Components n i , chem. potentials of the components m, etc.), used in Ch. arr. to describe thermodynamic equilibrium. Each thermodynamic potential corresponds to a set of state parameters , called. natural variables.
The most important thermodynamic potentials: internal energy U (natural variables S, V, n i); enthalpy H \u003d U - (- pV) (natural variables S, p, n i); Helmholtz energy (Helmholtz free energy, Helmholtz function) F = = U - TS (natural variables V, T, n i); Gibbs energy (free Gibbs energy, Gibbs function) G=U - - TS - (- pV) (natural variables p, T, n i); large thermodynamic potential(naturalvenous variables V, T, m i).
T thermodynamic potentials can be represented by a common f-loy
where L k are intensive parameters that do not depend on the mass of the system (these are T, p, m i), X k are extensive parameters proportional to the mass of the system (V, S, n i). Index l = 0 for internal energy U, 1 for H and F, 2 for G and W. Thermodynamic potentials are functions of the state of a thermodynamic system, i.e. their change in any transition process between two states is determined only by the initial and final states and does not depend on the transition path. The total differentials of thermodynamic potentials have the form:
Ur-tion (2) called. Gibbs fundamental equation in energetic. expression. All thermodynamic potentials have the dimension of energy.
Thermodynamic equilibrium conditions. systems are formulated as the equality to zero of the total differentials of thermodynamic potentials with the constancy of the corresponding natural variables:
Thermodynamic the stability of the system is expressed by the inequalities:
The decrease in thermodynamic potentials in an equilibrium process with constant natural variables is equal to the maximum useful work of the process A:
In this case, work A is performed against any generalized force L k acting on the system, except for external. pressure (see Maximum work of reaction).
T thermodynamic potentials, taken as functions of their natural variables, are the characteristic functions of the system. This means that any thermodynamic sv-in (compressibility, heat capacity, etc.) m. b. expressed by a relation that includes only the given thermodynamic potential, its natural variables, and derivatives of thermodynamic potentials of different orders with respect to natural variables. In particular, with the help of thermodynamic potentials one can obtain the equations of state of a system.
Derivatives of thermodynamic potentials have important properties. The first partial derivatives with respect to natural extensive variables are equal to intensive variables, for example:
[generally: (9 Y l /9 X i) = L i ]. Conversely, derivatives with respect to natural intensive variables are equal to extensive variables, for example:
[generally: (9 Y l /9 L i) = X i ]. The second partial derivatives with respect to natural variables determine the fur. and ter-mitch. holy systems, for example:
Because differentials of thermodynamic potentials are complete, cross second partial derivatives of thermodynamic potentials are equal, for example. for G(T, p, n i):
Relations of this type are called Maxwell's relations.
T thermodynamic potentials can also be represented as functions of variables other than natural ones, for example. G(T, V, n i), however, in this case, St-va thermodynamic potentials as a characteristic. functions will be lost. In addition to thermodynamic potentials characteristic. the functions are the entropy S (natural variables U, V, n i), the Massier function F 1= (natural variables 1/T, V, n i), functionplank 
(natural variables 1/T, p/T, n i).
T the thermodynamic potentials are interconnected by the Gibbs-Helmholtz equations. For example, for H and G
In general:
T thermodynamic potentials are homogeneous functions of the first degree of their natural extensive variables. For example, with an increase in the entropy S or the number of moles n i, the enthalpy H increases proportionally. According to Euler's theorem, the homogeneity of thermodynamic potentials leads to relations of the type:
In chem. thermodynamics, in addition to the thermodynamic potentials recorded for the system as a whole, average molar (specific) quantities are widely used (for example, 
,
Lecture on the topic: “Thermodynamic potentials”
Plan:
The group of potentials “E F G H ”, having the dimension of energy.
Dependence of thermodynamic potentials on the number of particles. Entropy as a thermodynamic potential.
Thermodynamic potentials of multicomponent systems.
Practical implementation of the method of thermodynamic potentials (on the example of the problem of chemical equilibrium).
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 initially 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;
Gibbs thermodynamic potential;
Enthalpy.
The potentiality of internal energy was shown in the previous topic. It implies the potentiality of the remaining quantities.
The differentials of thermodynamic potentials take the form:
From relations (3.1) it can be seen that the corresponding thermodynamic potentials characterize the same thermodynamic system with different methods .... descriptions (methods of setting the state of a thermodynamic system). So, 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 conjugated to the potentials, are determined from the relations:
,
,
,
(3.2)
If the “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:
,
,
,
(3.3)
Next, we will choose the “system under the piston” model as a way of describing it. 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:
,
,
,
(3.4)
And in the case of an “adiabatic system over a piston” given by state functions, the role of the thermodynamic potential is played by the enthalpy H . Then the system parameters take the form:
,
,
,
(3.5)
Since relations (3.1) define the total differentials of thermodynamic potentials, we can equate their second derivatives.
For example, 
Given that
we get
(3.6a)
Similarly, for the remaining parameters of the system related to the thermodynamic potential , we write:
(3.6b-e)
Similar identities can also 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 the system “above the piston” with the Gibbs potential, the equalities will be valid:
And, finally, for a system with an adiabatic piston with potential H , we get:
Equalities of the form (3.6) - (3.9) are called thermodynamic identities and in some cases turn out to be convenient for practical calculations.
The use of thermodynamic potentials makes it quite easy to determine the operation of the system and the thermal effect.
Thus, relations (3.1) imply:
From the first part of the equality follows the well-known position that the work of a thermally insulated system () is carried out due to the decrease in its internal energy. The second equality means that free energy is that part of the internal energy, which in the isothermal process is completely converted into work (respectively, 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. Gibbs specific potential does not depend on. Then from (3.4) it follows:
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The physical quantity, the elementary change of which during the transition of the system from one state to another is equal to the amount of heat received or given away, divided by the temperature at which this transition occurred, is called entropy.
For an infinitesimal change in the state of the system:
When a system transitions from one state to another, the change in entropy can be calculated as follows:
Based on the first law of thermodynamics, one can obtain
dS=dQ/T=C V dT/T+RdV/V, and
In an isothermal process T=const, i.e. T1=T2:
DS=R×ln(V 2 /V 1).
With an isobaric process, p=const, i.e. V 2 / V 1 \u003d T 2 / T 1:
DS \u003d (C V + R) × ln (T 2 / T 1) \u003d C p × ln (T 2 / T 1) \u003d C p × ln (V 2 / V 1).
With an isochoric process, V=const, i.e. V1=V2:
DS=C V ×ln(T 2 /T 1).
With an adiabatic process, dQ=0, i.e. DS=0:
S 1 =S 2 = const.
Changes in the entropy of a system that performs a Carnot cycle:
DS=-(Q 1 /T 1 +Q 2 /T 2).
The entropy of a closed system performing a reversible Carnot cycle does not change:
dS=0 or S=const.
If the system performs an irreversible cycle, then dS>0.
Thus, the entropy of a closed (isolated) system for any processes occurring in it cannot decrease:
where the equal sign is valid for reversible processes, and the inequality sign is valid for irreversible ones.
The second law of thermodynamics: "In an isolated system, only such processes are possible in which the entropy of the system increases." That is
dS³0 or dS³dQ/T.
The second law of thermodynamics determines the direction of thermodynamic processes and indicates the physical meaning of entropy: entropy is a measure of energy dissipation, i.e. characterizes that part of the energy that cannot be converted into work.
Thermodynamic potentials are called certain functions of volume V, pressure p, temperature T, entropy S, the number of particles of the system N and other macroscopic parameters x that characterize the state of the thermodynamic system. These include: internal energy U=U(S,V,N,x), enthalpy H=H(S,p,N,x); free energy – F=F(V,T,N,x), Gibbs energy G=G(p,T,N,x).
The change in the internal energy of the system in some process is defined as the algebraic sum of the amount of heat Q, which the system exchanges with the environment during the process, and the work A done by the system or produced on it. This reflects the first law of thermodynamics:
The change in U is determined only by the values of the internal energy in the initial and final states:
For any closed process that returns the system to its original state, the change in internal energy is zero (U 1 =U 2 ; DU=0; Q=A).
The change in the internal energy of the system in an adiabatic process (when Q=0) is equal to the work done on the system or done by the system DU=A.
In the case of the simplest physical system with a small intermolecular interaction (ideal gas), the change in internal energy is reduced to a change in the kinetic energy of molecules:
where m is the mass of gas;
c V is the specific heat capacity at constant volume.
Enthalpy (heat content, Gibbs thermal function) characterizes the state of a macroscopic system in thermodynamic equilibrium when entropy S and pressure p – H(S,p,N,x) are chosen as the main independent variables.
Enthalpy is an additive function (that is, the enthalpy of the entire system is equal to the sum of the enthalpies of its constituent parts). The enthalpy is related to the internal energy U of the system by the relation:
where V is the volume of the system.
The total enthalpy differential (for constant N and x) is:
From this formula, you can determine the temperature T and volume V of the system:
T=(dH/dS), V=(dH/dp).
At constant pressure, the heat capacity of the system
These properties of enthalpy at constant pressure are similar to the properties of internal energy at constant volume:
T=(dU/dS), p=-(dU/dV), c V=(dU/dT).
Free energy is one of the names of the isochoric-isothermal thermodynamic potential or Helmholtz energy. It is defined as the difference between the internal energy of a thermodynamic system (U) and the product of its entropy (S) and temperature (T):
where TS is the bound energy.
Gibbs energy - isobaric-isothermal potential, free enthalpy, characteristic function of a thermodynamic system for independent parameters p, T and N - G. It is determined through the enthalpy H, entropy S and temperature T by the equation
With free energy - the Helmholtz energy, the Gibbs energy is related by the relation:
The Gibbs energy is proportional to the number of particles N, per particle, is called the chemical potential.
The work performed by a thermodynamic system in any process is determined by the decrease in the thermodynamic potential corresponding to the conditions of the process. So, with a constant number of particles (N=const) under conditions of thermal insulation (adiabatic process, S=const), the elementary work dA is equal to the loss of internal energy:
With isothermal process (T=const)
In this process, work is done not only due to internal energy, but also due to the heat entering the system.
For systems in which the exchange of matter with the environment is possible (change in N), processes are possible at constant p and T. In this case, the elementary work dA of all thermodynamic forces, except for pressure forces, is equal to the loss of the Gibbs thermodynamic potential (G), i.e.
According to the Nernst theorem, the change in entropy (DS) in any reversible isothermal processes performed between two equilibrium states at temperatures approaching absolute zero tends to zero
Another equivalent formulation of Nernst's theorem is: "It is impossible to reach a temperature equal to absolute zero by means of a sequence of thermodynamic processes."
The change in entropy uniquely determines the direction and limit of the spontaneous flow of the process only for the simplest systems - isolated ones. In practice, for the most part, one has to deal with systems that interact with the environment. To characterize the processes occurring in closed systems, new thermodynamic state functions were introduced: isobaric-isothermal potential (Gibbs free energy) And isochoric-isothermal potential (Helmholtz free energy).
The behavior of any thermodynamic system in the general case is determined by the simultaneous action of two factors - enthalpy, which reflects the system's desire to minimize thermal energy, and entropy, which reflects the opposite trend - the system's desire for maximum disorder. If for isolated systems (ΔH = 0) the direction and limit of the spontaneous flow of the process is uniquely determined by the value of the change in the entropy of the system ΔS, and for systems at temperatures close to absolute zero (S = 0 or S = const), the criterion for the direction of the spontaneous process is the change enthalpy ΔH, then for closed systems at temperatures not equal to zero, it is necessary to simultaneously take into account both factors. The direction and limit of the spontaneous flow of the process in any systems is determined by the more general principle of the minimum free energy:
Only those processes that lead to a decrease in the free energy of the system can proceed spontaneously; the system comes to a state of equilibrium when the free energy reaches its minimum value.
For closed systems that are in isobaric-isothermal or isochoric-isothermal conditions, the free energy takes the form of isobaric-isothermal or isochoric-isothermal potentials (the so-called Gibbs and Helmholtz free energy, respectively). These functions are sometimes called simply thermodynamic potentials, which is not quite strict, since internal energy (isochorically isentropic) and enthalpy (isobaric isentropic potential) are also thermodynamic potentials.
Consider a closed system in which an equilibrium process is carried out at constant temperature and volume. We express the work of this process, which we denote by A max (since the work of the process carried out in equilibrium is maximum), from equations (I.53, I.54):
(I.69)
We transform expression (I.69) by grouping terms with the same indices:
Entering the designation:
we get:
(I.72) (I.73)
The function is an isochoric-isothermal potential (Helmholtz free energy), which determines the direction and limit of the spontaneous flow of the process in a closed system under isochoric-isothermal conditions.
A closed system under isobaric-isothermal conditions is characterized by the isobaric-isothermal potential G:
|
(I.74) Since –ΔF = A max , we can write:
The value A "max is called maximum useful work(maximum work minus expansion work). Based on the principle of minimum free energy, it is possible to formulate the conditions for the spontaneous flow of the process in closed systems.
Conditions for spontaneous processes in closed systems:
Isobaric-isothermal(P=const, T=const):
ΔG<0.dG<0
Isochoric-isothermal(V=const, T=const):
ΔF<0.dF< 0
Processes that are accompanied by an increase in thermodynamic potentials occur only when work is performed on the system from the outside. In chemistry, the isobaric-isothermal potential is most commonly used, since most chemical (and biological) processes occur at constant pressure. For chemical processes, the value of ΔG can be calculated, knowing the ΔH and ΔS of the process, according to equation (I.75), or using tables of standard thermodynamic potentials for the formation of substances ΔG ° arr; in this case, ΔG° of the reaction is calculated similarly to ΔH° according to equation (I.77):
The value of the standard change in the isobaric-isothermal potential during any chemical reaction ΔG° 298 is a measure of the chemical affinity of the starting substances. Based on equation (I.75), it is possible to estimate the contribution of enthalpy and entropy factors to the value of ΔG and make some generalizing conclusions about the possibility of spontaneous occurrence of chemical processes, based on the sign of ΔН and ΔS.
1. exothermic reactions; ΔH<0.
a) If ΔS > 0, then ΔG is always negative; exothermic reactions accompanied by an increase in entropy always proceed spontaneously.
b) If ΔS< 0, реакция будет идти самопроизвольно при ΔН >TΔS (low temperatures).
2. Endothermic reactions; ΔH >0.
a) If ΔS > 0, the process will be spontaneous at ΔН< TΔS (высокие температуры).
b) If ΔS< 0, то ΔG всегда положительно; самопроизвольное протекание эндотермических реакций, сопровождающихся уменьшением энтропии, невозможно.
CHEMICAL EQUILIBRIUM
As shown above, the occurrence of a spontaneous process in a thermodynamic system is accompanied by a decrease in the free energy of the system (dG< 0, dF < 0). Очевидно, что рано или поздно (напомним, что понятие "время" в термодинамике отсутствует) система достигнет минимума свободной энергии. Условием минимума некоторой функции Y = f(x) является равенство нулю первой производной и положительный знак второй производной: dY = 0; d 2 Y >0. Thus, the condition for thermodynamic equilibrium in a closed system is the minimum value of the corresponding thermodynamic potential:
Isobaric-isothermal(P=const, T=const):
ΔG=0dG=0, d 2 G>0
Isochoric-isothermal(V=const, T=const):
ΔF=0dF=0, d 2 F>0
The state of the system with the minimum free energy is the state of thermodynamic equilibrium:
Thermodynamic equilibrium is such a thermodynamic state of a system that, under constant external conditions, does not change in time, and this invariability is not due to any external process.
The doctrine of equilibrium states is one of the branches of thermodynamics. Next, we will consider a special case of a thermodynamic equilibrium state - chemical equilibrium. As is known, many chemical reactions are reversible, i.e. can flow simultaneously in both directions - forward and reverse. If a reversible reaction is carried out in a closed system, then after a while the system will come to a state of chemical equilibrium - the concentrations of all reacting substances will cease to change with time. It should be noted that the achievement of a state of equilibrium by the system does not mean the termination of the process; chemical equilibrium is dynamic, i.e. corresponds to the simultaneous flow of the process in opposite directions at the same speed. The chemical equilibrium is mobile– any infinitely small external influence on the equilibrium system causes an infinitely small change in the state of the system; upon termination of the external influence, the system returns to its original state. Another important property of chemical equilibrium is that the system can spontaneously come to a state of equilibrium from two opposite sides. In other words, any state adjacent to the equilibrium one is less stable, and the transition to it from the equilibrium state is always associated with the need to expend work from outside.
The quantitative characteristic of chemical equilibrium is the equilibrium constant, which can be expressed in terms of equilibrium concentrations C, partial pressures P, or mole fractions X of the reactants. For some reaction
the corresponding equilibrium constants are expressed as follows:
(I.78) (I.79) (I.80)
The equilibrium constant is a characteristic quantity for every reversible chemical reaction; the value of the equilibrium constant depends only on the nature of the reactants and temperature. The expression for the equilibrium constant for an elementary reversible reaction can be derived from kinetic concepts.
Consider the process of establishing equilibrium in a system in which at the initial moment of time only initial substances A and B are present. The rate of the direct reaction V 1 at this moment is maximum, and the rate of the reverse reaction V 2 is equal to zero:
(I.81)
(I.82)
As the concentration of the starting substances decreases, the concentration of the reaction products increases; accordingly, the rate of the forward reaction decreases, the rate of the reverse reaction increases. Obviously, after some time, the rates of the forward and reverse reactions will become equal, after which the concentrations of the reactants will stop changing, i.e. chemical equilibrium is established.
Assuming that V 1 \u003d V 2, we can write:
(I.84)
Thus, the equilibrium constant is the ratio of the rate constants of the forward and reverse reactions. This implies the physical meaning of the equilibrium constant: it shows how many times the rate of the forward reaction is greater than the rate of the reverse at a given temperature and concentrations of all reacting substances equal to 1 mol / l.
Now consider (with some simplifications) a more rigorous thermodynamic derivation of the expression for the equilibrium constant. For this, it is necessary to introduce the concept chemical potential. Obviously, the value of the free energy of the system will depend both on external conditions (T, P or V), and on the nature and amount of substances that make up the system. If the composition of the system changes with time (i.e., a chemical reaction occurs in the system), it is necessary to take into account the effect of the change in composition on the value of the free energy of the system. Let us introduce into some system an infinitely small number dn i moles of the i-th component; this will cause an infinitesimal change in the thermodynamic potential of the system. The ratio of an infinitesimal change in the value of the free energy of the system to an infinitesimal amount of a component introduced into the system is the chemical potential μ i of this component in the system:
(I.85) 
(I.86)
The chemical potential of a component is related to its partial pressure or concentration by the following relationships:
(I.87) 
(I.88)
Here μ ° i is the standard chemical potential of the component (P i = 1 atm., С i = 1 mol/l.). Obviously, the change in the free energy of the system can be related to the change in the composition of the system as follows:
Since the equilibrium condition is the minimum free energy of the system (dG = 0, dF = 0), we can write:
In a closed system, a change in the number of moles of one component is accompanied by an equivalent change in the number of moles of the remaining components; i.e., for the above chemical reaction, the following relation holds: If the system is in a state of chemical equilibrium, then the change in the thermodynamic potential is zero; we get:
(I.98) 
(I.99)
Here with i And p i – equilibrium concentrations and partial pressures of the initial substances and reaction products (in contrast to non-equilibrium С i and Р i in equations I.96 - I.97).
Since for each chemical reaction the standard change in the thermodynamic potential ΔF° and ΔG° is a strictly defined value, the product of equilibrium partial pressures (concentrations) raised to a power equal to the stoichiometric coefficient for a given substance in the chemical reaction equation (stoichiometric coefficients for starting substances are considered to be negative) there is a certain constant called the equilibrium constant. Equations (I.98, I.99) show the relationship between the equilibrium constant and the standard change in free energy during a reaction. The equation of the isotherm of a chemical reaction relates the values of the real concentrations (pressures) of the reactants in the system, the standard change in the thermodynamic potential during the reaction, and the change in the thermodynamic potential during the transition from a given state of the system to equilibrium. The sign of ΔG (ΔF) determines the possibility of spontaneous flow of the process in the system. In this case, ΔG° (ΔF°) is equal to the change in the free energy of the system during the transition from the standard state (P i = 1 atm., С i = 1 mol/l) to the equilibrium state. The equation of the isotherm of a chemical reaction makes it possible to calculate the value of ΔG (ΔF) during the transition from any state of the system to equilibrium, i.e. answer the question whether the chemical reaction will proceed spontaneously at given concentrations C i (pressures P i) of the reagents:
If the change in the thermodynamic potential is less than zero, the process under these conditions will proceed spontaneously.
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