Chemical potential

Consider systems in which the quantities of substances change. These changes can occur as a result of chemical reactions or phase transitions. In this case, the values ​​of thermodynamic potentials change U, H, F, G systems.

To characterize the ability of substances to chemical transformations (or phase transitions) is used chemical potential R. It is found as a partial derivative of thermodynamic potentials with respect to the number of moles. Depending on the process conditions, the chemical potential of the i-th component is expressed through the corresponding thermodynamic potential Gj, F t , me or Uj. So, at constant temperature, pressure and the number of moles of all components, except for the i-th, the chemical potential of the i-th component is equal to the partial derivative of the Gibbs energy with respect to the number of moles of the i-th component:

Chemical potentials are introduced in a similar way under other conditions:

Change in thermodynamic potential (for example, g) when changing the amount of only the /-th component is equal to

Usually, in a system, during chemical reactions, the concentration of several components changes or there is a change in the number of components in several phases. Therefore, the total change in the thermodynamic potential in the system d G equals

For a spontaneous process

If in a system in which a chemical reaction occurs

there are only two components A and B, then for a spontaneous process we can write

Since in this case

In this case, the chemical potential of the starting substance A is greater than the chemical potential of the product B.

In equilibrium, the ratio must be observed

The thermodynamic potentials considered in this section will be further used to study physical and chemical processes in other areas of physical chemistry - phase and chemical equilibria, chemical kinetics, colloidal systems, etc.

Free Energy Calculation Examples

Using the example of two specific processes, we will consider how the free energy (Gibbs energy) is calculated and, based on the results obtained, we will draw conclusions about the possibility and conditions for the processes to occur.

Example 4.1. Is it possible under normal (standard) conditions to reduce iron oxide (III) with hydrogen? The process proceeds in accordance with the equation

Solution. To answer the question of the problem, it is necessary to know the change in the isobaric isothermal potential under standard conditions for the above reaction, i.e.

To do this, apparently, you need to know the standard change in enthalpy and entropy in the process. From the thermodynamic reference book, we write out the enthalpies of formation and entropy of all substances included in the reaction equation, and summarize the data in Table. 4.3.

Let us calculate the change in the enthalpy and entropy of the reaction under standard conditions in accordance with the Hess law.

Table 4.3

Thermodynamic Characteristics of Starting Substances and Reaction Products

Thus, the change in enthalpy in the reaction under standard conditions is ΔНр of the reaction = 95.74 kJ/mol.

The change in entropy ΔSр eacc|1i will also be calculated in accordance with the Hess law:

The change in entropy in the reaction under standard conditions turned out to be equal to

At a temperature of 298 K, the change in the Gibbs energy will be

A large positive value ΔGр of the reaction = +54.48 kJ/mol indicates impossibility reduction of Fe 2 0 3 (cr) with hydrogen to metallic iron under standard conditions.

On the contrary, the opposite process

characterized by a negative change in the Gibbs energy

From which it follows that such a reaction is possible. Indeed, this process proceeds spontaneously, and its result is the oxidation (corrosion) of iron, which we observe daily. And under what conditions (at what temperature) will equilibrium be observed in this system? In the equilibrium state, the change in the Gibbs energy of the system is equal to zero, i.e.

At this temperature, both reactions - the reduction and oxidation of iron - are equally likely, their rates are the same. When calculating the equilibrium temperature, we assume that the change in enthalpy and entropy does not depend on temperature, and use their standard values ​​for 298 K. At temperatures below 691 K, iron is spontaneously oxidized by water vapor to iron oxide with the release of hydrogen, and at temperatures above 691 K. vice versa , hydrogen reduces iron oxide to metallic.

Thus, using the basic concepts and thermodynamic laws, we have estimated the fundamental possibility of the occurrence of both direct and reverse reactions, as well as the temperature of the equilibrium state of the reaction system.

Example 4.2. Is the process of water evaporation possible under standard conditions? What is the effect of temperature on this process?

Solution. To resolve the issue of the possibility of this process, it is necessary to know the change in the Gibbs energy during its course. Therefore, we first compose the thermochemical equation of the required transition:

To calculate the change in the Gibbs energy of this transition, which is equal to

ΔS use this process.

Let us calculate the change in the enthalpy of this transition under standard conditions:

and the change in entropy in this process under the same conditions:

Substituting tabular data (see appendix), we get:

Knowing these quantities, we calculate the change in the Gibbs energy during this transition for standard conditions:

The resulting value of the Gibbs energy Δ G° mn = +8.6 kJ is clearly a positive value (Δ G exp > 0), and, consequently, the process of water evaporation at 25°C is impossible. But our everyday experience suggests otherwise: water evaporates under room conditions. What's the matter?

We performed the calculation for standard conditions, when water vapor has a partial pressure of 101325 Na at a temperature of 25°C. In real conditions, the partial pressure of water vapor is much less (only 3647 Pa) at this temperature, which corresponds to the equilibrium state of the system: "liquid water - steam". If, under real conditions, the partial pressure of water vapor would be equal to 101 325 Pa, then, naturally, no spontaneous evaporation of liquid water would occur under these conditions (therefore, Δ Gncn > 0), but the reverse process - the condensation of water vapor - would be observed. The above example shows that it is necessary to carefully handle the concept of "standard conditions" and the results of thermodynamic calculations.

Let us calculate at what temperature equilibrium will come between the rate of evaporation and condensation of water vapor if their partial pressure is 101 325 Pa. At equilibrium, the change in Gibbs energy is zero

Hence, the temperature of the equilibrium state Г |Х1НМ is determined by the expression:

The value of 97.4°C is close to the boiling point of water, which is 100°C. The difference of 2.6°C is due to the fact that we used the standard values ​​of enthalpy and entropy for 25°C and did not take into account their dependence on temperature.

If the temperature of the system exceeds its equilibrium value, the rate of evaporation of water will predominate compared to the rate of condensation, and then the change in the Gibbs energy will turn out to be a negative value. And when the temperature of the system decreases relative to its equilibrium value, the rate of condensation of water vapor will prevail, and therefore Δ G NSP will be a positive value.

Chemical potential? is one of the thermodynamic parameters of the system; the energy of adding one particle to a system without doing work.
Thermodynamic systems in many cases can exchange atoms and molecules with the environment. In addition to thermal equilibrium brought into contact, thermodynamic systems try to establish an equilibrium in composition. The processes of establishing equilibrium in composition are slower than the processes of establishing equilibrium with temperature. Their speed depends on the nature of the substance. Gases mix quickly, liquids are slower, and diffusion in solids can take many years or millennia.
The desire of thermodynamic systems to establish equilibrium in composition is quantitatively characterized by a quantity called chemical potential. The processes of establishing equilibrium in thermodynamic systems occur in such a way as to equalize the chemical potential in each area. At first glance, it may seem that concentrations are leveling off, but this is true only for a certain class of substances and processes. There are systems in which a certain heterogeneity is inherent. For example, the density of air in the earth's atmosphere decreases with height. This decrease is due to the forces of attraction. Therefore, the concentration of molecules in the air is not uniform. The chemical potential remains constant, which takes into account the potential energy of molecules at different heights.
To determine the chemical potential, you need to find the difference between the energy of a system with N +1 shares and a system with N particles. Chemical potential is the energy that must be provided to a fraction in order to place it in a thermodynamic system. It is important to remember that the fraction is placed in the system in such a way as to be in thermal equilibrium with other particles.
For example, the particles of an ideal gas do not interact with each other, so the minimum energy required to throw one atom of an ideal gas into a system of N atoms is zero. But in order for this new atom to be in thermal equilibrium with other atoms, it is necessary to provide it with energy that would be equal to the average kinetic energy of other atoms. As a consequence of these considerations, the chemical potential of an ideal gas is not zero.
The definition of the chemical potential in terms of other thermodynamic potentials can be written as:

d E = T d S – P d V + ? d N

Where E is the total energy of the system, S is its entropy, N is the number of particles in the system.
This formula determines, in addition to the chemical potential?, also the pressure P and the temperature T.
It can be proved that the chemical potential is given by the formula

If the energy of the system does not depend on volume, but on other thermodynamic parameters A 1, A 2 …, the initial formula becomes

If there are several different types of particles in a system, there are just as many different chemical potentials. Are they usually denoted by different indescams? i. The internal energy differential is written:

Where N i is the number of particles of the i-th type. This relation can be rewritten in terms of concentrations

– The total number of particles in the system.

If there can be several phases in a thermodynamic system, then under conditions of thermodynamic equilibrium, each of the types of chemical potentials must be the same for all phases. This requirement leads to the phase rule.

3.20. Thermodynamic meaning of chemical potential

Let us consider the general relations of thermodynamic systems (potentials) with a variable number of particles N. If the number of particles N in a thermodynamic system can change, then in the formula of the first law of thermodynamics

The same term will be added in the expressions for other thermodynamic potentials: enthalpy H, free energy F, Gibbs energy G.

The value of μ* in thermodynamics is called the chemical potential. Then it follows from this definition that

(3.80)

Let us find the general form of the dependence, for example, of the Gibbs energy G (thermodynamic potential) on the number of particles in a thermodynamic system. For systems with a variable number of particles, the Gibbs energy is a function of temperature T, pressure P, and the number of particles N, i.e., G = G(T, P, N). Leaving T and P unchanged, we increase the number of particles by a factor of σ. Then G also increases by a factor of s, i.e., σG = G (T, P, σN). If we take σ such that σN = 1, then σ = 1/N.

Then the Gibbs energies G = NG (T, P, 1). Therefore, the chemical potential

Therefore, the physical meaning of the chemical potential can be interpreted as the thermodynamic potential (Gibbs energy) related to one particle. Let us express the chemical potential μ* in terms of the Helmholtz free energy F.

For free energy, we can write that F = F (V, T, N). With an increase in the number of particles N by σ times, not only F will increase, but also the volume V, i.e., σF = F (sV, T, sN).

If σ = 1/N, then

F = NF(V/N, T, 1).

(3.83)

Since the number of particles N stands not only as a factor, but also under the sign of the function F (V/N, T, 1).Therefore, the chemical potential

.

Thus, the thermodynamic definition of the chemical potential μ* is not unique. Since the internal energy U and entropy S are determined up to arbitrary constants U 0 and S 0 , then G and F are determined up to a linear function of temperature U 0 – S 0 T. Since U 0 and S 0 depend on the number of particles, then this affects the value of the chemical potential μ*. Therefore, in order to uniquely determine μ*, it is necessary to fix the origins of the internal energy and entropy.

To find the entropy of the gas of fermions S f and bosons S B, according to the quantum statistics of Fermi - Dirac and Bose - Einstein, and applying the Stirling formula, we have:

(3.84)

,

(3.85)

where const do not depend on the number of fermion and boson particles; Z i is the number of quantum states in the i-th energy layer. Now let's calculate the chemical potential μ*, for example, for a Fermi gas.

To do this, consider the Helmholtz free energy F. Change the number of particles N in the system while keeping T and V unchanged and find the free energy increment dF. Since the volume V remains unchanged, and the particles do not interact with each other, then when N changes, the energy levels e i and the corresponding numbers Z i will not change. Only the filling numbers N i will change.

Then for the entropy increment of the fermion gas, from (3.84) we obtain

In a state of equilibrium

.

Thus,

Where - speaks about the average value of the numbers N i , more precisely about their values ​​in the most probable state. Sum Se i – gives an increment of internal energy dU. And the sum

SdN i = d SN i = dN.

As a result, we get TdS = – μ dN + dU or dF = μ dN.

Hence,

The same is true in Bose–Einstein statistics. The chemical potential is used, for example, in chemical reactions and when chemical equilibrium conditions are met. Chemical reactions always proceed in two opposite directions. In chemical equilibrium, the rates of reactions in both directions are the same. In the case of a plasma with equilibrium ionization, the temperatures of its constituent particles of electrons, ions, neutral molecules, etc. are the same.

CHEMICAL POTENTIAL

CHEMICAL POTENTIAL

Thermodynamic , which determines the change in thermodynamic potentials with a change in the number of h-ts in the system and is necessary to describe the St-in open systems (with a variable number of h-ts).

X. p. mi of the i-th component of a multicomponent system is equal to the partial derivative of any of the thermodynamic. potentials by the number (number of hours) of this component at post. the values ​​of the remaining thermodynamic. variables that determine this thermodynamic. ; e.g. mi \u003d (dF / dNi) T, V, Ni (F -, T - temperature, V - volume, j? i). Thus, in systems with alternating the number of h-ts in the expression for the differential thermodynamic. potential should add the value of SimidNi, eg. dF=-SdT-pdV+SimidNi, where p - , S - . Most simply, X. p. is associated with thermodynamic. potential G (see GIBBS ENERGY) G=SimiNi;. For a one-component X. system, m = G / N, i.e., is the Gibbs energy, referred to one hour. Due to the additivity of G X. p., in addition to pressure and temperature, it depends only on the concentrations of otd. components, but not on the number of h-ts in each component. In the simplest case of ideal gases, m depends only on the concentration of the i-ro component mi=m=i+kTln(Ni/N), where N=SiNi is the total number of h-c, (m=i- X. p. of pure i- ro component It is often convenient to use mi as independent thermodynamic variables instead of Ni. In variables T, V, mi, the state of the system characterizes the thermodynamic potential W=F-SimiNi.

X. p. yavl. parameter in the Gibbs grand canonical distribution for systems with variable. the number of h-ts. As a normalization constant X. p. is included in the distributions of Boltzmann, Bose - Einstein and Fermi - Dirac for p-ts ideal gases (see STATISTICAL PHYSICS). In systems in which Boltzmann or Bose-Einstein statistics are applicable, X. p. is always negative. For a Fermi gas, the Xp at zero temperature is positive and determines the boundary Fermi energy (see FERMI SURFACE) and the degeneracy temperature. If the total number of h-ts in the system is not fixed, but must be determined from the thermodynamic condition. equilibrium, as, for example, for phonons in TV. body or for photons in the case of equilibrium thermal radiation, then the equilibrium is characterized by zero X. p.

The concept of X. p. allows us to formulate the conditions for thermodynamic equilibrium. One of the conditions is that the CV of any component is the same in different phases and in different places of one phase. This is due to the possibility of redistribution of h-ts, leading to the alignment of X. p. For systems in a spatially inhomogeneous external equilibrium means that mi = m0i + Ui (r) = const, where m0i- X. p. in the absence of a field, Ui (r ) - potent. h-ts of the i-th component in the external field. For a gas in a gravitational field, this condition leads to a barometric formula for the density of the gas. In case of charging h-c in electric. field (eg, in PP) the value of mi is called electrochemical. potential, leaving the name X. p. for m0i. The equality of the values ​​of X. p. for the h-c of one component that are in different phases determines the conditions for equilibrium during phase transitions (the Gibbs rule of phases) and chemical. reactions (the law of mass action), ionization. equilibrium (see IONIZATION), the properties of solutions (the laws of Van't Hoff, Henry, Raoult), etc. If for one of the components the transition from one part of the system to another is impossible, then for this component the conditions of constancy X are violated and arise in the system. X. p. was introduced by Amer. physicist J. W. Gibbs (1875) when considering chem. equilibrium in multicomponent systems, hence its name. Numerically X. p. is expressed in units. energy per unit mass (J / kg), per unit. number of in-va (J / mol) or 1 hour.

Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .

CHEMICAL POTENTIAL

Thermodynamic state function, determining change thermodynamic potentials when changing the number of particles in the system and necessary to describe the properties of open systems (with a variable number of particles).

X. p. m i i th component of a multicomponent system is equal to the partial derivative of any of the thermodynamic. potentials according to the number (number of particles) of this component at post. the values ​​of the remaining thermodynamic. variables that determine this thermodynamic. potential, eg. m i = (dF/dN i) T,V,N (F- free energy, T- temp-ra, V- volume, ji). Thus, in systems with alternating the number of particles in the expression for the differential, for example. dF, value should be added:

Where R - pressure, S-entropy. Naib. just X. p. is associated with thermodynamic. potential G(cm. Gibbs energy): . For a one-component system X. n. m =G/N,

i.e., is the Gibbs energy per particle. Because of the additivity G, in addition to pressure and temperature-ry, X. p. depends only on the concentration of otd. components, but not on the number of particles in each component. In the simplest case ideal gases m i depends only on the concentration i-th component:

where is the total number of particles, m ~ i-X. n. pure i th

component. Often the values ​​m i convenient to use as independent thermodynamic. variables instead N i . In variables T, V, m i the state of the system characterizes thermodynamic. potential

X. p. is thermodynamic. parameter in Gibbs grand canonical distribution for AC systems the number of particles. As a normalization constant, xp is included in the Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for particles of ideal gases (see Fig. Statistical Physics). In systems to which the statistics of Boltzmann or Bose-Einstein are applicable, X. p. is always negative. For Fermi gas X. p. at zero temperature is positive and determines the boundary Fermi energy(cm. Fermi surface)And temperature degeneration. If

the total number of particles in the system is not fixed, but must be determined from the thermodynamic condition. equilibrium, such as for phonons in solid state or photons in the case of equilibrium thermal radiation, then the equilibrium is characterized by the equality to zero X. p.

The concept of x.p. allows us to formulate the conditions thermodynamic equilibrium. One of the conditions is that the X. p. of any component is the same in decomp. phases and in different places of one phase. This is due to the possibility of a redistribution of particles, leading to an equalization of X. p. For systems in a spatially inhomogeneous ext. the equilibrium field means that

where m i 0 - X. p. in the absence of a field, U i (r)- potent. particle energy i th component in ext. field. For a gas in a gravitational field, this condition leads to a barometric f-le for the density of the gas. In case of charging particles in electric. field (e.g. in semiconductors) value m i called electrochemical potential, leaving the name X. p. for m i 0 . The equality of the values ​​of X.p. for particles of the same component that are in different phases determines the equilibrium conditions for phase transitions ( Gibbs phase rule) and chem. reactions (law of mass action), ionization equilibrium, properties solutions(laws of van't Hoff, Henry, Raoult), etc. If for the particles of one of the components the transition from one part of the system to another is impossible, then for this component the conditions of constancy of X. p. are violated and osmotic pressure arises in the system (see. Osmosis).

X. p. was introduced by J. W. Gibbs (J. W. Gibbs) in 1875 when considering the chemical. equilibrium in multicomponent systems, hence its name. Numerically, X. p. is expressed in units of energy per unit mass (J / kg), or per unit quantity of a substance (J / mol), or per 1 particle.

Lit. see at Art. Thermodynamics. A. E. Meyerovich.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .

See what "CHEMICAL POTENTIAL" is in other dictionaries:

A concept used to describe thermodynamic equilibrium in multicomponent systems. Usually, the chemical potential of a system component is calculated as a partial derivative of the Gibbs energy with respect to the number of particles (or moles) of this component at ... ... Big Encyclopedic Dictionary

chemical potential- is the partial molar Gibbs energy of the substance. The chemical potential of a given substance reflects the change in the Gibbs energy when 1 mole of this substance is added to the system under study. General chemistry: textbook / A. V. Zholnin ... Chemical terms

chemical potential- Gibbs... Dictionary of chemical synonyms I

chemical potential- Partial derivative of the characteristic function with respect to the mass of the component at constant corresponding independent parameters and masses of the remaining components. [Collection of recommended terms. Issue 103. Thermodynamics. USSR Academy of Sciences. Committee… … Technical Translator's Handbook

chemical potential- thermodynamic function used in describing the states of systems with a variable number of components. In a system of i components, the chemical potential is defined as the increase in its internal energy when adding infinitely ... ... Encyclopedic Dictionary of Metallurgy

A concept used to describe thermodynamic equilibrium in multicomponent and (or) multiphase systems. Usually, the chemical potential of a system component is calculated as a partial derivative of the Gibbs energy with respect to the number of particles (or moles) of this ... ... encyclopedic Dictionary

This term has other meanings, see Potential. Thermodynamic quantities ... Wikipedia

chemical potential- cheminis potencialas statusas T sritis Standartizacija ir metrologija apibrėžtis Bendrąja prasme - intensyvusis dydis. Tai termodinaminė jėga μ, apibūdinanti sistemos pusiausviruosius ir nepusiausviruosius medžiagos mainus: medžiaga visada… … Penkiakalbis aiskinamasis metrologijos terminų žodynas

chemical potential- cheminis potencialas statusas T sritis chemija apibrėžtis Intensyvusis dydis, apibūdinantis fizikocheminius termodinaminės sistemos virsmus ir pusiausvyrą. atitikmenys: engl. chemical potential chemical potential... Chemijos terminų aiskinamasis žodynas

chemical potential- cheminis potencialas statusas T sritis fizika atitikmenys: angl. chemical potential vok. chemisches Potential, n rus. chemical potential, m pranc. potentiel chimique, m … Fizikos terminų žodynas

When changing the number of particles in the system and necessary to describe the properties of open systems (with a variable number of particles).

X. p. m i i th component of a multicomponent system is equal to the partial derivative of any of the thermodynamic. potentials according to the number (number of particles) of this component at post. the values ​​of the remaining thermodynamic. variables that determine this thermodynamic. potential, eg. m i = (dF/dN i) T,V,N (F-free energy, T-temp-ra, V-volume, ji). Thus, in systems with alternating the number of particles in the expression for the differential, for example. dF, add the value :

Where R- pressure, S-entropy. Naib. just X. p. is associated with thermodynamic. potential G(cm. Gibbs energy):. For a one-component system X. n. m = G/N,

i.e., is the Gibbs energy per particle. Because of the additivity G, in addition to pressure and temperature-ry, X. p. depends only on the concentration of otd. components, but not on the number of particles in each component. In the simplest case ideal gases m i depends only on the concentration i-th component:

where is the total number of particles, m ~ i-X. n. pure i th

component. Often the values ​​m i convenient to use as independent thermodynamic. variables instead N i. In variables T, V, m i the state of the system characterizes thermodynamic. potential

X. p. is thermodynamic. parameter in Gibbs grand canonical distribution for AC systems the number of particles. As a normalization constant, xp is included in the Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for particles of ideal gases (see Fig. statistical physics). In systems to which Boltzmann or Bose-Einstein statistics are applicable, X. p. is always negative. For Fermi gas X. p. at zero temperature is positive and determines the boundary Fermi energy(cm. Fermi surface)And degeneration temperature. If

the total number of particles in the system is not fixed, but must be determined from the thermodynamic condition. equilibrium, such as for phonons in solid state or photons in the case of equilibrium, then the equilibrium is characterized by the equality to zero X. p.

The concept of x.p. allows us to formulate the conditions thermodynamic equilibrium. One of the conditions is that the X. p. of any component is the same in decomp. phases and in different places of one phase. This is due to the possibility of a redistribution of particles, leading to an equalization of X. p. For systems in a spatially inhomogeneous ext. the equilibrium field means that

where m i 0 - X. p. in the absence of a field, U i (r)-potential particle energy i th component in ext. field. For a gas in a gravitational field, this condition leads to a barometric filter for gas. In case of charging particles in electric. field (e.g. in semiconductors) value m i called electrochemical potential, leaving the name X. p. for m i 0 . The equality of the values ​​of X.p. for particles of the same component that are in different phases determines the equilibrium conditions for phase transitions (