15 thermodynamic potentials transformation of derivative thermodynamic quantities. Thermodynamic potentials. Systems with a variable number of particles. Large thermodynamic potential

Lecture on the topic: “Thermodynamic potentials”

Plan:

Group of potentials “E F G H”, having the dimension of energy.

Dependence of thermodynamic potentials on the number of particles. Entropy as thermodynamic potential.

Thermodynamic potentials of multicomponent systems.

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, thanks 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 in different 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:

, , , (3.2)

If the “system in a thermostat” defined by the variables is used as a description method, it is most convenient to use free energy as the potential . Accordingly, for the system parameters we obtain:

, , , (3.3)

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:

, , , (3.4)

And in the case of an “adiabatic system over a piston”, defined by state functions, the role of thermodynamic potential is played by 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, Considering 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 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 produced due to the 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:

(3.14) gas parameters (T, P, V) ... system neutral molecular gas with high potential ionization + free electrons emitted by particles...

Thermodynamic basics of thermoelasticity

Coursework >> Physics

And thermoelasticity introduced generalized complex potentials thermoelasticity, which made it possible to solve various problems... Kozionov V.A., Ispulov N.A., Bayaubaev E.K. Seytkhanova A.K. Dynamic and thermodynamic processes in rocky soils and building structures...

Thermodynamic characteristics (H, S, G) and the possibility of spontaneous occurrence of the process

Coursework >> Chemistry

University Department of Chemistry Coursework " Thermodynamic characteristics (H, S, G) and the possibility of spontaneous...). Find potentials oxidizer and reducer indicate the direction of the process. Define thermodynamic characteristics...

Thermodynamic characteristics of reaction sites

Test >> Chemistry

CaCO4 = CaO + CO2 Standard thermodynamic characteristics of reaction sites: kJ ∆ ... element difference between electrode potentials cathode and anode. ...with a more positive electrode potential, and the anode is an electrode with a more negative potential. EMF = E...

The change in entropy unambiguously determines the direction and limit of spontaneous occurrence of the process only for the simplest systems - isolated ones. In practice, most of the time we have to deal with systems that interact with the environment. To characterize 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, reflecting the system's desire for a minimum of thermal energy, and entropy, reflecting the opposite tendency - the system's desire for maximum disorder. If for isolated systems (ΔН = 0) the direction and limit of the spontaneous occurrence of the process is uniquely determined by the magnitude of the change in the entropy of the system ΔS, and for systems located 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 spontaneous occurrence of the process in any system is determined by the more general principle of minimum free energy:

Only those processes that lead to a decrease in the free energy of the system can occur spontaneously; the system reaches a state of equilibrium when the free energy reaches a minimum value.

For closed systems under isobaric-isothermal or isochoric-isothermal conditions, 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 entirely strict, since internal energy (isochoric-isentropic) and enthalpy (isobaric-isentropic potential) are also thermodynamic potentials.

Let us consider a closed system in which an equilibrium process occurs at constant temperature and volume. Let us express the work of this process, which we denote as A max (since the work of a process carried out in equilibrium is maximum), from equations (I.53, I.54):

(I.69)

Let's transform expression (I.69) by grouping terms with the same indices:

By 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 occurrence of a process in a closed system under isochoric-isothermal conditions.

A closed system under isobaric-isothermal conditions is characterized by an isobaric-isothermal potential G:

(1.75)

(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 conditions for the spontaneous occurrence of a process in closed systems.

Conditions for the spontaneous occurrence of 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 from outside on the system. In chemistry, the isobaric-isothermal potential is most often used, since most chemical (and biological) processes occur at constant pressure. For chemical processes, the value of ΔG can be calculated by knowing ΔH and ΔS of the process, using equation (I.75), or using tables of standard thermodynamic potentials for the formation of substances ΔG°arr; in this case, ΔG° reaction is calculated similarly to ΔH° using equation (I.77):

The magnitude 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 general conclusions about the possibility of spontaneous occurrence of chemical processes based on the sign of the values ​​of ΔH 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 occur 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 ΔH< TΔS (высокие температуры).

b) If ΔS< 0, то ΔG всегда положительно; самопроизвольное протекание эндотермических реакций, сопровождающихся уменьшением энтропии, невозможно.

CHEMICAL EQUILIBRIUM

As was 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=0 dG=0, d 2 G>0

Isochoric-isothermal(V = const, T = const):

ΔF=0 dF=0, d 2 F>0

The state of the system with minimal free energy is the state of thermodynamic equilibrium:

Thermodynamic equilibrium is a thermodynamic state of a system that, given constant external conditions, does not change over time, and this invariability is not caused by any external process.

The study 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 simultaneously flow in both directions - forward and reverse. If a reversible reaction is carried out in a closed system, then after some time the system will reach a state of chemical equilibrium - the concentrations of all reacting substances will cease to change over time. It should be noted that the achievement of a state of equilibrium by the system does not mean the cessation of the process; chemical equilibrium is dynamic, i.e. corresponds to the simultaneous occurrence of a process in opposite directions at the same speed. Chemical equilibrium is mobile– any infinitesimal external influence on an equilibrium system causes an infinitesimal 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 a system can spontaneously reach a state of equilibrium from two opposite sides. In other words, any state adjacent to the equilibrium state is less stable, and the transition to it from the equilibrium state is always associated with the need to expend work from the outside.

A 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 reactants. For some reaction

the corresponding equilibrium constants are expressed as follows:

(I.78) (I.79) (I.80)

The equilibrium constant is a characteristic value for each 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.

Let us consider the process of establishing equilibrium in a system in which at the initial moment of time only the starting substances A and B are present. The rate of the forward reaction V 1 at this moment is maximum, and the rate of the reverse reaction V 2 is 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. It is obvious that after some time the rates of the forward and reverse reactions will become equal, after which the concentrations of the reacting substances will cease to change, i.e. chemical equilibrium will be established.

Assuming that V 1 = 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 leads to 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 reaction at a given temperature and concentrations of all reactants equal to 1 mol/l.

Now let us consider (with some simplifications) a more rigorous thermodynamic derivation of the expression for the equilibrium constant. To do this, it is necessary to introduce the concept chemical potential. It is obvious that 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 over 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 free energy of the system. Let us introduce into some system an infinitesimal 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 a system to an infinitesimal amount of a component introduced into the system is the chemical potential μ i of a given 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., C i = 1 mol/l.). Obviously, a change in the free energy of the system can be associated with a 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 thermodynamic potential is zero; we get:

(I.98) (I.99)

Here with i And p i – equilibrium concentrations and partial pressures of starting substances and reaction products (in contrast to nonequilibrium C i and P i in equations I.96 - I.97).

Since for each chemical reaction the standard change in 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 equation of a chemical reaction (stoichiometric coefficients for starting substances are usually considered negative) there is a certain constant called the equilibrium constant. Equations (I.98, I.99) show the relationship of the equilibrium constant with the standard change in free energy during the 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 occurrence 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., C 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 a chemical reaction will proceed spontaneously at given concentrations C i (pressures P i) of reagents:

If the change in thermodynamic potential is less than zero, the process under these conditions will proceed spontaneously.

Related information.

$S$ and generalized coordinates $x\_1,x\_2,...$(volume of the system, interface area, length of the elastic rod or spring, polarization of the dielectric, magnetization of the magnet, masses of the system components, etc.), and thermodynamic characteristic functions obtained by applying the Legendre transformation to the internal energy

$U=U(S,x\_1,x\_2,...)$.

The purpose of introducing thermodynamic potentials is to use such a set of natural independent variables that describe the state of a thermodynamic system, which is most convenient in a particular situation, while maintaining the advantages that the use of characteristic functions with the dimension of energy gives. In particular, the decrease in thermodynamic potentials in equilibrium processes occurring at constant values ​​of the corresponding natural variables is equal to useful external work.

Thermodynamic potentials were introduced by W. Gibbs, who spoke of “fundamental equations”; term thermodynamic potential belongs to Pierre Duhem.

The following thermodynamic potentials are distinguished:

Definitions (for systems with a constant number of particles)

Internal energy

Defined in accordance with the first law of thermodynamics, as the difference between the amount of heat imparted to the system and the work done by the system above external bodies:

$U=Q\; -\; A$.

Enthalpy

Defined as follows:

$H=U+PV$,

Since in an isothermal process the amount of heat received by the system is $T\backslash Delta\; S$, That 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):

$G\; =\; H\; -\; TS\; =\; F\; +\; PV\; =\; U+PV-TS$.

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:

$A^f\_(max)=-\backslash Delta\; F$,

Where $F$- Helmholtz free energy.

In this sense $F$ 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

$A^u\_(max)=-\backslash Delta\; G$

Where $G$- 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

$dU=\; \backslash delta\; Q\; -\; \backslash delta\; A=T\; dS\; -\; P\; dV$,

• for enthalpy

$dH\; =\; dU\; +\; d(PV)\; =\; T\; dS\; -\; P\; dV\; +\; P\; dV\; +\; V\; dP\; =\; T\; dS\; +\; V\; dP$,

• for Helmholtz free energy

$dF\; =\; dU\; -\; d(TS)\; =\; T\; dS\; -\; P\; dV\; -\; T\; dS\; -\; S\; dT\; =\; -P\; dV\; -\; S\; dT$,

• for the Gibbs potential

$dG\; =\; dH\; -\; d(TS)\; =\; T\; dS\; +\; V\; dP\; -\; T\; dS\; -\; S\; dT\; =\; V\; dP\; -\; S\; dT$.

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:

$U\; =\; U(S,V)$, $H\; =\; H(S,P)$, $F\; =\; F(T,V)$, $G\; =\; G(T,P)$.

Specifying any of these four dependencies - that is, specifying the type of functions $U(S,V)$, $H(S,P)$, $F(T,V)$, $G(T,P)$- allows you to obtain all the information about the properties of the system. So, for example, if we are given the internal energy $U$ as a function of entropy $S$ and volume $V$, the remaining parameters can be obtained by differentiation:

$T=(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; S)\backslash right))\_V$ $P=-(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; V)\backslash right))\_S$

Here are the indices $V$ And $S$ mean the constancy of the second variable on which the function depends. These equalities become obvious if we consider that $dU\; =\; T\; dS\; -\; P\; dV$.

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.

Transition from one thermodynamic potential to another. Gibbs - Helmholtz formulas

The values ​​of all thermodynamic potentials in certain variables can be expressed in terms of a potential whose differential is complete in these variables. For example, for simple systems in variables $V$, $T$ thermodynamic potentials can be expressed in terms of Helmholtz free energy:

$U\; =\; -\; T^2\; \backslash left(\backslash frac(\backslash partial)(\backslash partial\; T\; )\backslash frac(F)(T)\; \backslash right)\_(V)$,

$H\; =\; -\; T^2\; \backslash left(\backslash frac(\backslash partial)(\backslash partial\; T\; )\backslash frac(F)(T)\; \backslash right)\_(V)\; -\; V\; \backslash left(\backslash frac(\backslash partial\; F)(\backslash partial\; V)\backslash right)\_(T)$,

$G=\; F-\; V\; \backslash left(\backslash frac(\backslash partial\; F)(\backslash partial\; V)\backslash right)\_(T)$.

The first of these formulas is called Gibbs-Helmholtz formula, but sometimes the term is applied to all such formulas in which temperature is the only independent variable.

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:

$dU\; =\; T\; dS\; -\; P\; dV$.

It is known that if mixed derivatives exist and are continuous, then they do not depend on the order of differentiation, that is

$\backslash frac(\backslash partial^2\; U)(\backslash partial\; V\; \backslash partial\; S)=\backslash frac(\backslash partial^2\; U)(\backslash partial\; S\; \backslash partial\; V)$.

But $(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; V)\backslash right))\_S=-P$ And $(\backslash left(\backslash frac(\backslash partial\; U)(\backslash partial\; S)\backslash right))\_V=T$, That's why

$(\backslash left(\backslash frac(\backslash partial\; P)(\backslash partial\; S)\backslash right))\_V=-(\backslash left(\backslash frac(\backslash partial\; T)(\backslash partial\; V)\backslash right))\_S$.

Considering the expressions for other differentials, we obtain:

$(\backslash left(\backslash frac(\backslash partial\; T)(\backslash partial\; P)\backslash right))\_S=(\backslash left(\backslash frac(\backslash partial\; V)(\backslash partial\; S)\backslash right))\_P$, $(\backslash left(\backslash frac(\backslash partial\; S)(\backslash partial\; V)\backslash right))\_T=(\backslash left(\backslash frac(\backslash partial\; P)(\backslash partial\; T)\backslash right))\_V$, $(\backslash left(\backslash frac(\backslash partial\; S)(\backslash partial\; P)\backslash right))\_T=-(\backslash left(\backslash frac(\backslash partial\; V)(\backslash partial\; T)\backslash right))\_P$.

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

Chemical potential ( $\backslash mu$ ) 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:

$dU\; =\; T\; dS\; -\; P\; dV\; +\; \backslash mu\; dN$, $dH\; =\; T\; dS\; +\; V\; dP\; +\; \backslash mu\; dN$, $dF\; =\; -S\; dT\; -\; P\; dV\; +\; \backslash mu\; dN$, $dG\; =\; -S\; dT\; +\; V\; dP\; +\; \backslash mu\; dN$.

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 the fact that $S$ And $V$ are additive quantities, and $T$ And $P$- No):

$U\; =\; U(S,V,N)\; =\; N\; f\; \backslash left(\backslash frac(S)(N),\backslash frac(V)(N)\backslash right)$, $H\; =\; H(S,P,N)\; =\; N\; f\; \backslash left(\backslash frac(S)(N),P\backslash right)$, $F\; =\; F(T,V,N)\; =\; N\; f\; \backslash left(T,\backslash frac(V)(N)\backslash right)$, $G\; =\; G(T,P,N)\; =\; N\; f\; \backslash left(T,P\backslash right)$.

And, since $\backslash frac(d\; G)(dN)=\backslash mu$, from the last expression it follows that

$G\; =\; \backslash mu\; N$,

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:

$\backslash Omega\; =\; F\; -\; \backslash mu\; N\; =\; -\; P\; V$; $d\backslash Omega\; =\; -S\; dT\; -\; N\; d\backslash mu\; -\; P\; dV$

It is easy to verify that the so-called bound energy $T\; S$ is the thermodynamic potential for a system specified with constants $S\; P\backslash mu$.

Potentials and thermodynamic equilibrium

In a state of equilibrium, the dependence of thermodynamic potentials on the corresponding variables is determined by the canonical equation of state of this system. However, in states other than equilibrium, these relationships lose their validity. However, thermodynamic potentials also exist for nonequilibrium states.

Thus, with fixed values ​​of its variables, the potential can take on different values, one of which corresponds to the state of thermodynamic equilibrium.

It can be shown that in a state of thermodynamic equilibrium the corresponding potential value is minimal. Therefore, the equilibrium is stable.

The table below shows the minimum of which potential corresponds to the state of stable equilibrium of a system with given fixed parameters.

fixed parameters	thermodynamic potential
S,V,N	internal energy
S,P,N	enthalpy
T,V,N	Helmholtz free energy
T,P,N	Gibbs potential
T,V, $\backslash mu$	Large thermodynamic potential
S,P, $\backslash mu$	bound energy

Write a review about the article "Thermodynamic potentials"

Notes

Literature

• Duhem P.. - Paris: A. Hermann, 1886. - XI + 247 pp.
• Gibbs J. Willard. The Collected Works. - N. Y. - London - Toronto: Longmans, Green and Co., 1928. - T. 1. - XXVIII + 434 p.
• Bazarov I. P.- M.: Higher School, 1991. 376 p.
• Bazarov I. P. Misconceptions and errors in thermodynamics. Ed. 2nd revision - M.: Editorial URSS, 2003. 120 p.
• Gibbs J.W. Thermodynamics. Statistical mechanics. - M.: Nauka, 1982. - 584 p. - (Classics of science).
• Gukhman A. A. On the foundations of thermodynamics. - 2nd ed., rev. - M.: Publishing house LKI, 2010. - 384 p. - ISBN 978-5-382-01105-9.
• Zubarev D.N. Nonequilibrium statistical thermodynamics. M.: Nauka, 1971. 416 p.
• Kvasnikov I. A. Thermodynamics and statistical physics. Equilibrium Systems Theory, vol. 1. - M.: Moscow State University Publishing House, 1991. (2nd ed., revised and supplemented. M.: URSS, 2002. 240 p.)
• Krichevsky I. R. Concepts and fundamentals of thermodynamics. - 2nd ed., revision. and additional - M.: Chemistry, 1970. - 440 p.
• Kubo R. Thermodynamics. - M.: Mir, 1970. - 304 p.
• Landau, L. D., Lifshits, E. M. Statistical physics. Part 1. - 3rd edition, supplemented. - M.: Nauka, 1976. - 584 p. - (“Theoretical Physics”, Volume V).
• Mayer J., Geppert-Mayer M. Statistical mechanics. M.: Mir, 1980.
• Munster A. Chemical thermodynamics. - M.: Mir, 1971. - 296 p.
• Sivukhin D.V. General physics course. - M.: Science, 1975. - T. II. Thermodynamics and molecular physics. - 519 p.
• Sychev V.V. Complex thermodynamic systems. - 4th ed., revised. and additional.. - M: Energoatomizdat, 1986. - 208 p.
• Thermodynamics. Basic concepts. Terminology. Letter designations of quantities. Collection of definitions, vol. 103/ Committee of Scientific and Technical Terminology of the USSR Academy of Sciences. M.: Nauka, 1984

An excerpt characterizing Thermodynamic potentials

She looked to where she knew he was; but she could not see him otherwise than as he was here. She saw him again the same as he was in Mytishchi, at Trinity, in Yaroslavl.
She saw his face, heard his voice and repeated his words and her words spoken to him, and sometimes she came up with new words for herself and for him that could then be said.
Here he lies on an armchair in his velvet fur coat, resting his head on his thin, pale hand. His chest is terribly low and his shoulders are raised. The lips are firmly compressed, the eyes shine, and a wrinkle jumps up and disappears on the pale forehead. One of his legs is trembling almost noticeably quickly. Natasha knows that he is struggling with excruciating pain. “What is this pain? Why pain? How does he feel? How it hurts!” - Natasha thinks. He noticed her attention, raised his eyes and, without smiling, began to speak.
“One terrible thing,” he said, “is to bind yourself forever to a suffering person. This is eternal torment." And he looked at her with a searching look—Natasha now saw this look. Natasha, as always, answered then before she had time to think about what she was answering; she said: “This cannot go on like this, this will not happen, you will be healthy - completely.”
She now saw him first and now experienced everything that she had felt then. She remembered his long, sad, stern look at these words and understood the meaning of the reproach and despair of this long look.
“I agreed,” Natasha was now telling herself, “that it would be terrible if he remained always suffering. I said it that way only because it would have been terrible for him, but he understood it differently. He thought it would be terrible for me. He still wanted to live then - he was afraid of death. And I told him so rudely and stupidly. I didn't think that. I thought something completely different. If I had said what I thought, I would have said: even if he were dying, dying all the time before my eyes, I would be happy compared to what I am now. Now... Nothing, no one. Did he know this? No. Didn't know and never will. And now it will never, never be possible to correct this.” And again he spoke to her the same words, but now in her imagination Natasha answered him differently. She stopped him and said: “Terrible for you, but not for me. You know that I have nothing in life without you, and suffering with you is the best happiness for me.” And he took her hand and squeezed it as he had squeezed it on that terrible evening, four days before his death. And in her imagination she told him other tender, loving speeches that she could have said then, which she said now. “I love you... you... I love you, I love you...” she said, convulsively squeezing her hands, gritting her teeth with fierce effort.
And sweet grief overwhelmed her, and tears were already welling up in her eyes, but suddenly she asked herself: to whom is she telling this? Where is he and who is he now? And again everything was clouded with dry, hard bewilderment, and again, tensely knitting her eyebrows, she peered at where he was. And so, it seemed to her that she was penetrating the secret... But at that moment, just as something incomprehensible was opening up to her, the loud knock of the door lock handle painfully struck her ears. Quickly and carelessly, with a frightened, uninterested expression on her face, the maid Dunyasha entered the room.
“Come to daddy, quickly,” Dunyasha said with a special and animated expression. “It’s a misfortune, about Pyotr Ilyich... a letter,” she said, sobbing.

In addition to the general feeling of alienation from all people, Natasha at this time experienced a special feeling of alienation from her family. All her own: father, mother, Sonya, were so close to her, familiar, so everyday that all their words and feelings seemed to her an insult to the world in which she had lived lately, and she was not only indifferent, but looked at them with hostility . She heard Dunyasha’s words about Pyotr Ilyich, about misfortune, but did not understand them.
“What kind of misfortune do they have there, what kind of misfortune can there be? Everything they have is old, familiar and calm,” Natasha mentally said to herself.
When she entered the hall, the father was quickly leaving the countess's room. His face was wrinkled and wet with tears. He apparently ran out of that room to give vent to the sobs that were crushing him. Seeing Natasha, he desperately waved his hands and burst into painful, convulsive sobs that distorted his round, soft face.
- Pe... Petya... Come, come, she... she... is calling... - And he, sobbing like a child, quickly mincing with weakened legs, walked up to the chair and fell almost on it, covering his face with his hands.
Suddenly, like an electric current ran through Natasha’s entire being. Something hit her terribly painfully in the heart. She felt terrible pain; It seemed to her that something was being torn away from her and that she was dying. But following the pain, she felt an instant release from the ban on life that lay on her. Seeing her father and hearing her mother’s terrible, rude cry from behind the door, she instantly forgot herself and her grief. She ran up to her father, but he, helplessly waving his hand, pointed to her mother’s door. Princess Marya, pale, with a trembling lower jaw, came out of the door and took Natasha by the hand, saying something to her. Natasha didn’t see or hear her. She entered the door with quick steps, stopped for a moment, as if in a struggle with herself, and ran up to her mother.
The Countess lay on an armchair, stretching out strangely awkwardly, and banging her head against the wall. Sonya and the girls held her hands.
“Natasha, Natasha!..” shouted the countess. - It’s not true, it’s not true... He’s lying... Natasha! – she screamed, pushing those around her away. - Go away, everyone, it’s not true! Killed!.. ha ha ha ha!.. not true!
Natasha knelt on the chair, bent over her mother, hugged her, lifted her with unexpected strength, turned her face towards her and pressed herself against her.
- Mama!.. darling!.. I’m here, my friend. “Mama,” she whispered to her, without stopping for a second.
She did not let her mother go, gently struggled with her, demanded a pillow, water, unbuttoned and tore her mother’s dress.
“My friend, my dear... mamma, darling,” she whispered incessantly, kissing her head, hands, face and feeling how uncontrollably her tears flowed in streams, tickling her nose and cheeks.
The Countess squeezed her daughter's hand, closed her eyes and fell silent for a moment. Suddenly she stood up with unusual speed, looked around senselessly and, seeing Natasha, began squeezing her head with all her might. Then she turned her face, wrinkled in pain, towards her and peered at it for a long time.
“Natasha, you love me,” she said in a quiet, trusting whisper. - Natasha, won’t you deceive me? Will you tell me the whole truth?
Natasha looked at her with tear-filled eyes, and in her face there was only a plea for forgiveness and love.
“My friend, mamma,” she repeated, straining all the strength of her love in order to somehow relieve her of the excess grief that was oppressing her.
And again, in a powerless struggle with reality, the mother, refusing to believe that she could live when her beloved boy, blooming with life, was killed, fled from reality in a world of madness.
Natasha did not remember how that day, that night, the next day, the next night went. She did not sleep and did not leave her mother. Natasha’s love, persistent, patient, not as an explanation, not as a consolation, but as a call to life, every second seemed to embrace the countess from all sides. On the third night, the Countess fell silent for a few minutes, and Natasha closed her eyes, resting her head on the arm of the chair. The bed creaked. Natasha opened her eyes. The Countess sat on the bed and spoke quietly.
– I’m so glad you came. Are you tired, do you want some tea? – Natasha approached her. “You have become prettier and more mature,” the countess continued, taking her daughter by the hand.
- Mama, what are you saying!..
- Natasha, he’s gone, no more! “And, hugging her daughter, the countess began to cry for the first time.

Princess Marya postponed her departure. Sonya and the Count tried to replace Natasha, but they could not. They saw that she alone could keep her mother from insane despair. For three weeks Natasha lived hopelessly with her mother, slept on an armchair in her room, gave her water, fed her and talked to her incessantly - she talked because her gentle, caressing voice alone calmed the countess.
The mother's mental wound could not be healed. Petya's death took away half of her life. A month after the news of Petya’s death, which found her a fresh and cheerful fifty-year-old woman, she left her room half-dead and not taking part in life - an old woman. But the same wound that half killed the countess, this new wound brought Natasha to life.
A mental wound that comes from a rupture of the spiritual body, just like a physical wound, no matter how strange it may seem, after a deep wound has healed and seems to have come together at its edges, a mental wound, like a physical one, heals only from the inside with the bulging force of life.
Natasha’s wound healed in the same way. She thought her life was over. But suddenly love for her mother showed her that the essence of her life - love - was still alive in her. Love woke up and life woke up.
The last days of Prince Andrei connected Natasha with Princess Marya. The new misfortune brought them even closer together. Princess Marya postponed her departure and for the last three weeks, like a sick child, she looked after Natasha. The last weeks Natasha spent in her mother’s room had strained her physical strength.
One day, Princess Marya, in the middle of the day, noticing that Natasha was trembling with a feverish chill, took her to her place and laid her on her bed. Natasha lay down, but when Princess Marya, lowering the curtains, wanted to go out, Natasha called her over.
– I don’t want to sleep. Marie, sit with me.
– You’re tired, try to sleep.
- No no. Why did you take me away? She will ask.
- She's much better. “She spoke so well today,” said Princess Marya.
Natasha lay in bed and in the semi-darkness of the room looked at the face of Princess Marya.
“Does she look like him? – thought Natasha. – Yes, similar and not similar. But she is special, alien, completely new, unknown. And she loves me. What's on her mind? All is good. But how? What does she think? How does she look at me? Yes, she is beautiful."
“Masha,” she said, timidly pulling her hand towards her. - Masha, don’t think that I’m bad. No? Masha, my dear. I love you so much. We will be completely, completely friends.
And Natasha, hugging and kissing the hands and face of Princess Marya. Princess Marya was ashamed and rejoiced at this expression of Natasha’s feelings.
From that day on, that passionate and tender friendship that only happens between women was established between Princess Marya and Natasha. They kissed constantly, spoke tender words to each other and spent most of their time together. If one went out, then the other was restless and hurried to join her. The two of them felt greater agreement among themselves than apart, each with itself. A feeling stronger than friendship was established between them: it was an exceptional feeling of the possibility of life only in the presence of each other.
Sometimes they were silent for hours; sometimes, already lying in bed, they began to talk and talked until the morning. They talked mostly about the distant past. Princess Marya talked about her childhood, about her mother, about her father, about her dreams; and Natasha, who had previously turned away with calm incomprehension from this life, devotion, humility, from the poetry of Christian self-sacrifice, now, feeling herself bound by love with Princess Marya, fell in love with Princess Marya’s past and understood a side of life that was previously incomprehensible to her. She did not think of applying humility and self-sacrifice to her life, because she was accustomed to looking for other joys, but she understood and fell in love with this previously incomprehensible virtue in another. For Princess Marya, listening to stories about Natasha’s childhood and early youth, a previously incomprehensible side of life, faith in life, in the pleasures of life, also opened up.
They still never spoke about him in the same way, so as not to violate with words, as it seemed to them, the height of feeling that was in them, and this silence about him made them forget him little by little, not believing it.
Natasha lost weight, turned pale and became so physically weak that everyone constantly talked about her health, and she was pleased with it. But sometimes she was suddenly overcome not only by the fear of death, but by the fear of illness, weakness, loss of beauty, and involuntarily she sometimes carefully examined her bare arm, surprised at its thinness, or looked in the mirror in the morning at her elongated, pitiful, as it seemed to her , face. It seemed to her that this was how it should be, and at the same time she became scared and sad.
Once she quickly went upstairs and was out of breath. Immediately, involuntarily, she came up with something to do downstairs and from there she ran upstairs again, testing her strength and observing herself.
Another time she called Dunyasha, and her voice trembled. She called her again, despite the fact that she heard her steps, called her in the chest voice with which she sang, and listened to him.
She didn’t know this, she wouldn’t have believed it, but under the seemingly impenetrable layer of silt that covered her soul, thin, tender young needles of grass were already breaking through, which were supposed to take root and so cover with their life shoots the grief that had crushed her that it would soon not be visible and not noticeable. The wound was healing from the inside. At the end of January, Princess Marya left for Moscow, and the Count insisted that Natasha go with her in order to consult with doctors.

thermodynamic potentials, thermodynamic potentials of elements

Thermodynamic potentials- internal energy, considered as a function of entropy and generalized coordinates (volume of the system, interface area, length of an elastic rod or spring, polarization of the dielectric, magnetization of the magnet, masses of the system components, etc.), and thermodynamic characteristic functions obtained by applying the Legendre transformation to internal energy

.

The purpose of introducing thermodynamic potentials is to use such a set of natural independent variables that describe the state of a thermodynamic system, which is most convenient in a particular situation, while maintaining the advantages that the use of characteristic functions with the dimension of energy gives. in particular, the decrease in thermodynamic potentials in equilibrium processes occurring at constant values ​​of the corresponding natural variables is equal to useful external work.

Thermodynamic potentials were introduced by W. Gibbs, who spoke of “fundamental equations”; The term thermodynamic potential belongs to Pierre Duhem.

The following thermodynamic potentials are distinguished:

• internal energy
• enthalpy
• Helmholtz free energy
• Gibbs potential
• high thermodynamic potential

• 1 Definitions (for systems with a constant number of particles)
  • 1.1 Internal energy
  • 1.2 Enthalpy
  • 1.3 Helmholtz free energy
  • 1.4 Gibbs potential
• 2 Thermodynamic potentials and maximum work
• 3 Canonical equation of state
• 4 Transition from one thermodynamic potential to another. Gibbs - Helmholtz formulas
• 5 Method of thermodynamic potentials. Maxwell's relations
• 6 Systems with a variable number of particles. Large thermodynamic potential
• 7 Potentials and thermodynamic equilibrium
• 8 Notes
• 9 Literature

Definitions (for systems with a constant number of particles)

Internal energy

Defined in accordance with the first law of thermodynamics, as the difference between the amount of heat imparted to the system and the work done by the system on external bodies:

.

Enthalpy

Defined as follows:

,

where is pressure and is volume.

Since the work is equal in an isobaric process, the increment in enthalpy in a quasi-static isobaric process is equal to the amount of heat received by the system.

Helmholtz free energy

Also often called simply free energy. Defined as follows:

,

where is temperature and is entropy.

Since in an isothermal process the amount of heat received by the system is equal, the loss of free energy in a quasi-static isothermal process is equal to the work done by the system on 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 total work (both on the environment and on external bodies) that can be obtained from a 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 bound.

In some applications, you have to distinguish between complete and useful work. The latter represents the work of the system on external bodies, excluding the environment in which it is immersed. The maximum useful work of the system is

where is the Gibbs energy.

In this sense, 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, represents the canonical equation of state of the system. Like other equations of state, it is valid only for states of thermodynamic equilibrium. In nonequilibrium states, these dependencies may not hold.

Transition from one thermodynamic potential to another. Gibbs - Helmholtz formulas

The values ​​of all thermodynamic potentials in certain variables can be expressed in terms of a potential whose differential is complete in these variables. For example, for simple systems in variables, thermodynamic potentials can be expressed in terms of Helmholtz free energy:

The first of these formulas is called the Gibbs-Helmholtz formula, but the term is sometimes applied to all similar formulas in which temperature is the only independent variable.

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 that relates the free energy to the chemical potential:

;

It is easy to verify that the so-called bound energy is a thermodynamic potential for a system given with constants.

Potentials and thermodynamic equilibrium

In a state of equilibrium, the dependence of thermodynamic potentials on the corresponding variables is determined by the canonical equation of state of this system. However, in states other than equilibrium, these relationships lose their validity. However, thermodynamic potentials also exist for nonequilibrium states.

Thus, with fixed values ​​of its variables, the potential can take on different values, one of which corresponds to the state of thermodynamic equilibrium.

It can be shown that in a state of thermodynamic equilibrium the corresponding potential value is minimal. Therefore, the equilibrium is stable.

The table below shows the minimum of which potential corresponds to the state of stable equilibrium of a system with given fixed parameters.

Notes

1. Krichevsky I.R., Concepts and fundamentals of thermodynamics, 1970, p. 226–227.
2. Sychev V.V., Complex thermodynamic systems, 1970.
3. Kubo R., Thermodynamics, 1970, p. 146.
4. Munster A., ​​Chemical thermodynamics, 1971, p. 85–89.
5. Gibbs J. W., The Collected Works, Vol. 1, 1928.
6. Gibbs J.W., Thermodynamics. Statistical Mechanics, 1982.
7. Duhem P., Le potentiel thermodynamique, 1886.
8. Gukhman A. A., On the foundations of thermodynamics, 2010, p. 93.

Literature

• Duhem P. Le potentiel thermodynamique et ses applications à la mécanique chimique et à l "étude des phénomènes électriques. - Paris: A. Hermann, 1886. - XI + 247 pp.
• Gibbs J. Willard. The Collected Works. - N. Y. - London - Toronto: Longmans, Green and Co., 1928. - T. 1. - XXVIII + 434 p.
• Bazarov I. P. Thermodynamics. - M.: Higher School, 1991. 376 p.
• Bazarov I. P. Misconceptions and errors in thermodynamics. Ed. 2nd revision - M.: Editorial URSS, 2003. 120 p.
• Gibbs J. W. Thermodynamics. Statistical mechanics. - M.: Nauka, 1982. - 584 p. - (Classics of science).
• Gukhman A. A. On the foundations of thermodynamics. - 2nd ed., rev. - M.: Publishing house LKI, 2010. - 384 p. - ISBN 978-5-382-01105-9.
• Zubarev D.N. Nonequilibrium statistical thermodynamics. M.: Nauka, 1971. 416 p.
• Kvasnikov I. A. Thermodynamics and statistical physics. Equilibrium Systems Theory, vol. 1. - M.: Moscow State University Publishing House, 1991. (2nd ed., revised and supplemented. M.: URSS, 2002. 240 pp.)
• Krichevsky I. R. Concepts and fundamentals of thermodynamics. - 2nd ed., revision. and additional - M.: Chemistry, 1970. - 440 p.
• Kubo R. Thermodynamics. - M.: Mir, 1970. - 304 p.
• Landau, L. D., Lifshits, E. M. Statistical physics. Part 1. - 3rd edition, supplemented. - M.: Nauka, 1976. - 584 p. - (“Theoretical Physics”, Volume V).
• Mayer J., Geppert-Mayer M. Statistical mechanics. M.: Mir, 1980.
• Munster A. Chemical thermodynamics. - M.: Mir, 1971. - 296 p.
• Sivukhin D.V. General course in physics. - M.: Nauka, 1975. - T. II. Thermodynamics and molecular physics. - 519 p.
• Sychev V.V. Complex thermodynamic systems. - 4th ed., revised. and additional.. - M: Energoatomizdat, 1986. - 208 p.
• Thermodynamics. Basic concepts. Terminology. Letter designations of quantities. Collection of definitions, vol. 103/ Committee of Scientific and Technical Terminology of the USSR Academy of Sciences. M.: Nauka, 1984

thermodynamic potentials, thermodynamic potentials of elements, thermodynamic potentials

Thermodynamic potentials, or characteristic functions, are thermodynamic functions that contain all the thermodynamic information about the system. The four main thermodynamic potentials are of greatest importance:

1) internal energy U(S,V),

2) enthalpy H(S,p) = U + pV,

3) Helmholtz energy F(T,V) = U - T.S.,

4) Gibbs energy G(T,p) = H - T.S. = F+ pV.

Thermodynamic parameters, which are called natural variables for thermodynamic potentials, are indicated in brackets. All these potentials have the dimension of energy and all of them do not have an absolute value, since they are determined to within a constant, which is equal to the internal energy at absolute zero.

The dependence of thermodynamic potentials on their natural variables is described by the basic thermodynamic equation, which combines the first and second principles. This equation can be written in four equivalent forms:

dU = TdS - pdV (5.1)

dH = TdS + Vdp (5.2)

dF = - pdV - SDT (5.3)

dG = Vdp - SDT (5.4)

These equations are written in a simplified form - only for closed systems in which only mechanical work is performed.

Knowing any of the four potentials as a function of natural variables, you can use the basic equation of thermodynamics to find all the other thermodynamic functions and parameters of the system (see Example 5-1).

Another important meaning of thermodynamic potentials is that they allow one to predict the direction of thermodynamic processes. So, for example, if the process occurs at constant temperature and pressure, then the inequality expressing the second law of thermodynamics:

is equivalent to the inequality dG p,T 0 (we took into account that at constant pressure Qp = dH), where the equal sign refers to reversible processes, and inequalities - to irreversible ones. Thus, during irreversible processes occurring at constant temperature and pressure, the Gibbs energy always decreases. The minimum Gibbs energy is achieved at equilibrium.

Similarly, any thermodynamic potential in irreversible processes with constant natural variables decreases and reaches a minimum at equilibrium:

Potential	Natural variables	Spontaneity condition	Conditions equilibrium
	S = const, V= const		dU = 0, d 2 U > 0
	S = const, p= const		dH = 0, d 2 H > 0
	T = const, V= const		dF = 0, d 2 F > 0
	T = const, p= const		dG = 0, d 2 G > 0

The last two potentials are of greatest importance in specific thermodynamic calculations - the Helmholtz energy F and Gibbs energy G, because their natural variables are most convenient for chemistry. Another (outdated) name for these functions is isochoric-isothermal and isobaric-isothermal potentials. They have an additional physical and chemical meaning. A decrease in the Helmholtz energy in any process when T= const, V= const is equal to the maximum mechanical work that the system can perform in this process:

F 1 - F 2 = A max (= A arr).

So the energy F equal to that part of the internal energy ( U = F + T.S.), which can turn into work.

Similarly, a decrease in the Gibbs energy in any process at T= const, p= const is equal to the maximum useful (i.e., non-mechanical) work that the system can do in this process:

G 1 - G 2 = A floor.

The dependence of the Helmholtz (Gibbs) energy on volume (pressure) follows from the basic equation of thermodynamics (5.3), (5.4):

. (5.5)

The dependence of these functions on temperature can be described using the basic equation of thermodynamics:

(5.6)

or using the Gibbs-Helmholtz equation:

(5.7)

Calculation of function changes F And G chemical reactions can be carried out in different ways. Let's consider two of them using the Gibbs energy as an example.

1) By definition, G = H - T.S.. If the reaction products and starting materials are at the same temperature, then the standard change in Gibbs energy in a chemical reaction is equal to:

2) Similar to the thermal effect of a reaction, the change in Gibbs energy can be calculated using the Gibbs energies of formation of substances:

Thermodynamic tables usually give absolute entropies and values ​​of thermodynamic functions for the formation of compounds from simple substances at a temperature of 298 K and a pressure of 1 bar (standard state). For calculation r G And r F under other conditions, relations (5.5) - (5.7) are used.

All thermodynamic potentials are functions of state. This property allows us to find some useful relations between partial derivatives, which are called Maxwell's relations.

Let us consider expression (5.1) for internal energy. Because dU- total differential, partial derivatives of internal energy with respect to natural variables are equal to:

If we differentiate the first identity by volume, and the second by entropy, we obtain cross second partial derivatives of the internal energy, which are equal to each other:

(5.10)

Three other relations are obtained by cross-differentiating equations (5.2) - (5.4).

(5.11)

(5.12)

(5.13)

EXAMPLES

Example 5-1. The internal energy of some system is known as a function of entropy and volume, U(S,V). Find the temperature and heat capacity of this system.

Solution. From the basic equation of thermodynamics (5.1) it follows that temperature is the partial derivative of internal energy with respect to entropy:

Isochoric heat capacity determines the rate of change of entropy with temperature:

Using the properties of partial derivatives, we can express the derivative of entropy with respect to temperature in terms of the second derivative of internal energy:

.

Example 5-2. Using the basic equation of thermodynamics, find the dependence of enthalpy on pressure at constant temperature: a) for an arbitrary system; b) for an ideal gas.

Solution. a) If the basic equation in the form (5.2) is divided by dp at constant temperature, we get:

.

The derivative of entropy with respect to pressure can be expressed using Maxwell’s relation for the Gibbs energy (5.13):

.

b) For an ideal gas V(T) = nRT / p. Substituting this function into the last identity, we get:

.

The enthalpy of an ideal gas does not depend on pressure.

Example 5-3. Express derivatives in terms of other thermodynamic parameters.

Solution. The basic equation of thermodynamics (5.1) can be rewritten as:

,

by representing entropy as a function of internal energy and volume. Coefficients at dU And dV are equal to the corresponding partial derivatives:

.

Example 5-4. Two moles of helium (ideal gas, molar heat capacity C p = 5/2 R) heated from 100 o C to 200 o C at p= 1 atm. Calculate the change in Gibbs energy in this process if the entropy of helium is known, = 131.7 J/(mol. K). Can this process be considered spontaneous?

Solution. The change in the Gibbs energy when heated from 373 to 473 K can be found by integrating the partial derivative with respect to temperature (5.6):

.

The dependence of entropy on temperature at constant pressure is determined by the isobaric dark capacity:

Integrating this expression from 373 K to T gives:

Substituting this expression into the entropy integral, we find:

The heating process does not have to be spontaneous, because a decrease in the Gibbs energy serves as a criterion for the spontaneous occurrence of a process only when T= const and p= const.

Answer. G= -26850 J.

Example 5-5. Calculate the change in Gibbs energy in the reaction

CO + SO 2 = CO 2

at a temperature of 500 K and partial pressures of 3 bar. Will this reaction be spontaneous under these conditions? Gases are considered ideal. Take the necessary data from the directory.

Solution. Thermodynamic data at a temperature of 298 K and a standard pressure of 1 bar are tabulated:

Substance	Enthalpy of formation , kJ/mol	Entropy , J/(mol. K)	Heat capacity , J/(mol. K)
	KJ/mol	J/(mol. K)	J/(mol. K)
CO + SO 2 = = CO 2

Let's assume that C p= const. Changes in thermodynamic functions as a result of the reaction are calculated as the difference between the functions of the reactants and products:

f = f(CO2) - f(CO)-S f(O2).

The standard thermal effect of the reaction at 500 K can be calculated using the Kirchhoff equation in integral form (3.8):

The standard entropy change in a reaction at 500 K can be calculated using formula (4.9):

Standard change in Gibbs energy at 500 K:

To calculate the change in the Gibbs energy at partial pressures of 3 atm, it is necessary to integrate formula (5.5) and use the gas ideality condition ( V= n RT / p, n - change in the number of moles of gases in the reaction):

This reaction can occur spontaneously under these conditions.

Answer. G= -242.5 kJ/mol.

TASKS

5-1. Express internal energy as a function of variables G, T, p.

5-2. Using the basic equation of thermodynamics, find the dependence of internal energy on volume at constant temperature: a) for an arbitrary system; b) for an ideal gas.

5-3. It is known that the internal energy of a certain substance does not depend on its volume. How does the pressure of a substance depend on temperature? Justify your answer.

5-4. Express derivatives in terms of other thermodynamic parameters and functions.

5-5. Write an expression for the infinitesimal change in entropy as a function of internal energy and volume. Find the partial derivatives of entropy with respect to these variables and construct the corresponding Maxwell equation.

5-6. For some substance the equation of state is known p(V, T). How does heat capacity change? C v with volume change? Solve the problem: a) in general form; b) for any specific equation of state (except for an ideal gas).

5-7. Prove the identity: .

5-8. The Helmholtz energy of one mole of some substance is written as follows:

F= a + T(b - c - b ln T - d ln V),

Where a, b, c, d- constants. Find pressure, entropy and heat capacity C V of this body. Give a physical interpretation to the constants a, b, d.

5-9. Draw a graph of the Gibbs energy of an individual substance as a function of temperature in the range from 0 to T > T kip.

5-10. For some system the Gibbs energy is known:

G( T,p) = aT(1-ln T) + RT ln p - T.S. 0 + U 0 ,

Where a, R, S 0 , U 0 - constant. Find the equation of state p(V,T) and dependence U(V,T) for this system.

5-11. The dependence of the molar Helmholtz energy of a certain system on temperature and volume has the form:

Where a, b, c, d- constants. Derive the equation of state p(V,T) for this system. Find the dependence of internal energy on volume and temperature U(V,T). What is the physical meaning of the constants a, b, c?

5-12. Find the dependence of the molar internal energy on volume for a thermodynamic system, which is described by the equation of state (for one mole)

,

Where B(T) is a known function of temperature.

5-13. For a certain substance, the dependence of heat capacity on temperature has the form: C V= aT 3 at a temperature of 0 - 10 K. Find the dependence of the Helmholtz energy, entropy and internal energy on temperature in this range.

5-14. For some substance, the dependence of internal energy on temperature has the form: U = aT 4 + U 0 at a temperature of 0 - 10 K. Find the relationship between the Helmholtz energy, entropy and heat capacity C V on temperature in this range.

5-15. Derive the relationship between the heat capacities:

.

5-16. Based on the identity , prove the identity:

.

5-17. One mole of van der Waals gas expands isothermally with volume V 1 to volume V 2 at temperature T. Find U, H, S, F And G for this process.