Coordinates and potentials of a thermodynamic system. Thermodynamic potentials
All calculations in thermodynamics are based on the use of state functions called thermodynamic potentials. Each set of independent parameters has its own thermodynamic potential. Changes in potentials that occur during any process determine either the work performed by systole or the heat received by the system.
When considering thermodynamic potentials, we will use relation (103.22), presenting it in the form
The equal sign refers to reversible processes, the inequality sign refers to non-reversible processes.
Thermodynamic potentials are functions of state. Therefore, the increment of any of the potentials is equal to the total differential of the function by which it is expressed. The total differential of the function of the variables and y is determined by the expression
Therefore, if during the transformations we receive an expression of the form for the increment of a certain value
it can be argued that this quantity is a function of the parameters, and the functions are partial derivatives of the function
Internal energy. We are already very familiar with one of the thermodynamic potentials. This is the internal energy of the system. The expression of the first law for a reversible process can be represented as
(109.4)
Comparison with (109.2) shows that the so-called natural variables for the potential V are the variables S and V. From (109.3) it follows that
From the relationship it follows that in the case when the body does not exchange heat with the external environment, the work performed by it is equal to
or in integral form:
Thus, in the absence of heat exchange with the external environment, work is equal to the loss of internal energy of the body.
At constant volume
Therefore, - the heat capacity at constant volume is equal to
(109.8)
Free energy. According to (109.4), the work done by heat during a reversible isothermal process can be represented in the form
Status function
(109.10)
called the free energy of the body.
In accordance with formulas (109.9) and (109.10), in a reversible isothermal process, work is equal to the decrease in the free energy of the body:
Comparison with formula (109.6) shows that in isothermal processes free energy plays the same role as internal energy in adiabatic processes.
Note that formula (109.6) is valid for both reversible and irreversible processes. Formula (109.12) is valid only for reversible processes. In irreversible processes (see). Substituting this inequality into the relation it is easy to obtain that for irreversible isothermal processes
Consequently, the loss of free energy determines the upper limit on the amount of work that can be done by the system during an isothermal process.
Let's take the differential of function (109.10). Taking into account (109.4) we obtain:
From comparison with (109.2) we conclude that the natural variables for free energy are T and V. In accordance with (109.3)
Let us replace: in (109.1) dQ by and divide the resulting relationship by ( - time). As a result we get that
If the temperature and volume remain constant, then relation (109.16) can be transformed into the form
From this formula it follows that an irreversible process occurring at constant temperature and volume is accompanied by a decrease in the free energy of the body. Once equilibrium is reached, F stops changing with time. Thus; at constant T and V, the equilibrium state is the state for which the free energy is minimal.
Enthalpy. If the process "occurs at constant pressure, then the amount of heat received by the body can be represented as follows:
Status function
called enthalpy or heat function.
From (109.18) and (109.19) it follows that the amount of heat received by the body during the isobathic process is equal to
or in integral form
Consequently, in the case when the pressure remains constant, the amount of heat received by the body is equal to the increase in enthalpy. Differentiation of expression (109.19) taking into account (109.4) gives
From here we conclude. enthalpy is the thermodynamic potential in variables Its partial derivatives are equal
Lecture outline: Thermodynamic potential. Isochoric-isothermal potential or Helmholtz free energy. Application of Helmholtz energy as a criterion for the direction of a spontaneous process and equilibrium in closed systems. Isobaric-isothermal potential or Gibbs free energy. Application of the Gibbs energy as a criterion for the direction of a spontaneous process and equilibrium in closed systems. Characteristic functions: internal energy, enthalpy, Helmholtz free energy, Gibbs free energy. Gibbs-Helmholtz equations. Chemical potential.
Thermodynamic potential – this is a function of the state of the system, the loss of which in a process occurring with two parameters constant is equal to the maximum useful work.
Helmholtz energy as isochoric-isothermal potential.
For isochoric-isothermal conditions V = const, T = const. Let us recall that the combined equation expressing the first and second laws of thermodynamics has the following form: .
Since when V = const, = 0, we get . (6.1) Let's integrate given equation:
Let us introduce the notation F– This is Helmholtz energy. F = U - TS (6.2)
Then F 2 = U 2 - TS 2 And F 1 = U 1 - TS 1.
That is, Helmholtz energy is a thermodynamic potential, since its change is equal to useful work during a reversible process in the system. For an irreversible process: In general, for reversible and irreversible processes the following expression is valid:
The Helmholtz energy is equal to , hence U=F+TS. (6.4)
That is F – this is that part of the internal energy that can be converted into work, which is why it is called free energy; work T.S. is energy that is released in the form of heat, which is why it is called bound energy.
Helmholtz energy as a criterion for the possibility of a process occurring. Differentiating the expression we get dF = dU – TdS - SdT. Substituting for the product TdS its expression from the "unified" equation TdS ≥ dU+pdV we get
dF ≤ - SdT - pdV. (6.5)
Because SdT = 0 And pdV= 0(at T = cons t and V= const), then for isochoric-isothermal conditions
(dF) v , T ≤ 0. (6.6)
In closed (closed) systems under isochoric-ichothermal conditions:
· If dF< 0 , then the process proceeds spontaneously;
· If dF > 0, then the process does not proceed;
· If dF = 0, then the system is in a state of equilibrium.
Gibbs energy as an isobaric-isothermal potential. For isobaric-isothermal conditions p = const, T = const. Let us transform the combined equation of the first and second laws of thermodynamics:
Let's integrate this expression:
Let us introduce the notation - this is the Gibbs energy. (6.8)
That is, Gibbs energy G is the thermodynamic potential, since its change is equal to the useful work during the occurrence of a reversible process in the system. For an irreversible process In the case of a reversible and irreversible process, the following expression is valid:
Thermodynamic potentials (thermodynamic functions) - characteristic functions in thermodynamics, the decrease of which in equilibrium processes occurring at constant values of the corresponding independent parameters is equal to the useful external work.
Since in an isothermal process the amount of heat received by the system is equal to , then decline free energy in a quasi-static isothermal process is equal to the work done by the system above external bodies.
Gibbs potential
Also called Gibbs energy, thermodynamic potential, Gibbs free energy and even just free energy(which can lead to mixing of the Gibbs potential with the Helmholtz free energy):
.Thermodynamic potentials and maximum work
Internal energy represents the total energy of the system. However, the second law of thermodynamics prohibits converting all internal energy into work.
It can be shown that the maximum full work (both on the environment and on external bodies) that can be obtained from the system in an isothermal process, is equal to the decrease in Helmholtz free energy in this process:
,where is the Helmholtz free energy.
In this sense it represents free energy that can be converted into work. The remaining part of the internal energy can be called related.
In some applications it is necessary to distinguish full And useful work. The latter represents the work of the system on external bodies, excluding the environment in which it is immersed. Maximum useful the system's work is equal to
where is the Gibbs energy.
In this sense, the Gibbs energy is also free.
Canonical equation of state
Specifying the thermodynamic potential of a certain system in a certain form is equivalent to specifying the equation of state of this system.
The corresponding thermodynamic potential differentials are:
- for internal energy
- for enthalpy
- for Helmholtz free energy
- for the Gibbs potential
These expressions can be considered mathematically as complete differentials of functions of two corresponding independent variables. Therefore, it is natural to consider thermodynamic potentials as functions:
, , , .Specifying any of these four dependencies - that is, specifying the type of functions , , , - allows you to obtain all the information about the properties of the system. So, for example, if we are given internal energy as a function of entropy and volume, the remaining parameters can be obtained by differentiation:
Here the indices mean the constancy of the second variable on which the function depends. These equalities become obvious if we consider that .
Setting one of the thermodynamic potentials as a function of the corresponding variables, as written above, is canonical equation of state systems. Like other equations of state, it is valid only for states of thermodynamic equilibrium. In nonequilibrium states, these dependencies may not hold.
Method of thermodynamic potentials. Maxwell's relations
The method of thermodynamic potentials helps to transform expressions that include basic thermodynamic variables and thereby express such “hard-to-observe” quantities as the amount of heat, entropy, internal energy through measured quantities - temperature, pressure and volume and their derivatives.
Let us again consider the expression for the total differential of internal energy:
.It is known that if mixed derivatives exist and are continuous, then they do not depend on the order of differentiation, that is
.But also, therefore
.Considering the expressions for other differentials, we obtain:
, , .These relations are called Maxwell's relations. Note that they are not satisfied in the case of discontinuity of mixed derivatives, which occurs during phase transitions of the 1st and 2nd order.
Systems with a variable number of particles. Large thermodynamic potential
The chemical potential () of a component is defined as the energy that must be expended in order to add an infinitesimal molar amount of this component to the system. Then the expressions for the differentials of thermodynamic potentials can be written as follows:
, , , .Since thermodynamic potentials must be additive functions of the number of particles in the system, the canonical equations of state take the following form (taking into account that S and V are additive quantities, but T and P are not):
, , , .And, since , from the last expression it follows that
,that is, the chemical potential is the specific Gibbs potential (per particle).
For a large canonical ensemble (that is, for a statistical ensemble of states of a system with a variable number of particles and an equilibrium chemical potential), a large thermodynamic potential can be defined, relating free energy to chemical potential:
;It is easy to verify that the so-called bound energy is a thermodynamic potential for a system given with constants.
The potential considered in thermodynamics is associated with the energy required for the reversible transfer of ions from one phase to another. This potential, of course, is the electrochemical potential of the ionic component. The electrostatic potential, except for the problems associated with its determination in condensed phases, is not directly related to reversible work. Although in thermodynamics it is possible to dispense with the electrostatic potential by using the electrochemical potential instead, the need to describe the electrical state of the phase remains.
Often the electrochemical potential of an ionic component is represented as the sum of the electrical and “chemical” terms:
where Ф is the “electrostatic” potential, and the activity coefficient, assumed here, is independent of the electrical state of a given phase. Let us note first of all that such an expansion is not necessary, since the corresponding formulas, which are significant from the point of view of thermodynamics, have already been obtained in Chapter. 2.
The electrostatic potential Ф can be defined so that it is measurable or immeasurable. Depending on how Φ is defined, the quantity will also be either uniquely determined or completely undefined. It is possible to develop the theory even without having such a clear definition of the electrostatic potential as is provided by electrostatics, and without worrying about carefully defining its meaning. If the analysis is carried out correctly, then physically meaningful results can be obtained in the end by compensating for the uncertain terms.
Any chosen definition of Φ must satisfy one condition. It should be reduced to the definition (13-2) used for the electrical potential difference between phases with the same composition. So, if the phases have the same composition, then
Thus, Ф is a quantitative measure of the electrical state of one phase relative to another, having the same composition. This condition is satisfied whole line possible definitions of F.
Instead of Ф, an external potential can be used, which is, in principle, measurable. Its disadvantage is the difficulty of measurement and use in thermodynamic calculations. The advantage is that it gives some meaning to Φ, and this potential does not appear in the final results, so there is virtually no need to measure it.
Another possibility is to use the potential of a suitable reference electrode. Since the reference electrode is reversible for some ion present in the solution, this is equivalent to using the electrochemical potential of the ion or The arbitrariness of this definition is evident from the need to select a specific reference electrode or ionic component. An additional disadvantage of this choice is that in a solution that does not contain component i, the value turns to minus infinity. Thus, the electrochemical potential is not consistent with our usual concept of electrostatic potential, which is explained by its connection with reversible work. This choice of potential has the advantage that it is associated with measurements using reference electrodes commonly used in electrochemistry.
Let us now consider the third possibility. Let us select the ionic component and determine the potential Ф as follows:
Then the electrochemical potential of any other component can be expressed as
It should be noted that the combinations in parentheses are precisely defined and do not depend on the electrical state in accordance with the rules outlined in Sect. 14. In this case, we can write down the gradient of the electrochemical potential
The arbitrariness of this definition of Ф is again visible, associated with the need to choose the ionic component n. The advantage of this definition of Φ is its unambiguous connection with electrochemical potentials and consistency with our usual idea of electrostatic potential. Due to the presence of a term in equation (26-3), the latter can be used for a solution with a vanishing concentration of the component.
In the limit of infinitely dilute solutions, terms with activity coefficients disappear due to the choice of a secondary standard state (14-6). In this limit, the determination of Ф becomes independent of the choice of standard ion n. This creates the basis of what should be called the theory of dilute electrolyte solutions. At the same time, equations (26-4) and (26-5) show how to make corrections for the activity coefficient in the theory of dilute solutions, without resorting to the activity coefficients of individual ions. The absence of dependence on the type of ion in the case of infinitely dilute solutions is due to the possibility of measuring electrical potential differences between phases with the same composition. Such solutions have essentially the same compositions in the sense that the ion in the solution interacts only with the solvent and even the long-range interaction from the other ions is not felt by it.
The introduction of such an electric potential is useful in the analysis of transport processes in electrolyte solutions. For a potential thus defined, Smerle and Newman use the term quasi-electrostatic potential.
We discussed possible ways use of electrical potential in electrochemical thermodynamics. The application of potential in transfer theory is essentially the same as
and in thermodynamics. When working with electrochemical potentials, you can do without an electric potential, although its introduction may be useful or convenient. In the kinetics of electrode processes as driving force reaction can use a change in free energy. This is equivalent to using the surface overvoltage defined in Sect. 8.
Electric potential also finds application in microscopic models, such as the Debye-Hückel theory mentioned above and presented in the next chapter. It is impossible to always strictly determine such potential. One should clearly distinguish between macroscopic theories - thermodynamics, theory of transport processes and fluid mechanics - and microscopic theories - statistical mechanics and kinetic theory of gases and liquids. Based on the properties of molecules or ions, microscopic theories make it possible to calculate and relate such macroscopic characteristics as, for example, activity coefficients and diffusion coefficients. In this case, it is rarely possible to obtain satisfactory quantitative results without the use of additional experimental information. Macroscopic theories, on the one hand, create the basis for the most economical measurement and tabulation of macroscopic characteristics, and on the other hand, they make it possible to use these results to predict the behavior of macroscopic systems.
1. Group of potentials “E F G H”, having the dimension of energy.
2. Dependence of thermodynamic potentials on the number of particles. Entropy as thermodynamic potential.
3. Thermodynamic potentials of multicomponent systems.
4. Practical implementation of the method of thermodynamic potentials (using the example of a chemical equilibrium problem).
One of the main methods of modern thermodynamics is the method of thermodynamic potentials. This method arose largely due to the use of potentials in classical mechanics, where its change was associated with the work performed, and the potential itself is an energy characteristic of a thermodynamic system. Historically, the originally introduced thermodynamic potentials also had the dimension of energy, which determined their name.
The mentioned group includes the following systems:
Internal energy;
Free energy or Helmholtz potential;
Thermodynamic Gibbs potential;
Enthalpy.
The potential of internal energy was shown in the previous topic. The potentiality of the remaining quantities follows from it.
The thermodynamic potential differentials take the form:
From relations (3.1) it is clear that the corresponding thermodynamic potentials characterize the same thermodynamic system at in various ways…. descriptions (methods of specifying the state of a thermodynamic system). Thus, for an adiabatically isolated system described in variables, it is convenient to use internal energy as a thermodynamic potential. Then the parameters of the system, thermodynamically conjugate to the potentials, are determined from the relations:
If a “system in a thermostat” defined by variables is used as a description method, it is most convenient to use free energy as a potential. Accordingly, for the system parameters we obtain:
Next, we will choose the “system under the piston” model as a description method. In these cases, the state functions form a set (), and the Gibbs potential G is used as the thermodynamic potential. Then the system parameters are determined from the expressions:
And in the case of an “adiabatic system above the piston”, specified by the state functions, the role of the thermodynamic potential is played by the enthalpy H. Then the system parameters take the form:
From the fact that relations (3.1) define full differentials thermodynamic potentials, we can equate their second derivatives.
For example, given that
we get
Similarly, for the remaining parameters of the system related to the thermodynamic potential, we write:
Similar identities can be written for other sets of parameters of the thermodynamic state of the system based on the potentiality of the corresponding thermodynamic functions.
So, for a “system in a thermostat” with potential, we have:
For a system “above the piston” with a Gibbs potential, the following equalities will be valid:
And finally, for a system with an adiabatic piston with potential H, we obtain:
Equalities of the form (3.6) - (3.9) are called thermodynamic identities and in a number of cases turn out to be convenient for practical calculations.
The use of thermodynamic potentials makes it possible to quite simply determine the operation of the system and the thermal effect.
Thus, from relations (3.1) it follows:
From the first part of the equality follows the well-known proposition that the work of a thermally insulated system () is carried out due to a decrease in its internal energy. The second equality means that free energy is that part of the internal energy that, during an isothermal process, is completely converted into work (accordingly, the “remaining” part of the internal energy is sometimes called bound energy).
The amount of heat can be represented as:
From the last equality it is clear why enthalpy is also called heat content. During combustion and other chemical reactions occurring at constant pressure (), the amount of heat released is equal to the change in enthalpy.
Expression (3.11), taking into account the second law of thermodynamics (2.7), allows us to determine the heat capacity:
All thermodynamic potentials of the energy type have the property of additivity. Therefore we can write:
It is easy to see that the Gibbs potential contains only one additive parameter, i.e. the specific Gibbs potential does not depend on. Then from (3.4) it follows:
That is, the chemical potential is the specific Gibbs potential, and the equality holds
Thermodynamic potentials (3.1) are interconnected by direct relationships, allowing a transition from one potential to another. For example, let's express all thermodynamic potentials in terms of internal energy.
In this case, we obtained all thermodynamic potentials as functions (). In order to express them in other variables, use the re... procedure.
Let pressure be specified in variables ():
Let us write the last expression in the form of an equation of state, i.e. let's find the view
It is easy to see that if the state is specified in variables (), then the thermodynamic potential is the internal energy. By virtue of (3.2), we find
Considering (3.18) as an equation for S, we find its solution:
Substituting (3.19) into (3.17) we get
That is, from variables () we moved to variables ().
The second group of thermodynamic potentials arises if, in addition to those discussed above, chemical potential is included as thermodynamic variables. The potentials of the second group also have the dimension of energy and can be related to the potentials of the first group through the relations:
Accordingly, potential differentials (3.21) have the form:
Just like for thermodynamic potentials of the first group, for potentials (3.21) one can construct thermodynamic identities, find expressions for the parameters of a thermodynamic system, etc.
Let us consider the characteristic relations for the “omega potential”, which expresses quasi-free energy and is used in practice most often among other potentials of the group (3.22).
The potential is specified in variables () that describe a thermodynamic system with imaginary walls. The system parameters in this case are determined from the relations:
Thermodynamic identities following from potentiality have the form:
The additive properties of the thermodynamic potentials of the second group are quite interesting. Since in this case the number of particles is not among the system parameters, volume is used as an additive parameter. Then for the potential we get:
Here is the specific potential by 1. Taking into account (3.23), we obtain:
Accordingly, (3.26)
The validity of (3.26) can also be proven on the basis of (3.15):
The potential can also be used to recalculate thermodynamic functions written in form. For this, relation (3.23) for N:
permitted regarding:
Not only the energy characteristics of the system, but also any other quantities included in relation (3.1) can act as thermodynamic potentials. As important example Let's consider entropy as a thermodynamic potential. The initial differential relation for entropy follows from the generalized notation of the I and II principles of thermodynamics:
Thus, entropy is the thermodynamic potential for a system given by parameters. Other system parameters are:
By resolving the first of relations (3.28), the transition from variables to variables is relatively possible.
The additive properties of entropy lead to the well-known relationships:
Let's move on to determining thermodynamic potentials based on given macroscopic states of a thermodynamic system. To simplify calculations, let us assume that there are no external fields (). This does not reduce the generality of the results, since additional systems simply appear in the resulting expressions.
As an example, we will find expressions for free energy using as initial the equation of state, the caloric equation of state and the features of the behavior of the system at. Taking into account (3.3) and (3.12), we find:
Let us integrate the second equation of system (3.30) taking into account the boundary condition at:
Then system (3.30) takes the form:
Solving system (3.31) allows us to find the specific free energy in the form
The origin of the specific free energy can also be found from the conditions at:
Then (3.32) takes the form:
and the expression of the entire free energy of the system, up to an additive constant, takes the form:
Then the system’s response to the inclusion of an external field is specified by an additional state equation, which, depending on the set of state variables, has the form:
Then the change in the corresponding thermodynamic potential associated with the inclusion of zero from zero to is determined from the expressions:
Thus, setting the thermodynamic potential in macroscopic theory is possible only on the basis of using given equations thermodynamic state, which in turn are themselves obtained on the basis of setting thermodynamic potentials. Break this “ vicious circle” is possible only on the basis of microscopic theory, in which the state of the system is specified on the basis of distribution functions taking into account statistical features.
Let us generalize the results obtained to the case of multicomponent systems. This generalization is achieved by replacing the parameter with a set. Let's look at what has been said using specific examples.
Let us assume that the thermodynamic state of the system is specified by parameters, i.e. we are considering a system in a thermostat consisting of several components, the number of particles in which is equal to Free energy, which is the thermodynamic potential in this description, has the form:
As an additive parameter in (3.37), not the number of particles, but the volume of the system V is introduced. Then the density of the system is denoted by. The function is a non-additive function of non-additive arguments. This is quite convenient because when the system is divided into parts, the function will not change for each part.
Then for the parameters of the thermodynamic system we can write:
Considering that we have
For the chemical potential of an individual component we write:
There are other ways to take into account the additive properties of free energy. Let us introduce the relative densities of particle numbers for each component:
independent of the volume of the system V. Here - total number particles in the system. Then
The expression of the chemical potential in this case takes on a more complex form:
Let's calculate the derivatives and and substitute them into the last expression:
The expression for pressure, on the contrary, will be simplified:
Similar relations can be obtained for the Gibbs potential. So, if volume is specified as an additive parameter, then taking into account (3.37) and (3.38) we write:
the same expression can be obtained from (3.yu), which in the case of many particles takes the form:
Substituting expression (3.39) into (3.45), we find:
which completely coincides with (3.44).
In order to move on to the traditional notation of the Gibbs potential (through state variables ()), it is necessary to resolve equation (3.38):
Relative to the volume V and substitute the result in (3.44) or (3.45):
If the total number of particles in the system N is given as an additive parameter, then the Gibbs potential, taking into account (3.42), takes the following form:
Knowing the type of specific quantities: , we obtain:
In the last expression, the summation over j replace it with summation over i. Then the second and third terms add up to zero. Then for the Gibbs potential we finally obtain:
The same relation can be obtained in another way (from (3.41) and (3.43)):
Then for the chemical potential of each component we obtain:
When deriving (3.48), transformations similar to those used in deriving (3.42) were performed using imaginary walls. The system state parameters form a set ().
The role of the thermodynamic potential is played by the potential, which takes the form:
As can be seen from (3.49), the only additive parameter in in this case is the volume of the system V.
Let us determine some thermodynamic parameters of such a system. The number of particles in this case is determined from the relation:
For free energy F and Gibbs potential G can be written:
Thus, the relationships for thermodynamic potentials and parameters in the case of multicomponent systems are modified only due to the need to take into account the number of particles (or chemical potentials) of each component. At the same time, the very idea of the method of thermodynamic potentials and calculations carried out on its basis remains unchanged.
As an example of using the method of thermodynamic potentials, let us consider the problem of chemical equilibrium. Let us find the conditions of chemical equilibrium in a mixture of three substances that enter into a reaction. Additionally, let us assume that the initial reaction products are rarefied gases (this allows us to ignore intermolecular interactions), and constant temperature and pressure are maintained in the system (this process is the easiest to implement in practice, therefore the condition of constant pressure and temperature is created in industrial installations for chemical reaction).
The equilibrium condition of a thermodynamic system, depending on the method of its description, is determined by the maximum entropy of the system or the minimum energy of the system (for more details, see Bazarov Thermodynamics). Then you can get following conditions system equilibrium:
1. The equilibrium state of an adiabatically isolated thermodynamic system, specified by parameters (), is characterized by a maximum of entropy:
The second expression in (3.53a) characterizes the stability of the equilibrium state.
2. The equilibrium state of an isochoric-isothermal system, specified by parameters (), is characterized by a minimum of free energy. The equilibrium condition in this case takes the form:
3. The equilibrium of the isobaric-isothermal system, specified by parameters (), is characterized by the conditions:
4. For a system in a thermostat with a variable number of particles, defined by parameters (), the equilibrium conditions are characterized by potential minima:
Let's move on to using chemical equilibrium in our case.
In general, the equation of a chemical reaction is written as:
Here are the symbols chemical substances, - the so-called stoichiometric numbers. Yes, for reaction
Because the system parameters are pressure and temperature, which are assumed to be constant. It is convenient to consider the Gibbs potential as the state of the thermodynamic potential G. Then the condition for equilibrium of the system will be the requirement of constant potential G:
Since we are considering a three-component system, let us put. In addition, taking into account (3.54), we can write the balance equation for the number of particles ():
Introducing the chemical potentials for each of the components: and taking into account the assumptions made, we find:
Equation (3.57) was first obtained by Gibbs in 1876. and is the desired chemical equilibrium equation. It is easy to notice by comparing (3.57) and (3.54) that the chemical equilibrium equation is obtained from the chemical reaction equation by simply replacing the symbols of the reacting substances with their chemical potentials. This technique can also be used when writing the chemical equilibrium equation for an arbitrary reaction.
In the general case, solving equation (3.57) even for three components is quite busy. This is due, firstly, to the fact that even for a one-component system it is very difficult to obtain explicit expressions for the chemical potential. Secondly, relative concentrations are not small values. That is, it is impossible to perform a series expansion on them. This makes the problem of solving the chemical equilibrium equation even more difficult.
Physically, the difficulties noted are explained by the need to take into account the rearrangement of the electronic shells of the atoms entering the reaction. This leads to certain difficulties in the microscopic description, which also affects the macroscopic approach.
Since we agreed to limit ourselves to the study of gas rarefaction, we can use the ideal gas model. We will assume that all reacting components are ideal gases filling the total volume and creating pressure p. In this case, any interaction (except chemical reactions) between the components of the gas mixture can be neglected. This allows us to assume that the chemical potential i of the th component depends only on the parameters of the same component.
Here is the partial pressure i-th component, and:
Taking into account (3.58), the equilibrium condition of the three-component system (3.57) will take the form:
For further analysis, we will use the equation of state of an ideal gas, which we write in the form:
Here, as before, thermodynamic temperature is denoted by. Then the notation known from school takes the form: , which is written in (3.60).
Then for each component of the mixture we get:
Let us determine the form of expression for the chemical potential of an ideal gas. As follows from (2.22), the chemical potential has the form:
Taking into account equation (3.60), which can be written in the form, the problem of determining the chemical potential is reduced to determining the specific entropy and specific internal energy.
The system of equations for specific entropy follows from thermodynamic identities (3.8) and the heat capacity expression (3.12):
Taking into account the equation of state (3.60) and moving on to the specific characteristics, we have:
Solution (3.63) has the form:
The system of equations for the specific internal energy of an ideal gas follows from (2.23):
The solution to this system will be written as:
Substituting (3.64) - (3.65) into (3.66) and taking into account the equation of state of an ideal gas, we obtain:
For a mixture of ideal gases, expression (3.66) takes the form:
Substituting (3.67) into (3.59), we get:
Performing transformations, we write:
Performing potentiation in the last expression, we have:
Relationship (3.68) is called the law of mass action. The quantity is a function of temperature only and is called the component of a chemical reaction.
Thus chemical equilibrium and the direction of a chemical reaction is determined by the magnitude of pressure and temperature.