundings) must always increase or remain constant. This
statement implies that the only "stationary" thermodynamic state is
where the entropy has reached its maximum, henceforth known as
the *equilibrium* state. The equilibrium state is of
particular interest as all thermodynamic systems approach it and, if
left undisturbed, remain there indefinitely.

It is often the basis of process calculations that a particular thermodynamic system has reached equilibrium, thus determining the equilibrium state (via a maximization of the total entropy) is our primary goal. Starting from some initial non-equilibrium state, some unconstrained internal parameters (e.g., composition, reaction progression) are varied such that the total entropy is maximized.

Although the universe's entropy must be maximized at equilibrium, our interest is in a smaller thermodynamic system contained within it. The total entropy is the sum of the entropy of this system within the universe and the rest of the universe, i.e., It is clear that both $S_{sys.}$ and $S_{surr.}$ may increase or decrease, provided the overall change results in an increase of $S_{total}$.

It is henceforth assumed that the surroundings are at equilibrium,
they remain at equilibrium, and any interaction with the
surroundings is reversible. The author considers these the largest
assumptions they have ever made, both physically and in terms of
approximation; however, it is equivalent to a “worst
case” estimate for the generation of entropy. In this case,
there can be no “external” process driving changes
within the thermodynamic system. Anything that the system does must
happen “spontaneously”. In this case, the only possible
mechanism by which the universe's entropy may change is via heat
transfer from the system (and the heat flux becomes an exact differential).
This makes it clear that the entropy change of the system must be
balanced against the entropy it is generating in the surroundings
through heat transfer (the surroundings are also so large the other
effects of the heat transfer are negligble). Inserting the
fundamental thermodynamic equation
(Eq.\eqref{eq:fundamentalThermoRelation}),
To simplify the remainder of this section, the thermodynamic system
is now assumed to be *closed* which allows the elimination of
the chemical potential term,
The subscript $C$ on the parenthesis is used to indicate that the
system is closed. This equation makes it clear that, in closed
systems interacting reversibly with its surroundings which are at
local equilibrium, the overall equilibrium is not solely linked to
the entropy of the system itself but is the *minimisation* of
the RHS of Eq.\eqref{eq:totalentropy} (due to the negative sign on
the entropy change). The RHS often corresponds to
some *thermodynamic potential* which arise under different
constraints and these are now derived in the sections below.

Consider a system which is completely isolated from its surroundings. It cannot exchange heat $\left({\partial}Q=0\right)$ or work $\left({\partial}W=0\right)$. As we're in the zero-work reversible limit, the system must be at constant volume $\left({\rm d}V=0\right)$ and the molar amounts $\left\{N_i\right\}$ may individually vary but only such that $\left(\sum_i^{N_C} \mu_i\,{\rm d} N_{i,A}=0\right)$. Examining the original balance in Eq.\eqref{eq:initialebalance}, thus it is clear that the isolated constraint is also equivalent to ${\rm d}V=0$ and ${\rm d}U=0$. Examining the total entropy under these constraints, where the subscripts on the brackets indicate that the $U$, $V$, are held constant in the closed system. It is clear from this expression (and our own intuition) that an isolated system, with surroundings already at equilibrium, all changes in the total entropy must arise from changes in the system entropy.

To put this in terms of minimising a thermodynamic potential
we define the negative of the entropy (sometimes
called *negentropy*),
To determine the equilibrium state the potential, $f_{U,V,C}$,
must be minimised and this action is equivalent to maximising
the total entropy.

Isolated systems are interesting in certain cases; however, in process engineering, we often have a closed system at a fixed temperature $T$ and pressure $p$. Under these conditions the system is free to transfer heat and change its volume. For the interaction of the system with its surroundings to be reversible, the surroundings must have the same temperature $T_{sys.}=T_{surr.}$ and pressure $p_{sys.}=p_{surr.}$.

If we now define a new thermodynamic potential called the Gibbs free energy, and look at changes in $G$ while holding $T$ and $p$ constant, we have, Comparing with this expression against Eq.\eqref{eq:totalentropy} it is immediately apparent that, Thus, maximisation of $S_{total}$ is equivalent to minimisation of $G$ when $T$ and $p$ are held constant. Writing this in the same notation as before we have,

Again, reversibility requires that $T_{surr.}=T_{sys.}$. No pressure-volume work occurs as the volume is fixed and so the pressure of the surroundings is actually irrelevant. Material work is also ignored as the system is closed. In the final equality, another thermodynamic potential is introduced: the Helmholtz free energy, It is now clear that under these constraints the maximum total entropy is reached at the minimum Helmholtz free energy. I.e.,

The enthalpy is defined as follows, For constant pressure and enthalpy we have, Examining the total entropy under these constraints (Eq.\eqref{eq:totalentropy}), Even though the surroundings temperature is unknown, it is merely a scaling factor and again the maximisation of the total entropy is equivalent to the minimisation of the system's negentropy,

Again, starting with the total entropy of a closed system, Eq.\eqref{eq:totalentropy}, We note that, Thus, we have The surroundings can be at some arbitrary constant temperature (they are at equilibrium and very large thus unaffected by heat transfer), thus $T_{surr.}$ is simply a scaling factor. The thermodynamic potential to minimise is then $f_{p,S,C} = H_{sys.}$.

Finally, the simplest example, Thus, the thermodynamic potential to minimise for maximum total entropy is $f_{S,V,C} = U_{sys.}$ and the surroundings temperature is unimportant.

In summary, there are a number of relevant thermodynamic potentials for a closed system. These are defined below,

For each set of constrained thermodynamic states in closed systems, a particular thermodynamic potential is minimised at equilibrium. These are summarised in the table below:

Constants | Function to minimise, $f$ |
---|---|

$p,\,S$ | $H$ |

$p,\,T$ | $G$ |

$p,\,H$ | $-S$ |

$V,\,S$ | $U$ |

$V,\,T$ | $A$ |

$V,\,U$ | $-S$ |

The variables held constant correspond to the "natural" variables of each potential. Expressing the change in each thermodynamic potential in terms of these natural variables yields the following differential equations, The significance of the chemical potential cannot be overstated. It is the change of each thermodynamic potential per mole of each species exchanged when the other natural variables of the potential are held constant, The implication of this is that when dealing with systems exchanging mass, but constrained by two "natural" variables, the chemical potential for each species must be equal in all phases, regardless of which constrained variables are actually used (otherwise a change of mass between systems could change the value of the overall thermodynamic potential implying it is not at a minimum). It is also the partial molar Gibbs free energy ($G=\sum_i N_i\,\mu_i$) and thus calculation of the Gibbs free energy can be reduced to considering the chemical potential.

Now that equilibrium has been defined, how do we calculate the
equilibrium state? To determine the equilibrium state,
$\vec{X}_{equil.}$, of a closed system containing many sub-systems a
thermodynamic potential, $f$, is minimised.
where $\vec{X}$ represents all the variables used by SimCem to
describe the state of all the $N_p$ sub-systems within the closed
system. A particular sub-system/"model",
$\alpha\in[1,N_p]$, will typically have $N_{C,\,\alpha}$ molar amounts,
i.e., $\left\{N_{i,\alpha}\right\}^{N_{C,\,\alpha}}$, the
temperature $T_\alpha$, and either the pressure $p_\alpha$ or the
volume $V_\alpha$ depending on the model used.
To keep the minimisation in a physical region, constraints are added
to make sure all these variables remain positive, i.e.,
Finally, there are a number of constraints which we write in general
as follows,
where $k$ is the index of an equality constraint which holds some
function of the state variables, $g_k\left(\vec{X}\right)$, to a
value of zero. One example of these are the *material*
constraints arising from mass/mole balances (e.g. conservation of
elements) whereas the other constraints arise from constraints on
thermodynamic variables (e.g., constant enthalpy and
pressure). Before these are discussed, we review how constrained
minimisation is carried out.

To actually calclulate constrained minimisation problems they are
often transformed into unconstrained searches for stationary points
using the method of Lagrange multipliers. A new function called the
Lagrangian, $F$, is constructed like so,
where $\lambda_k$ is the Lagrange multiplier for the $k$th equality
constraint. The positivity constraints can be simply added using
bounds checking (other more advanced techniques are available but
are irrelevant for the discussion). The Lagrangian has the unique
property that the constrained minima's of $f$ now occur at the
(unconstrained) *extrema* of $F$. For example, it is easy to
see that the derivatives of $F$ with respect to each Lagrange
multiplier are zero if the constraints are satisfied,
Thus we are searching for a point where $\partial F/\partial
\lambda_k=0$ to ensure the constraints are satisfied. Taking a
derivative of the Lagrangian with respect to the state variables,
$\vec{X}$, yields the following,
or in vector notation,
Lets consider when $\partial F/\partial
\vec{X}=\nabla_\vec{X}\,F=\vec{0}$, at this point the following must be
true,
This makes it clear that at the point where $\partial F/\partial
\vec{X}=\vec{0}$, the downhill direction of $f$ can be decomposed into
directions where the constraint functions also change
($\nabla_\vec{X}\,c_k$ are basis vectors of $\nabla_\vec{X}\,f$, and
$-\lambda_k$ are the coordinate values in this basis). Thus,
attempting to lower $f$ any further will cause the constraint values
to alter away from zero if they are already satisfied at this point
(as guaranteed by $\frac{\partial F}{\partial \lambda_k}=0$).

As a result, the new strategy to find equilibrium is to find a stationary point of the Lagrangian, This is a stationary point and not a maximum or minimum as no statements on the second derivatives of the Lagrangian have been made. In fact, most stationary points of the Lagrangian turn out to be saddle points therefore minimisation of the Lagrangian is not suitable, instead a root search for the first derivative of the Lagrangian may be attempted. Even this approach may converge to a stationary point of $F$ which is not a minimum but a maximisation of $f$. Implementation of a suitable routine is therefore an art but fortunately general algorithms are available which perform this analysis, such as NLopt, Opt++, and SLSQP.

The purpose of introducing the Lagrangian is to demonstrate that derivatives of $f$ and $g_k$ are needed, but also to introduce the Lagrangian multipliers $\lambda_k$, which will later be shown to correspond to physical properties of the system. We will summarise the values that must be calculated before discussing how to calculate these values.

To determine the extreema of the Lagrangian, the minimisation algorithms require the derivatives of the Lagrangian. As illustrated above, the derivatives with respect to the lagrangian multipliers are given by the constraint functions themselves, thus no additional calculation is required there. Ignoring any contribution from the interfaces between phases, the thermodynamic potentials of the overall system can be broken down into the contribution from each phase, where $f_\alpha$ is the contribution arising from a single phase, $\alpha$. This allows us to rewrite the derivative of the Lagrangian with respect to the state variables as follows, It should be noted that the following is true, Thus, each individual phase's derivatives can be considered separately as only the $\partial f_\alpha/\partial \vec{X}_\alpha$ terms are nonzero. Later, when models are considered, these derivatives will be generated. Now we must consider the general constraints.

The Gibbs phase rule states that the number of independent intensive variables (AKA degrees of freedom), $F$, required to completely specify the equilibrium state of a thermodynamic system is, where $N_P$ is the number of phases and $C$ is the number of independent components in the system. It should be noted that in general $C\neq N_C$, as components may be linked by the constraints of elemental or molecular balances.

In SimCem, $\sum_\alpha^{N_P}\left(N_{C,\alpha}+2\right)$ state variables are always used to describe a system (there are $N_{C,\alpha}$ molar amounts $\left\{N_{i,\alpha}\right\}^{N_{C,\alpha}}$, the subsystem temperature $T_\alpha$, and either the subsystem pressure $p_\alpha$ or volume $V_\alpha$ for each subsystem). In general, $\sum_\alpha^{N_P}\left(N_{C,\alpha}+2\right)\ge C+2-N_P$, thus the state of a multi-phase and/or reactive system, $\vec{X}$, is typically over specified and constraints must be added to the minimisation to eliminate the additional degrees of freedom.

Two systems in equilibrium must have equal temperature, pressure, and chemical potential. These constraints will arise naturally from the minimisation; however, it is efficient to remove any variables of the minimisation wherever we can (and also helps with numerical accuracy/stability).

As the temperatures of all phases are equal at equilibrium and all models used in SimCem have a temperature variable, $\left\{T_\alpha\right\}^{N_{p}}$, these individual values are eliminated from $\vec{X}$ and set equal to a single system temperature, $T$, which is inserted into $\vec{X}$.

If a constant temperature is being considered, then the system temperature, $T$, is simply set to the constrained value and eliminated from $\vec{X}$ entirely. If temperature is free, then a constraint on $S$, $H$, or $U$ is added, i.e.,

Not all models used in SimCem have a pressure variable, thus it is a little more challenging to reduce it to a single value in $\vec{X}$, so this is not done (yet); however, if constant pressure is required, then the pressure of the first phase is constrained only and the other phases then equilibrate to this pressure via the minimisation, If the volume is held constant then a different constraint function is used, Any phase volumes appearing as independent variables must remain in $\vec{X}$ as it is the overall volume of the system which is constrained to $V_{target}$, thus individual phases themselves have unconstrained volumes.

SimCem has a reactive and non-reactive mode which selects between
the conservation of elements or species respectively. For example,
consider the water/steam equilibrium system,
The variables for this system in SimCem are typically as follows,
H_{2}O may be present in both the steam and water phase, but
the total amount is constrained to the initial amount,
$N_{H_2O}^0$. Thus we'd like to add a **non-reactive** species
constraint,
In **reactive** systems, the types of molecules are no-longer
conserved but the elements are. For example, consider the combustion
of graphite,
The state variables are,
Selecting a **reactive** system, SimCem will generate elemental
constraints,
Elemental constraints allow any and all possible
reactions/rearrangements of elements to minimise the free
energy. For example, the following reaction is also allowed by these
constraints,
Sometimes this is not desired, and only specific reaction paths are
fast enough to be considered (e.g. in catalysed reactions). In this
case, custom constraints will need to be implemented. At the moment,
Simcem will only automatically generate elemental and
molecular constraints.

Consider the water-steam system again. Imagine that a user selects
this as a **reactive** system. SimCem will attempt to use a
hydrogen and oxygen balance constraint:
These two constraints are identical aside from a multiplicative
factor. This leads to ambiguity in the values of the lagrange
multipliers as either constraint can apply and this indeterminacy
can cause issues with solvers, so we need to eliminate them.

Both elemental and species constraints are linear functions of the molar amounts, $N_i$, thus they can be expressed in matrix form, where $\vec{C}$ is a matrix where each row has an entry for the amount of either an element or molecule represented by a particular molar amount and $\vec{N}^0$ is the initial molar amounts.

To determine which rows of the matrix $\vec{C}$ are redundant, we perform singular-value decomposition on the matrix $\vec{C}=\vec{U}\,\vec{S}\,\vec{V}^T$. The benefit of this is the rows of $V^T$ corresponding to the non-zero singular values form a set of orthagonal constraints equivalent to the original constraint matrix $\vec{C}$, so we use the so-called "thin" $\vec{V}^{T}$ (only containing the non-zero singular rows) as our new constraint matrix. As we would like to extract the original lagrangian multipliers for later calculations, we need to be able to map back and forth between the original and reduced lagrangians. This mapping can be found by considering the original constraint and its lagrangians, where the final line implicitly defines the reduced set of lagrange multipliers $\vec{\lambda}_r$. Performing the minimisation using $\vec{V}^T$, we can then recover the original lagrange multipliers like so, As a side note, the matrices $\vec{U}$ and $\vec{T}$ are orthonormal/rotation matrices thus the tranpose is their inverse. Also, as the diagonal matrix $\vec{S}$ is singular, $\vec{S}^{-1}$ is the generalised inverse (i.e., only the non-zero diagonal values are inverted).

With the constraints outlined, the actual definition of thermodynamic models and the calculations may begin.

For the constraint functions, the only non-zero derivative of the element/molecular constraint function given in Eq.\eqref{eq:genMolConstraint} is as follows,

The thermodynamic potentials, when expressed as a function of their natural variables, provide a complete description of the state of a thermodynamic system. For example, consider the Gibbs free energy expressed in terms of its natural variables $G\left(T,\,p,\,\left\{N_i\right\}\right)$. If we know the values of the natural variables, we can calculate $G$. The derivatives of $G$ (see Eq.\eqref{eq:dG}) also allow us to calculate the following properties, If we evaluate the derivatives using the value of the natural variables then the values of all other thermodynamic potentials can be determined using Eqs.\eqref{eq:Urule}-\eqref{eq:Grule}. E.g. We can take further derivatives to evaluate properties such as the heat capacity $C_p = \left(\partial H/\partial T\right)_{p,\left\{N_i\right\}}$. In this way, an entire thermodynamic model can be created by only specifying the functional form of $G(T,\,p,\,\left\{N_i\right\})$. This is known as a generating function approach, and this can be applied to any thermodynamic potential in its natural variables. Most commonly either the Gibbs or Helmholtz $A(T,\,V,\,\left\{N_i\right\})$ free energy are used as generating functions due to the convenience of their natural variables.

It is sometimes convenient to generate enthalpy rather than entropy from the Gibbs free energy. Consider a reversible process in a closed system, This illustrates that the volume and enthalpy can also be directly obtained from the functional form of $G(T,p)/\,T$, by taking derivatives in its natural variables (while holding the other natural state variables, including $N_i$, constant).

Specifying the entire thermodynamic system using a single
thermodynamic potential guarantees *thermodynamic
consistency*. This is the requirement that all properties are
correctly related via their derivatives. This may be accidentally
violated if the system is specified another way, i.e., via any of
the derivatives. Many initial thermodynamic models are inconsistent
as they specify simple polynomials in temperature for both the heat
capacity and the density, but these cannot be integrated into a
consistent thermodynamic potential.

A large number of thermodynamic derivatives are required to implement the minimisation. This section presents a number of useful expressions and approaches which allow us to interrelate various derivatives to reduce the number which must be implemented to a minimal amount.

First, generalised partial properties are introduced to demonstrate that all extensive properties can be expressed in terms of their derivatives in their extensive variables. The Bridgman tables provide a convenient method of expressing any derivative of the thermodynamic potentials or their variables in terms of just three "material derivatives". A "generalised" product rule is then introduced to demonstrate how derivatives may be interchanged, particularly for the triple product rule. Finally, a key relationship between the partial molar and partial "volar" properties is derived.

As demonstrated in the section on solution for the internal energy, functions which are first-order homogeneous in their variables can be immediately expressed in terms of these derivatives. This is useful as it provided a relationship between the internal energy and the other thermodynamic potentials; however, a generalised rule would eliminate the need to generate expressions for thermodynamic properties IF their derivatives are available.

Unfortunately, the two variable sets considered here contain both
homogeneous first-order ($\left\{N_i\right\}$, $V$) and
inhomogeneous ($T$, $p$) variables. In this case, Euler's method
does not extend to expressions which are functions of both; However,
a similar solution can be derived for these expressions provided the
inhomogeneous variables are restricted to intensive
properties *and* the extensive variables together uniquely
specify the total size of the system.

The derivation presented here is a generalisation of the proof for
partial *molar* properties which you can find in any
thermodynamic text (e.g., Smith, van Ness, and Abbott, Sec.11.2, 7th
Ed). Consider an extensive thermodynamic property, $M$, which is a
function of extensive $\left\{X_i\right\}$ and intensive
$\left\{y_i\right\}$ variables. The total differential is as
follows,
The extensive property $M$ can be converted to a corresponding
intensive property, $m$, by dividing by the system amount, i.e.,
$M=m\,N$. If the extensive properties are held constant and *
they are sufficient to determine the total system amount*, $N$,
then the system size, $N$, is also constant and may be factored out
of $M$ in the intensive partial differential terms.
In addition, ${\rm d} M={\rm d} (N\,m) = m\,{\rm d} N+N\,{\rm
d}m$ and ${\rm d} X_i={\rm d} (N\,x_i) = x_i\,{\rm d} N+N\,{\rm
d}x_i$. Inserting these and factoring out the terms in $N$ and
${\rm d}N$ yields,
As ${\rm d}N$ and $N$ can vary independently this equation is only
satisfied if the terms in parenthesis are each zero. Multiplying the
first term in parenthesis by $N$ and setting it equal to zero yields
the required identity,
where $\bar{\bar{m}}_i=\left(\partial M/\partial
X_i\right)_{\left\{X_{i\neq j}\right\},\left\{y_{k}\right\}}$ is a
partial property.
This is a generalised solution for an extensive property in terms of
its partial properties which are its derivatives in its extensive
variables and is the primary objective of our derivation; however,
an additional important equation for the partial properties can be
easily obtained. The derivative of Eq.\eqref{eq:partialProperty}
(first divided by $N$) is,
This can be used to eliminate ${\rm d}m$ from the right term of
Eq.\eqref{eq:pmolarintermediate} which is also set equal to zero
(and multiplied by N) to give,
This is the generalised **Gibbs/Duhem** equation which
interrelates the intensive properties of the system to the partial
properties.

The most well known applications of Eqs.\eqref{eq:partialProperty}
and \eqref{eq:GibbsDuhem} are for the partial *molar*
properties when $p$, $T$, and $\left\{N_i\right\}$ are the state
variables. In this case Eq.\eqref{eq:partialProperty} is
where $\bar{m}_i=\left(\partial M/\partial
N_i\right)_{p,T,\left\{N_{j\neq i}\right\}}$ is the partial molar
property. The most important partial molar property is the chemical
potential,
and it has already been proven that
Eq.\eqref{eq:partialMolarProperty} applies in this case, i.e.,
Eq.\eqref{eq:Grule} gives $G=\sum_iN_i\,\mu_i$. The corresponding
Gibbs-Duhem equation for the chemical potential in
$T,p,\left\{N_i\right\}$ is the most well-known form,

As derivatives are distributive, and $T$ and $p$ are held constant, the partial molar properties for the thermodynamic potentials satisfy similar relationships as the original potentials (Eqs.\eqref{eq:Urule}-\eqref{eq:Grule}). Thus, the partial molar properties not only provide a derivative in the molar amounts, but also completely describe all thermodynamic potentials and many extensive quantities (such as $V$). Simcem therefore only requires the implementation of expressions for three partial molar amounts ($\mu_i$, $\bar{v}_i$, $\bar{h}_i$) to specify all the thermodynamic potentials and their first derivative in the molar amounts for the $\left\{T,p,\left\{N_i\right\}\right\}$ variable set.

For the $\left\{T,V,\left\{N_i\right\}\right\}$ variable set, Eq.\eqref{eq:partialProperty} is, where $\breve{m}_i = \left(\partial M/\partial N_i\right)_{T,V,\left\{N_{j\neq i}\right\}}$ is deemed here to be a partial "volar" quantity. A more useful form of this expression is Eq.\eqref{eq:molartovolarproperty} which is derived in the later section on the transformation between $\bar{m}_i$ and $\breve{m}_i$.

There are a number of interesting properties which are derivatives of the thermodynamic potentials. For example, consider the isobaric heat capacity,

Using partial molar properties allows us to illustrate a complication in the calculation of these material properties for multi-phases/sub-systems. For example, expressing the (extensive) heat capacity in terms of the partial molar (AKA specific) heat capacity using Eq.\eqref{eq:partialMolarProperty} yields the following expression, where $\bar{c}_{p,i}=\left(\partial \bar{h}_i/\partial T\right)_{p}$ is the partial molar isobaric heat capacity. The term on the LHS of Eq.\eqref{eq:mixCp} arises from changes to the enthalpy caused by species transferring in and out of the system as the equilibrium state changes. This results in additional contributions to the apparent heat capacity above the partial molar isobaric heat capacity. For example, when a single-component fluid is at its boiling point, the apparent heat capacity of the overall system is infinite as $\partial N_i/\partial T$ is infinite due to the discontinuous change from liquid to vapour causing the instananeous transfer of molecules from one phase to another.

To complement the "equilibrium" thermodynamic $C_p$ above, it is convenient to define "frozen" thermodynamic derivatives where there are no molar fluxes, i.e., the "frozen" isobaric heat capacity, The "frozen" properties are required while calculating the gradient of thermodynamic potentials during minimisation, and arise as all molar quantities are held constant while these derivatives are taken.

The $C_p$ is just one material property; however, there are many
other thermodynamic derivatives which may be
calculated. Fortunately,
the Bridgman
tables is a convenient method to express any material property
as a function of just three key material derivatives. The heat
capacity is one and the other two are the isothermal ($\beta_T$) and
isobaric ($\alpha$) expansivities,
where $\bar{\alpha}_i\,\bar{v}_i= \left(\partial
\bar{v}_{i}/\partial T\right)_{p,\left\{N_j\right\}}$ and
$\bar{\beta}_{T,i}\,\bar{v}_i=-\left(\partial v_i/\partial
p\right)_{T,\left\{N_j\right\}}$. Again, "frozen" material
derivatives are available,
The terms $\left(\partial N_i/\partial T\right)_{p,\left\{N_{j\neq
i}\right\}}$ and $\left(\partial N_i/\partial
p\right)_{T,\left\{N_{j\neq i}\right\}}$ which appear in the
material properties must be determined from the solution to the
thermodynamic minimisation problem. They quantify how
the *equilibrium* molar amounts change for a variation in
temperature and pressure and thus must account for the constraints
placed on the equilibrium state and the movement of the minimia.

The Bridgman table approach decomposes every thermodynamic derivative into a look-up table for the numerator and denominator expressed in terms of the three material derivatives, $C_p$, $\alpha$, and $\beta_T$. For example, consider the following "unnatural" derivative, Thus, to generate any derivative required for minimisation, only the three "material" derivatives and three partial molar properties are required.

This section is almost directly copied from this math.StackExchange.com post.

Consider any expression for a thermodynamic property, $M$, written in terms of the state variables, $M=M\left(\vec{X}\right)$. It is straightforward to transform this function into an implicit function $F=F\left(M,\,\vec{X}\right)=M(\vec{X}) - M=0$ which helps to illustrate that $M$ can be treated as a variable on an equal basis as $\vec{X}$. To allow a uniform representation, the arguments of $F$ are relabeled such that $F(x_1,x_2,\ldots,x_N)=0$. As the function $F$ remains at zero, holding all variables except two constant taking the total derivative and setting ${\rm d}F=0$ yields the following identity, This rule can be combined repeatedly and the terms on the RHS eliminated provided the variables loop back on themselves. For example, where the variables held constant were dropped for brevity on the right hand side. The first value $n=2$ yields the well known expression, or, more familiarly (and without the explicit variables held constant), Finally, $n=3$ yields the triple product rule, where it is implied that all other variables other than $x_1$, $x_2$, and $x_3$ are held constant. This rule has wide application but is particularly attractive when derivatives hold an "awkward" thermodynamic property constant. For example, consider the following case The LHS is difficult to directly calculate in the state variables chosen here; however, the RHS arising from the triple product rule is in terms of natural derivatives of $G$ which are straightforward to calculate. The triple product rule is used extensively in the following sections to express complex expressions in terms of natural derivatives.

Consider a property, $M$, which is a function for four variables $x_1,x_2,x_3,x_4$. The total derivative is, Holding two variables constant and taking a derivative wrt a third yields, Three relablings of this expression can be used to interrelate two derivatives which only differ in one variable which is held constant. where the final term in parenthesis is cancelled to zero using the triple product rule. This equation is particularly useful for changing between partial quantities while pressure or volume is held constant. For example, or, in the notation used so far, This is a more useful form of Eq.\eqref{eq:partialVolarProperty}. The partial molar volume is inconvenient to derive directly when volume is an explicit variable; however, it may be expressed more conveniently using the triple product rule, This allows us to obtain partial molar properties conveniently when working with volume as a state variable.

In this section, the calculation of the required properties for minimisation with phases specified by the following set of variables is considered, In this particular variable set, the required constraint derivatives to implement the derivative of the Lagrangian are as follows, The thermodynamic potential derivatives required to specify the derivatives of the Lagrangian (Eq.\eqref{eq:Fderiv}) as generated from Eqs.\eqref{eq:GradEqsStart}-\eqref{eq:GradEqsEnd} are where $f=\left\{H,G,-S,U,A\right\}$. These derivatives are easily expressed using the Bridgman tables in terms of the three standard material derivatives and the results are given in the table below.

$Y_1$ | $Y_2$ | $f$ | $\left(\frac{\partial f}{\partial N_i}\right)_{T,p,\left\{N_{j\neq i}\right\}}$ | $\left(\frac{\partial f}{\partial T}\right)_{p,\left\{N_j\right\}}$ | $\left(\frac{\partial f}{\partial p}\right)_{T,\left\{N_j\right\}}$ |
---|---|---|---|---|---|

$p$ | $T$ | $G$ | $\mu_i$ | $-S$ | $V$ |

$V$ | $T$ | $A$ | $\bar{a}_i$ | $-(S+p\,\alpha\,V)$ | $p\left(\beta_{T}\,V\right)_{\left\{N_i\right\}}$ |

$p$ | $S$ | $H$ | $\bar{h}_i$ | $C_{p,\left\{N_j\right\}}$ | $V(1-T\,\alpha)$ |

$p$ $V$ | $H$ $U$ | $-S$ | $-\bar{s}_i$ | $-T^{-1}\,C_{p,\left\{N_j\right\}}$ | $\left(\alpha\,V\right)_{\left\{N_i\right\}}$ |

$V$ | $S$ | $U$ | $\bar{u}_i$ | $C_{p,\left\{N_j\right\}}-p\left(\alpha\,V\right)_{\left\{N_i\right\}}$ | $p\left(\beta_{T}\,V\right)_{\left\{N_i\right\}}-T\,\left(\alpha\,V\right)_{\left\{N_i\right\}}$ |

In summary, a model using this variable set must provide implementations of $\mu_i$, $\bar{v}_i$, $\bar{s}_i$, $\bar{\alpha}_i$, $\bar{\beta}_{T,i}$, and $\bar{c}_{p,i}$. These are all straightforward to obtain by performing derivatives of a Gibbs free energy function in its natural variables or integration and differentiation if a mechanical equation of state, $V\left(T,p,\left\{N_i\right\}\right)$, is available.

All other partial molar properties are obtained using Eqs.\eqref{eq:partialmolarrelationstart}-\eqref{eq:partialmolarrelationend} expressed in the following form. Most other relevant thermodynamic properties are calculated using the Bridgman tables.

The ideal gas model is not only a good approximation for gases at low pressures, but it is also a key reference "state" for most complex equations. A thermodynamic path can be constructed to the ideal gas state at low pressures for most models, thus in this case a "ideal-gas" contribution can be factored out from these models.

The ideal gas chemical potential is defined as follows, where $p_0$ is the reference pressure at which the temperature-dependent term, $\mu^{(ig)}_{i}(T)$, is measured and $N_\alpha$ is the total moles of the phase $\alpha$.

Using the chemical potential as a generating function, the minimial set of partial molar properties is derived below. where the partial molar heat capacity has been implicitly defined as $C_{p,\left\{N_i\right\}}^{(ig)} = \sum_i N_i\,\bar{c}_{p,i}^{(ig)}$.

The model is not completely specified until the pure function of temperature, $\mu_{i}^{(ig)}(T)=h_{i}^{(ig)}(T) - T\,s_{i}^{(ig)}(T)$, is specified, and this is discussed in the following section.

This is not a proof of how mixing entropy arises (which is discussed below), but a demonstration that mixing entropy is consistent with Dalton's law holding true. Consider a single component ideal gas, we have, If Dalton's law is true, the pressure of a single species is reduced when mixed as it only exerts a partial pressure, ($p_i=x_i\,p$). Assuming the total pressure and temperature remains constant during mixing, As $x\le1$, $g_{mixed} - g_{pure}\le0$, which is consistent that two ideal gases will be mixed at equilibrium under constant $T,p$. Thus, Dalton's law is consistent with mixing entropy

First, it is noted that entropy is extensive. The entropy of
two separate systems is the sum of each,
Fundamentally, entropy is a measure of the number of states of
a system; however, the number of states
has *multiplicative* scaling. For example, consider a
collection of light switches. A single light switch has two
states but a collection of $N$ light switches has $2^N$
possible states. The number of states is clearly
multiplicative and yet entropy is additive. The entropy must
therefore be the logarithm of the number of states as $\ln 2^N
= N \ln 2$. This line of arguing leads to what is known as
Boltzmann's entropy,
where $W$ is the number of states and $R$ makes the connection
to the thermal scale.

Now consider a binary mixture where each location of the system is on a lattice. Each "site" of the lattice may be occupied by one of two end-members/molecules, $N_1$ or $N_2$. The total number of available sites is $N=N_1+N_2$ (the lattice is full). As each end-member/molecule is indistinguishable from other molecules of the same type, the number of ways we can arrange these molecules are, The entropy is then, Using Stirling's approximation $\lim_{n\to\infty}\ln(n!)=n\,\ln(n)-n$. For a gas, consider that the number of sites is actually the number of molecules in the system and all we are doing is allowing the gases to mix (each species may substitute for another species provided the overall molar balance is maintained), then it is clear that: In comparison with the ideal gas properties, this is the entropy of mixing as derived from Dalton's law!

The term $\mu_{i}^{(ig)}(T)$ is a function of temperature which is directly related to the thermodynamic properties of a single isolated molecule (molecules of ideal gases do not interact). There are many theoretical approaches to expressing these terms for solids, simple models (i.e., rotors), and quantum mechanics can even directly calculate values for real molecules. However these results are obtained, this term is typically parameterised using polynomial equations.

The most common parameterisation is to use a polynomial for the heat capacity. This is common to the NIST, NASA CEA, ChemKin, and many other thermodynamic databases. A heat capacity polynomial can be related to the enthalpy and entropy through two additional constants, To demonstrate how this is correct, consider a closed system, At constant pressure the following expression holds, Thus, where $T_{ref}$ is a reference Performing this integration for the polynomial of Eq.\eqref{eq:cppoly}, where the two additional constants $\bar{h}_i^0$ and $\bar{s}_i^0$ must be determined through construction of a thermodynamic path to a known value of the entropy and enthalpy. This is usually through heats of reaction, solution, and equilibria such as vapour pressure, linking to the elements at standard conditions.

To allow this data to be expressed as a single function in Simcem, it is expressed directly as $\mu_i^{(ig)}(T)$ using the following transformation, where,

The most comprehensive (and most importantly, free!) source of ideal gas heat capacities and constants is the NASA CEA database, but additional constants are widely available. When collecting $c_p$ data, a distinction must be made between the calculated ideal gas properties and the measured data. The measured data may include additional contributions from the interactions of the molecules and thus must be fitted with the full chemical potential model and not just the ideal gas model.

Comparison with Eq.\eqref{eq:Mu_shomate_form} yields the definitions of the coefficients.

Another useful reference state is the incompressible phase. This model is used to describe liquid and solid phases when an equation of state describing all phases at once is unavailable. The generating chemical potential is as follows, where $\mu^{(ip)}_{i}(T)$ is again a temperature-dependent term and $\bar{v}^0_{i}$ is the (constant) partial molar volume of the incompressible species $i$. Using the expression for the chemical potential as a generating function all required properties are recovered, As $\alpha$ and $\beta_T$ are zero, some thermodynamic properties are not well specified but can be determined by other means. For example, $\bar{c}_p^{(ip)}=\bar{c}_v^{(ip)}$. In Simcem, ideal solids still contain a mixing entropy term which is included to allow ideal solid solutions to be constructed and used as a base class for more complex models.

In this section, the calculation of the required properties for minimisation with phases specified by the following set of variables is considered, In this variable set, the required non-zero derivatives to compute the constraint functions are as follows, These derivatives are convenient to determine in this variable set and also indirectly specify $\alpha$ and $\beta_T$ which are two of the three required material derivatives to allow use of the Bridgman tables. The third molar derivative is also convenient to determine and provides a direct path to the molar volume as given in Eq.\eqref{eq:TVpartialmolarvolume} and reproduced below, Thus, these three derivatives are required to be implemented by all models and used to derive other properties. The derivatives of the potentials are as follows where $f=\left\{H,G,-S,U,A\right\}$. The derivatives for each potential are specified below in terms of the most convenient properties/derivatives for this variable set.

$Y_1$ | $Y_2$ | $f$ | $\left(\frac{\partial f}{\partial N_i}\right)_{T,V,\left\{N_{j\neq i}\right\}}$ | $\left(\frac{\partial f}{\partial T}\right)_{V,\left\{N_j\right\}}$ | $\left(\frac{\partial f}{\partial V}\right)_{T,\left\{N_j\right\}}$ |
---|---|---|---|---|---|

$p$ | $T$ | $G$ | $\breve{g}_i$ | $V\,\left(\frac{\partial p}{\partial T}\right)_{V,\left\{N_i \right\}} -S$ | $V \left(\frac{\partial p}{\partial V}\right)_{T,\left\{N_i\right\}}$ |

$V$ | $T$ | $A$ | $\breve{a}_i$ | $-S$ | $-p$ |

$p$ | $S$ | $H$ | $\breve{h}_i$ | $C_{V,\left\{N_i\right\}}+V\,\left(\frac{\partial p}{\partial T}\right)_{V, \left\{N_i\right\}}$ | $V\left(\frac{\partial p}{\partial V}\right)_{T,\left\{N_i\right\}}+T\left(\frac{\partial p}{\partial T}\right)_{V,\left\{N_i\right\}}$ |

$p$ $V$ | $H$ $U$ | $-S$ | $-\breve{s}_i$ | $-\frac{C_{V,\left\{N_i\right\}}}{T}$ | $-\left(\frac{\partial p}{\partial T}\right)_{V,\left\{N_i\right\}}$ |

$V$ | $S$ | $U$ | $\breve{u}_i$ | $C_{V,\left\{N_i\right\}}$ | $T\,\left(\frac{\partial p}{\partial T}\right)_{V,\left\{N_i\right\}} - p$ |

where $C_V = \left(\partial U/\partial T\right)_V$, and is related to the isobaric heat capacity using the following relationship, In summary, models should provide equations to calculate $p$, $\breve{a}_i$, $\breve{u}_i$, $C_{v,\left\{N_i\right\}}$, $\left(\partial p/\partial V\right)_{T,\left\{N_i\right\}}$, $\left(\partial p/\partial T\right)_{V,\left\{N_i\right\}}$, $\left(\partial p/\partial N_i\right)_{T,V,\left\{N_{j\neq i}\right\}}$. These derivatives are used to compute the frozen $\alpha$ and $\beta_T$ values using Eqs.\eqref{eq:VMatDeriv1} and \eqref{eq:VMatDeriv1}. The third material derivative, $C_{p,\left\{N_i\right\}}$, is then obtained from Eq.\eqref{eq:CpCv}. The partial molar volume is calculated from Eq.\eqref{eq:TVpartialmolarvolume}. The partial volar/molar quantities are related by Eq.\eqref{eq:molartovolarproperty}; however, the partial volar Helmholtz free energy is equal to the chemical potential, just like any molar derivative of a potential when its natural variables are held constant (see Eq.\eqref{eq:ChemPotDefinition}). Thus, the partial volar/molar Gibbs and Helmholtz free energies are closely related, All other required partial thermodynamic potential properties are derived using straightforward applications of Eq.\eqref{eq:molartovolarproperty} and the partial molar relations of Eqs.\eqref{eq:partialmolarrelationstart}-\eqref{eq:partialmolarrelationend}, the results of which are below,

As experimental ideal-gas data is usually presented in terms of isobaric polynomials, the ideal gas functions here are based on the previous form of the ideal gas model given in Eq.\eqref{eq:idealgasmuTP}. Transforming the pressure variable to phase volume yields the following definition of the chemical potential and partial volar Helmholtz free energy, The pressure derivatives are obtained from the well known relation $p=N\,R\,T/V$, Finally, the internal energy and heat capacity are as follows,

Assume a phase/model is infinite in quantity, therefore its extensive state variables must be excluded from the system under study. The infinite phase can exchange mass with the system under study, thus it has a chemical potential. The change in the Gibb's free energy of the (infinite) system can be defined in terms of the changes in the species within it: \begin{align} \Delta G = \sum_i \mu_i\,\Delta N_i \end{align} where $\mu_i$ is the constant chemical potential. We note then that the entropy change and volume change of this system due to variation in the system temperature/pressure must be zero.

The best introduction to automatic differentiation I found are here from the 2010 Sintef winter school and these notes are largely based on those notes.

Our goal is to evaluate a function and its derivatives at once. By considering how derivatives are propagated through operators/functions we see that the $k$th derivative of a function $f(g)$ can be expressed in terms of the first $k$ derivatives of $g$. To prove this, consider the Taylor expansion of a general function $f$: \begin{align*} f(a+x) &= f(a) + \frac{1}{1!}\left(\frac{\partial f}{\partial x}\right)(a)(x-a) + \frac{1}{2!}\left(\frac{\partial^2 f}{\partial x^2}\right)(a)(x-a)^2 + \ldots \\ &= \sum_{k=0}^\infty \frac{1}{k!}\left(\frac{\partial^k f}{\partial x^k}\right)(a)(x-a)^k \\ &= \sum_{k=0}^\infty f_k(a)(x-a)^k \end{align*} Each Taylor term $f_k$ corresponds to the $k$th derivative (divided by $k!$). Thus if we can determine how the Taylor coefficients of the result of an operation are related to the Taylor coefficients of its arguments, we can propogate derivative information through the operation.

To demonstrate this we start from two basic definitions: one for the Taylor coefficients of a constant, $c$: \begin{align*} (c)_k = \begin{cases} c & \text{for $k=0$}\\ 0 & \text{for $k>0$} \end{cases}, \end{align*} and another for the Taylor coefficients of a variable, $x$: \begin{align*} (x)_k = \begin{cases} x & \text{for $k=0$}\\ 1 & \text{for $k=1$}\\ 0 & \text{for $k>1$} \end{cases}. \end{align*} Our first operations are addition and subtraction, where it is easy to see the Taylor series of the result is trivially calcluated from the Taylor series of the arguments: \begin{align*} \left(f+g\right)_k &= f_k + g_k\\ \left(f-g\right)_k &= f_k - g_k \end{align*} For multiplication, consider Taylor expansions of both the result and the two functions: \begin{align*} \sum_{k=0}^\infty \left(f\,g\right)_k (x-a)^k &= \sum_{l=0}^\infty f_l (x-a)^l \sum_{m=0}^\infty g_m (x-a)^m \\ &= \sum_{l=0}^\infty\sum_{m=0}^\infty f_l\,g_m (x-a)^{l+m} \end{align*} We can then match terms in equal powers of $(x-a)$ on either side of the equation to yield the final result: \begin{align*} \left(f\,g\right)_k = \sum_{i=0}^k f_{i} g_{k-i} \end{align*} Division is slightly more challenging, instead consider that $f = (f\div g) g$, and insert the taylor expansions: \begin{align*} \sum_{k=0}^\infty f_k (x-a)^k &= \sum_{l=0}^\infty(f\div g)_l (x-a)^l \sum_{m=0}^\infty g_m (x-a)^m \\ &= \sum_{l=0}^\infty \sum_{m=0}^\infty (f\div g)_l g_m (x-a)^{l+m} \end{align*} Again equating terms with equal powers exactly as with multiplication: \begin{align*} f_k &= \sum_{l=0}^k(f\div g)_l g_{k-l} \\ &= \sum_{l=0}^{k-1} (f\div g)_l g_{k-l} + (f\div g)_k g_0 \\ (f\div g)_k &= \frac{1}{g_0}\left(f_k - \sum_{l=0}^{k-1} (f\div g)_l g_{k-l}\right) \end{align*} Where we removed our target term from the sum in the second line, then rearranged to make it the subject in the third line.

Now we focus on special functions. To solve these, we need the general derivative of a taylor series: \begin{align*} \frac{\partial f}{\partial x} = \sum_{k=1}^\infty k\,f_k (x-a)^{k-1} \end{align*} Considering $\ln$ first, we have the following differential relationship: \begin{align*} \frac{\partial}{\partial x} \ln f &= \frac{\partial f}{\partial x} \frac{1}{f} \\ f \frac{\partial}{\partial x} \ln f &= \frac{\partial f}{\partial x}, \end{align*} where we have multiplied by $f$ to avoid polynomial division when the Taylor series are inserted. Doing this now, \begin{align*} \sum_{i=0}^\infty f_i (x-a)^i \sum_{j=1}^\infty j\left(\ln f\right)_j (x-a)^{j-1} &= \sum_{k=1}^\infty k\, f_k (x-a)^{k-1} \end{align*} Multiplying both sides by $(x-a)$, factoring common terms as with multiplication: \begin{align*} \sum_{i=0}^\infty f_i \sum_{j=1}^\infty j\left(\ln f\right)_j (x-a)^{i+j} &= \sum_{k=1}^\infty k\,f_k (x-a)^k \\ \sum_{i=1}^k f_{k-i}\,i\left(\ln f\right)_i &= k\,f_k & \text{for $k>0$} \\ f_0\,k\,\left(\ln f\right)_k + \sum_{i=1}^{k-1} f_{k-i}\,i\left(\ln f\right)_i &= k\,f_k & \text{for $k>0$} \\ \left(\ln f\right)_k &= \frac{1}{f_0}\left(f_k - \frac{1}{k}\sum_{i=1}^{k-1} i\,f_{k-i}\,\left(\ln f\right)_i\right) & \text{for $k>0$} \end{align*} Where this expression only applies for $k>0$ as the first line has no constant terms within it; however, we have the trivial identity $(\ln f)_0 = \ln f_0$. Sine and Cosine have to be calcluated at the same time (regardless of which one is required). Starting from the general differentiation rules \begin{align*} \frac{\partial}{\partial x} \sin f &= \frac{\partial f}{\partial x} \cos f \\ \frac{\partial}{\partial x} \cos f &= -\frac{\partial f}{\partial x} \sin f \end{align*} Inserting the Taylor expansions and grouping terms with identical coefficients the result is found: \begin{align*} \left(\sin f\right)_k &= \frac{1}{k}\sum_{i=1}^k i\,f_i \left(\cos f\right)_{k-i} & \text{for $k>0$} \\ \left(\cos f\right)_k &= -\frac{1}{k}\sum_{i=1}^k i\,f_i \left(\sin f\right)_{k-i} & \text{for $k>0$}, \end{align*} again we have $\left(\sin f\right)_0 = \sin f_0$ and in general $\left(f(g)\right)_0 = f(g_0)$.

Finally, we calculate the generalized power rule: \begin{align*} \frac{\partial}{\partial x} f^g &= f^g \left(\frac{\partial f}{\partial x} \frac{g}{f} + \frac{\partial g}{\partial x} \ln f\right) \\ f \frac{\partial}{\partial x} f^g &= f^g \frac{\partial f}{\partial x} g + f^g \frac{\partial g}{\partial x} f\,\ln f \end{align*} Inserting Taylor expansions: \begin{align*} \sum_{i=0}^\infty \sum_{j=1}^\infty j\,f_i\left(f^g\right)_j \left(x-a\right)^{i+j-1} &= \sum_{i=0}^\infty \left(f^g\right)_i\sum_{j=1}^\infty j\,f_j \sum_{k=0}^\infty g_k \left(x-a\right)^{i+j-1+k} \\ &\qquad + \sum_{i=0}^\infty \left(f^g\right)_i\sum_{j=1}^\infty j\,g_j \sum_{k=0}^\infty f_k \sum_{l=0}^\infty \left(\ln f\right)_l\left(x-a\right)^{i+j-1+k+l} \end{align*} Multiplying both sides by $\left(x-a\right)$ and grouping common indices/terms: \begin{align*} \sum_{i=0}^\infty \sum_{j=1}^\infty j\,f_i\left(f^g\right)_j \left(x-a\right)^{i+j} &= \sum_{i=0}^\infty \sum_{j=1}^\infty \sum_{k=0}^\infty j\left(f^g\right)_i\left[f_j g_k \left(x-a\right)^{i+j+k} + g_j f_k \sum_{l=0}^\infty \left(\ln f\right)_l\left(x-a\right)^{i+j+k+l}\right] \end{align*} Selecting all terms with the $m$th power: \begin{align*} \sum_{j=1}^m j\,f_{m-j}\left(f^g\right)_j &= \sum_{j=1}^{m} \sum_{i=0}^{m-j} j\left(f^g\right)_i\left[f_j g_{m-i-j} + g_j \sum_{k=0}^{m-i-j} f_k \left(\ln f\right)_{m-i-j-k}\right] \end{align*} Factoring the $m$th term from the left-hand side: \begin{align*} \left(f^g\right)_m &= \frac{1}{m\,f_0}\left(\sum_{j=1}^{m} \sum_{i=0}^{m-j} j\left(f^g\right)_i\left[f_j g_{m-i-j} + g_j \sum_{k=0}^{m-i-j} f_k \left(\ln f\right)_{m-i-j-k}\right] - \sum_{j=1}^{m-1} j\,f_{m-j}\left(f^g\right)_j\right) \end{align*} If the power is a constant, i.e., $g=a$, then the following simplified expression is obtained: \begin{align*} \left(f^g\right)_m &= \frac{1}{m\,f_0}\left(\sum_{j=1}^{m} j\left(f^g\right)_{m-j} f_j\,a - \sum_{j=1}^{m-1} j\,f_{m-j}\left(f^g\right)_j\right) \\ &= \frac{1}{m\,f_0}\sum_{j=1}^{m}\left(f^g\right)_{m-j} f_j\left(j\,a - m+j\right) \\ &= \frac{1}{f_0}\sum_{j=1}^{m}\left(\frac{(a + 1)j}{m} - 1\right) f_j \left(f^g\right)_{m-j}. \end{align*}

The most precise and concise derivation of thermodynamics I have found is the review of thermodynamics which is part of the Statistical Physics Using Mathematica course by James J. Kelly.

All relevant thermodynamic equations are concisely summarized on Wikipedia's thermodynamic equations.

The SimCem interface and computational code was written
by **Marcus
Bannerman**. The database and its interface has
been developed with a number of students who are listed
below.

**Theodore Hanein** researched the literature and
reviewed the available thermodynamic data for phases
relevant to cement as part of his PhD studies. He later
assisted in entering this information into SimCem, as
well as helping in the testing of the code and website.

**Lewis McRae** provided the python cubic equation of
state implementation. He also added include molecular
structure information as well as common aliases using the
PubChem REST interface during a Carnegie Vacation
Scholarship.

This work could not exist without the free data provided
by the
NASA
CEA application database.

Without their
free collection of ideal gas contributions for
molecules, our research could not have begun.

Isotopic masses were obtained from the electronic file
at https://www-nds.iaea.org/amdc. These
are also published in Audi et. al., "The AME2012 atomic
mass evalulation", *Chinese Physics
C*, **36**,1287-2014 (2012).

Isotopic abundances were scraped from the table given by CIAAW. This data was scraped on 29/07/2015.

The PubChem PUG REST interface was used to scrape additional information on molecules, such as their structure, common names, and CAS identifiers.

Finally, a huge number of open-source libraries are used in this project. JQuery, Flot.js, SlickGrid.js, Hover.css, MathJax, NLOpt, Boost.Python, Tornado, JsTree, and more I've probably forgotten.

We stand on the shoulders of giants.