Thermodynamic potentials (thermodynamic functions) - characteristic functions in thermodynamics, the decrease of which in equilibrium processes occurring at constant values ​​of the corresponding independent parameters is equal to the useful external work.

Since in an isothermal process the amount of heat received by the system is equal to , then decline free energy in a quasi-static isothermal process is equal to the work done by the system above external bodies.

Gibbs potential

Also called Gibbs energy, thermodynamic potential, Gibbs free energy and even just free energy(which can lead to mixing of the Gibbs potential with the Helmholtz free energy):

Thermodynamic potentials and maximum work

Internal energy is full energy systems. However, the second law of thermodynamics prohibits converting all internal energy into work.

It can be shown that the maximum full work (both on the environment and on external bodies) that can be obtained from the system in an isothermal process, is equal to the decrease in Helmholtz free energy in this process:

where is the Helmholtz free energy.

In this sense it represents free energy that can be converted into work. The remaining part of the internal energy can be called related.

In some applications it is necessary to distinguish full And useful work. The latter represents the work of the system on external bodies, excluding the environment in which it is immersed. Maximum useful the system's work is equal to

where is the Gibbs energy.

In this sense, the Gibbs energy is also free.

Canonical equation of state

Specifying the thermodynamic potential of a certain system in a certain form is equivalent to specifying the equation of state of this system.

The corresponding thermodynamic potential differentials are:

• for internal energy

• for enthalpy

• for Helmholtz free energy

• for the Gibbs potential

These expressions can be considered mathematically as complete differentials of functions of two corresponding independent variables. Therefore, it is natural to consider thermodynamic potentials as functions:

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

Here the indices mean the constancy of the second variable on which the function depends. These equalities become obvious if we consider that .

Setting one of the thermodynamic potentials as a function of the corresponding variables, as written above, is canonical equation of state systems. Like other equations of state, it is valid only for states of thermodynamic equilibrium. In nonequilibrium states, these dependencies may not hold.

Method of thermodynamic potentials. Maxwell's relations

The method of thermodynamic potentials helps to transform expressions that include basic thermodynamic variables and thereby express such “hard-to-observe” quantities as the amount of heat, entropy, internal energy through measured quantities - temperature, pressure and volume and their derivatives.

Let us again consider the expression for the total differential of internal energy:

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

But also, therefore

Considering the expressions for other differentials, we obtain:

These relations are called Maxwell's relations. Note that they are not satisfied in the case of discontinuity of mixed derivatives, which occurs during phase transitions of the 1st and 2nd order.

Systems with a variable number of particles. Large thermodynamic potential

The chemical potential () of a component is defined as the energy that must be expended in order to add an infinitesimal molar amount of this component to the system. Then the expressions for the differentials of thermodynamic potentials can be written as follows:

Since thermodynamic potentials must be additive functions of the number of particles in the system, the canonical equations of state take the following form (taking into account that S and V are additive quantities, but T and P are not):

And, since , from the last expression it follows that

that is, the chemical potential is the specific Gibbs potential (per particle).

For a large canonical ensemble (that is, for a statistical ensemble of states of a system with a variable number of particles and an equilibrium chemical potential), a large thermodynamic potential can be defined, relating free energy to chemical potential:

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

Thermodynamic potentials, Pike, p.36

Thermodynamic potentials, Pike, p.36

For isolated systems, this relationship is equivalent to the classical formulation that entropy can never decrease. This conclusion was made by Nobel laureate I. R. Prigogine, analyzing open systems. He also put forward the principle that disequilibrium can serve as a source of order.

Third beginning thermodynamics describes the state of a system near absolute zero. In accordance with the third law of thermodynamics, it establishes the origin of entropy and fixes it for any system. At T 0 the coefficient of thermal expansion, the heat capacity of any process, goes to zero. This allows us to conclude that at absolute zero temperature, any changes in state occur without a change in entropy. This statement is called the theorem of Nobel laureate V. G. Nernst, or the third law of thermodynamics.

The third law of thermodynamics states :

absolute zero is fundamentally unattainable because when T = 0 And S = 0.

If there existed a body with a temperature equal to zero, then it would be possible to construct a perpetual motion machine of the second kind, which contradicts the second law of thermodynamics.

Modification of the third law of thermodynamics for calculations chemical equilibrium in system formulated by Nobel Prize laureate M. Planck in this way.

Planck's postulate : at absolute zero temperature, entropy takes the value S 0 , independent of pressure, state of aggregation, and other characteristics of the substance. This value can be set equal to zero, orS 0 = 0.

In accordance with statistical theory, the value of entropy is expressed as S = ln, where  – Boltzmann constant,  – statistical weight, or thermodynamic probability of macrostates. It is also called -potential. By statistical weight we mean the number of microstates with the help of which a given macrostate is realized. Entropy of an ideal crystal at T = 0 K, provided  = 1, or in the case where a macrostate can be realized by a single microstate, is equal to zero. In all other cases, the entropy value at absolute zero must be greater than zero.

3.3. Thermodynamic potentials

Thermodynamic potentials are functions of certain sets of thermodynamic parameters, allowing one to find all the thermodynamic characteristics of the system as a function of these same parameters.

Thermodynamic potentials completely determine the thermodynamic state of the system, and by differentiation and integration any parameters of the system can be calculated.

The main thermodynamic potentials include the following functions .

1. Internal energy U, which is a function of independent variables:

entropy S,

volume V,

number of particles N,

generalized coordinates x i

or U = U(S, V, N,x i).

2. Helmholtz free energy F is a function of temperature T, volume V, number of particles N, generalized coordinates x i So F = F(T, V, N, x t).

3. Thermodynamic Gibbs potential G = G(T, p, N, x i).

4. Enthalpy H =H(S, P, N, x i).

5. Thermodynamic potential , for which the independent variables are temperature T, volume V, chemical potential x,  =  (T, V, N, x i).

There are classical relationships between thermodynamic potentials:

U = F + T.S. = H – PV,

F = U – T.S. = H – T.S. – PV,

H = U + PV = F + T.S. + PV,

G = U – T.S. + PV = F + PV = H – T.S.,

 = U – T.S. – V = F – N = H – T.S. – N, (3.12)

U = G + T.S. – PV =  + T.S. + N,

F = G – PV =  + N,

H = G + T.S. =  + T.S. + N,

G =  + PV + N,

 = G – PV – N.

The existence of thermodynamic potentials is a consequence of the first and second laws of thermodynamics and shows that the internal energy of the system U depends only on the state of the system. The internal energy of the system depends on the full set of macroscopic parameters, but does not depend on the method of achieving this state. Let us write the internal energy in differential form

dU = TdS– PdV– X i dx i + dN,

T = ( U/ S) V, N, x= const,

P = –( U/ V) S, N, x= const,

 = ( U/ N) S, N, x= const.

Similarly we can write

dF = – SDT–PdV – X t dx t + dN,

dH= TdS+VdP– X t dx t + dN,

dG= – SdT+VdP – X i dx i + dN,

d = – SDT–PdV – X t dx t – NdN,

S = – ( F/ T) V ; P = –( F/ V) T ; T = ( U/ S) V ; V = ( U/ P) T ;

S = – ( G/ T) P ; V = ( G/ P) S ; T = ( H/ S;); P = – ( U/ V) S

S = – ( F/ T); N = – ( F/);  = ( F/ N); X = – ( U/ x).

These equations take place for equilibrium processes. Let us pay attention to the thermodynamic isobaric-isothermal potential G, called Gibbs free energy,

G = U – T.S. + PV = H –T.S., (3.13)

and isochoric-isothermal potential

F = U – T.S. (3.14)

which is called Helmholtz free energy.

IN chemical reactions, occurring at constant pressure and temperature,

G =  U – TS + PV = N, (3.15)

where  – chemical potential.

Under the chemical potential of some component of the system i we will understand the partial derivative of any of the thermodynamic potentials with respect to the amount of this component at constant values ​​of the remaining thermodynamic variables.

Chemical potential can also be defined as a quantity that determines the change in the energy of a system when one particle of a substance is added, for example,

 i = ( U/ N) S , V= cost , or G =  i N i .

From the last equation it follows that  = G/ N i , that is  represents the Gibbs energy per particle. Chemical potential is measured in J/mol.

Omega potential  is expressed in terms of a large statistical sum Z How

 = – Tln Z, (3.16)

Where [sum over N And k(N)]:

Z=   exp[( N – E k (N))/T].

Components n i, chemical. potentials of components m, etc.) used in Chap. 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 Н= 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(naturalvariable variables V, T, m i).

T thermodynamic potentials can be represented by a general f-loy

where L k are intensive parameters independent of 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 complete differentials of thermodynamic potentials have the form:

Level (2) called. fundamental Gibbs equation in energy. 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 at 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 at constant natural variables is equal to the maximum useful work of process A:

In this case, work A is performed against any generalized force L k acting on the system, except external. pressure (see Maximum reaction work).

T thermodynamic potentials, taken as functions of their natural variables, are characteristic functions of the system. This means that any thermodynamic. properties (compressibility, heat capacity, etc.) m.b. expressed by a relationship that includes only a 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 it is possible to obtain the equations of state of the system.

The derivatives of thermodynamic potentials have important properties. The first partial derivatives with respect to natural extensive variables are equal to intensive variables, for example:

[in general: (9 Y l /9 X i) = L i ]. Conversely, derivatives with respect to natural intensive variables are equal to extensive variables, for example:

[in general: (9 Y l /9 L i) = X i ]. The second partial derivatives with respect to natural variables determine the fur. and term-mich. system properties, 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 the properties of thermodynamic potentials as characteristic. functions will be lost. In addition to thermodynamic potentials, characteristic functions are entropy S (natural variables U, V, n i), Massier function F 1= (natural variables 1/T, V, n i), functionPlank (natural variables 1/T, p/T, n i).

T Thermodynamic potentials are interconnected by the Gibbs-Helmholtz equations. E.g. 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 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 relationships like:

In chem. thermodynamics, in addition to thermodynamic potentials recorded for the system as a whole, average molar (specific) values ​​are widely used (for example, ,

All calculations in thermodynamics are based on the use of state functions called thermodynamic potentials. Each set of independent parameters has its own thermodynamic potential. Changes in potentials that occur during any process determine either the work performed by systole or the heat received by the system.

When considering thermodynamic potentials, we will use relation (103.22), presenting it in the form

The equal sign refers to reversible processes, the inequality sign refers to non-reversible processes.

Thermodynamic potentials are functions of state. Therefore, the increment of any of the potentials is equal to the total differential of the function by which it is expressed. The total differential of the function of the variables and y is determined by the expression

Therefore, if during the transformations we obtain an expression of the form for the increment of a certain value

it can be argued that this quantity is a function of the parameters, and the functions are partial derivatives of the function

Internal energy. We are already very familiar with one of the thermodynamic potentials. This is the internal energy of the system. The expression of the first law for a reversible process can be represented as

(109.4)

Comparison with (109.2) shows that the so-called natural variables for the potential V are the variables S and V. From (109.3) it follows that

From the relationship it follows that in the case when the body does not exchange heat with the external environment, the work performed by it is equal to

or in integral form:

Thus, in the absence of heat exchange with the external environment, work is equal to the loss of internal energy of the body.

At constant volume

Therefore, - the heat capacity at constant volume is equal to

(109.8)

Free energy. According to (109.4), the work done by heat during a reversible isothermal process can be represented in the form

Status function

(109.10)

called the free energy of the body.

In accordance with formulas (109.9) and (109.10), in a reversible isothermal process, work is equal to the decrease in the free energy of the body:

Comparison with formula (109.6) shows that in isothermal processes free energy plays the same role as internal energy in adiabatic processes.

Note that formula (109.6) is valid for both reversible and irreversible processes. Formula (109.12) is valid only for reversible processes. In irreversible processes (see). Substituting this inequality into the relation it is easy to obtain that for irreversible isothermal processes

Consequently, the loss of free energy determines the upper limit on the amount of work that can be done by the system during an isothermal process.

Let's take the differential of function (109.10). Taking into account (109.4) we obtain:

From comparison with (109.2) we conclude that the natural variables for free energy are T and V. In accordance with (109.3)

Let us replace: in (109.1) dQ by and divide the resulting relationship by ( - time). As a result we get that

If the temperature and volume remain constant, then relation (109.16) can be transformed into the form

From this formula it follows that an irreversible process occurring at constant temperature and volume is accompanied by a decrease in the free energy of the body. Once equilibrium is reached, F stops changing with time. Thus; at constant T and V, the equilibrium state is the state for which the free energy is minimal.

Enthalpy. If the process “occurs at constant pressure, then the amount of heat received by the body can be represented as follows:

Status function

called enthalpy or heat function.

From (109.18) and (109.19) it follows that the amount of heat received by the body during the isobathic process is equal to

or in integral form

Consequently, in the case when the pressure remains constant, the amount of heat received by the body is equal to the increase in enthalpy. Differentiation of expression (109.19) taking into account (109.4) gives

From here we conclude. enthalpy is the thermodynamic potential in variables Its partial derivatives are equal

A physical quantity whose elementary change during the transition of a system from one state to another is equal to the amount of heat received or given 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, we can obtain

dS=dQ/T=C V dT/T+RdV/V, and

In an isothermal process T=const, i.e. T 1 =T 2:

DS=R×ln(V 2 /V 1).

In an isobaric process p=const, i.e. V 2 /V 1 =T 2 /T 1:

DS=(C V +R)×ln(T 2 /T 1)=C p ×ln(T 2 /T 1)=C p ×ln(V 2 /V 1).

For an isochoric process, V=const, i.e. V 1 = V 2:

DS=C V ×ln(T 2 /T 1).

In an adiabatic process dQ=0, i.e. DS=0:

S 1 =S 2 =const.

Changes in the entropy of a system performing 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 undergoes an irreversible cycle, then dS>0.

Thus, the entropy of a closed (isolated) system cannot decrease during any processes occurring in it:

where the equal sign is valid for reversible processes, and the inequality sign is valid for irreversible ones.

Second law of thermodynamics: "B 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 energy that cannot be converted into work.

Thermodynamic potentials are certain functions of volume V, pressure p, temperature T, entropy S, 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 a system in any process is defined as the algebraic sum of the amount of heat Q that the system exchanges during the process with environment, and work A, by the perfect system or performed 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 (at 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 small intermolecular interactions ( ideal gas) the change in internal energy is reduced to a change in the kinetic energy of the molecules:

where m is the gas mass;

c V – specific heat capacity at constant volume.

Enthalpy (heat content, Gibbs thermal function) – characterizes the state of a macroscopic system in thermodynamic equilibrium when choosing entropy S and pressure p – H(S,p,N,x) as the main independent variables.

Enthalpy is an additive function (i.e., the enthalpy of the entire system is equal to the sum of the enthalpies of its constituent parts). 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 (with constant N and x) has the form:

From this formula we 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 is

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 for 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 with independent parameters p, T and N – G. Determined through the enthalpy H, entropy S and temperature T by the equality

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, called the chemical potential.

The work performed by a thermodynamic system in any process is determined by the decrease in the thermodynamic potential that meets the conditions of the process. Thus, with a constant number of particles (N=const) under thermal insulation conditions (adiabatic process, S=const), the elementary work dA is equal to the loss of internal energy:

For an isothermal process (T=const)

In this process, work is performed 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 surrounding environment (change in N) is possible, processes are possible at constant p and T. In this case, the elementary work dA of all thermodynamic forces, except pressure forces, is equal to the decrease in the Gibbs thermodynamic potential (G), i.e.

According to Nernst's theorem, the change in entropy (DS) for any reversible isothermal processes occurring between two equilibrium states at temperatures approaching absolute zero tends to zero

Another equivalent formulation of Nernst's theorem is: "With the help of a sequence of thermodynamic processes it is impossible to achieve a temperature equal to absolute zero."