Chemical potential

The chemical potential or chemical potential is a thermodynamic state quantity that was introduced by Josiah Willard Gibbs for the analysis of heterogeneous , thermodynamic systems . Each material component (chemical element or chemical compound) of each homogeneous phase of a thermodynamic system is assigned a chemical potential. The conditions for the thermodynamic equilibrium and the stoichiometric balances in chemical material conversions lead to linear equations between the chemical potentials. These equations make it possible to determine the various proportions of substances in the individual phases (e.g. during distillation ), to calculate the pressure difference in osmosis , to determine the depression of the freezing point , to calculate the equilibrium constants of chemical reactions and to analyze the coexistence of different phases . In addition, the spontaneous development of the system back to a state of equilibrium can be determined by means of the chemical potentials for an external influence on a variable such as temperature, pressure or a substance amount. ${\ displaystyle \ mu}$

Clear interpretation

The chemical potential of a component indicates the change in the internal energy of a system if the number of particles in this component changes while volume and entropy remain constant. It describes, for example, the change in internal energy per added amount of substance of a certain component or the change in internal energy when a component is consumed during a reaction . The chemical potential can be seen in analogy to other intensive state variables such as pressure . The pressure describes the change in internal energy when the volume changes. Like pressure, the chemical potential of a component is a measure of the work done ; instead of mechanical work, it is about chemical work.

definition

The chemical potential describes the change in the energy stored in a system when the number of particles in a chemical species changes. To be more precise, the energy stored in a system must first be defined. It does this by changing the internal energy of a system. ${\ displaystyle \ mathrm {d} U}$

${\ displaystyle \ mathrm {d} U = \ delta Q + \ delta W}$

This means that the internal energy changes when you add or remove heat ( ) or the system does work or work is done on it ( ). The type of work can be mechanical, electrical or chemical in nature. In the following, only mechanical and chemical work are considered. The mechanical work can be written as If you compress a system ( ), you work on the system and the stored energy is increased. By definition, an increase in stored energy is defined as a positive change in internal energy. The change in the stored heat energy is defined by the change in entropy in the system, with the temperature being represented. It thus follows: ${\ displaystyle \ delta Q}$${\ displaystyle \ delta W}$${\ displaystyle -p \ mathrm {d} V}$${\ displaystyle \ mathrm {d} V <0}$${\ displaystyle \ delta Q = T \ mathrm {d} S}$${\ displaystyle S}$${\ displaystyle T}$

${\ displaystyle \ mathrm {d} U = T \ mathrm {d} Sp \ mathrm {d} V + \ delta W _ {\ text {chemical}}}$

The chemical potentials must now be found in the term , since they are supposed to characterize the change in internal energy when doing chemical work. Since chemical work is connected with the course of chemical reactions, a chemical reaction will now be formulated. ${\ displaystyle \ delta W _ {\ text {chemical}}}$

${\ displaystyle \ mathrm {A + B \ longrightarrow C + D}}$

Substances A and B are converted into substances C and D, with chemical work being carried out. When the reaction is voluntary, the internal energy of the system lowers, whereas the internal energy of the system increases when the reaction has to be forced. In the first case the system does chemical work, in the second case chemical work is done on it. In any case, the number of particles of every substance in the system changes as the reaction proceeds. In most cases, instead of changing the number of particles , it is common to use the change in the amount of substance , with the subscript designating the substance under consideration. Since the chemical potential is supposed to characterize the change in the internal energy per converted amount of substance, the following results for the chemical work: ${\ displaystyle \ mathrm {d} N_ {i}}$${\ displaystyle \ mathrm {d} n_ {i}}$${\ displaystyle i}$

${\ displaystyle \ delta W _ {\ text {chemical}} = \ mu _ {A} \ mathrm {d} n_ {A} + \ mu _ {B} \ mathrm {d} n_ {B} + \ mu _ { C} \ mathrm {d} n_ {C} + \ mu _ {D} \ mathrm {d} n_ {D}}$

If different reaction partners are involved, the result for the change in internal energy is: ${\ displaystyle i}$

${\ displaystyle \ mathrm {d} U = T \, \ mathrm {d} Sp \, \ mathrm {d} V + \ Sigma \ mu _ {i} \, \ mathrm {d} n_ {i} \! \ quad .}$

The internal energy is therefore a function of the entropy, the volume and the amount of substance of the components of the system . The chemical potential of a component can now be defined as the partial derivative of the internal energy according to its amount of substance. The other extensive state variables such as entropy, volume and the amount of substance of the other components are kept constant. ${\ displaystyle U (S, V, n_ {i})}$${\ displaystyle i}$${\ displaystyle \ mathrm {d} n_ {j}}$

${\ displaystyle \ Rightarrow \ mu _ {i}: = \ left ({\ frac {\ partial U (S, V, n_ {j})} {\ partial {n_ {i}}}} \ right) _ { S, V, n_ {j \ neq i}}}$

This shows that the chemical potential is defined as the change in internal energy per change in the amount of substance or number of particles.

This definition also shows the differences to other potentials, such as the electrical potential . Electric potential is the antiderivative of the electric field . However, the chemical potential is the derivative of a thermodynamic potential. In this comparison it is therefore analogous to a component of the electric field.

The chemical potential of a substance is composed of an ideal part and an interaction part (excess):

${\ displaystyle \ mu _ {i} = \ mu _ {i} ^ {\ text {ideal}} + \ mu _ {i} ^ {\ text {excess}},}$

where the ideal proportion is given by the chemical potential of an ideal gas. Occasionally the following definition is also made

${\ displaystyle \ mu _ {i} = \ mu _ {i} ^ {0} + \ mu _ {i} ^ {\ text {ideal}} + \ mu _ {i} ^ {\ text {excess}} ,}$

where the standard chemical potential of the particle type is i. ${\ displaystyle \ mu _ {i} ^ {0}}$

Alternative formulations

As an alternative to internal energy, other thermodynamic potentials can also be defined, which provide information about whether a chemical reaction takes place voluntarily or not. In the case of internal energy, a chemical reaction takes place voluntarily (the system releases energy to the environment) when . The same applies to enthalpy , free energy and free enthalpy . These thermodynamic potentials are obtained through successive Legendre transformations of the internal energy. However, the amount of substance remains unaffected during these variable transformations. Thus, the chemical potential of a component can also be defined as the partial derivative of the previously mentioned thermodynamic potentials according to the amount of substance in the component . In addition to the definitions of the chemical potential of the component , the transformation equation for the corresponding thermodynamic potential is also given. For example, the enthalpy is obtained directly from the internal energy. As you can see, the amount of substance does not enter into any transformation equation, as was already mentioned before. Thus the chemical potential can be written as follows: ${\ displaystyle \ mathrm {d} U <0}$${\ displaystyle i}$${\ displaystyle i}$${\ displaystyle i}$

${\ displaystyle \ mu _ {i}: = \ left ({\ frac {\ partial H (S, p, n_ {j})} {\ partial {n_ {i}}}} \ right) _ {S, p, n_ {j \ neq i}}}$from the enthalpy ${\ displaystyle H = U + pV}$
${\ displaystyle \ mu _ {i}: = \ left ({\ frac {\ partial F (T, V, n_ {j})} {\ partial {n_ {i}}}} \ right) _ {T, V, n_ {j \ neq i}}}$from free energy ${\ displaystyle F = U-TS}$
${\ displaystyle \ mu _ {i}: = \ left ({\ frac {\ partial G (T, p, n_ {j})} {\ partial {n_ {i}}}} \ right) _ {T, p, n_ {j \ neq i}}}$from the free enthalpy ${\ displaystyle G = U + pV-TS}$

Chemical equilibrium

The chemical potentials of the components involved in a chemical reaction are also suitable for describing the chemical equilibrium. First, the condition for chemical equilibrium is formulated with the help of the free enthalpy at constant pressure and constant temperature:

${\ displaystyle (\ mathrm {d} G) _ {T, p} = 0}$

A chemical reaction in equilibrium can be formulated as follows:

${\ displaystyle \ nu _ {\ mathrm {A}} \, \ mathrm {A} + \ nu _ {\ mathrm {B}} \, \ mathrm {B} \ rightleftharpoons \ nu _ {\ mathrm {C}} \, \ mathrm {C} + \ nu _ {\ mathrm {D}} \, \ mathrm {D}}$

The equilibrium condition mentioned above can be formulated as follows for this reaction:

${\ displaystyle \ mu _ {A} \ mathrm {d} n_ {A} + \ mu _ {B} \ mathrm {d} n_ {B} + \ mu _ {C} \ mathrm {d} n_ {C} + \ mu _ {D} \ mathrm {d} n_ {D} = 0}$

Every change in the amount of substance of a component can be described by the change in the amount of substance of another component. So the change in the amount of substance of component A can be expressed by. The negative sign comes from the fact that the amount of substance of component A decreases when the amount of substance of component D increases. ${\ displaystyle \ mathrm {d} n_ {A} = - {\ frac {\ nu _ {\ mathrm {A}}} {\ nu _ {\ mathrm {D}}}} \ mathrm {d} n_ {D }}$

If you now do this for all terms, you get to:

${\ displaystyle \ left (- \ mu _ {A} {\ frac {\ nu _ {\ mathrm {A}}} {\ nu _ {\ mathrm {D}}}} - \ mu _ {B} {\ frac {\ nu _ {\ mathrm {B}}} {\ nu _ {\ mathrm {D}}}} + \ mu _ {C} {\ frac {\ nu _ {\ mathrm {C}}} {\ nu _ {\ mathrm {D}}}} + \ mu _ {D} {\ frac {\ nu _ {\ mathrm {D}}} {\ nu _ {\ mathrm {D}}}} \ right) \ mathrm {d} n_ {D} = 0}$

Since the change in the amount of substance of component D cannot be zero, the following equilibrium condition is obtained by multiplying by : ${\ displaystyle \ nu _ {\ mathrm {D}}}$

${\ displaystyle - \ nu _ {\ mathrm {A}} \ mu _ {A} - \ nu _ {\ mathrm {B}} \ mu _ {B} + \ nu _ {\ mathrm {C}} \ mu _ {C} + \ nu _ {\ mathrm {D}} \ mu _ {D} = 0}$

This is an important finding. The chemical equilibrium can thus be fully described by the chemical potentials of the components and their reaction coefficients. In general, the following applies to chemical equilibrium:

${\ displaystyle \ Sigma _ {i, {\ text {Products}}} \ nu _ {\ mathrm {i}} \ mu _ {i} - \ Sigma _ {j, {\ text {Edukte}}} \ nu _ {\ mathrm {j}} \ mu _ {j} = 0}$

The insight that the chemical equilibrium can be expressed with the help of the chemical potentials of the components enables the formulation of the equilibrium constants for a chemical reaction. First of all, the concentration dependence of the chemical potential must be worked out. First remember that the following relationship holds for the change in the free enthalpy in equilibrium:

${\ displaystyle \ mathrm {d} G = \ Sigma \ mu _ {i} \, \ mathrm {d} n_ {i} \! \ quad = 0}$

If one integrates this equation and then forms the total differential , one arrives at the following expression:

${\ displaystyle \ mathrm {d} G = \ Sigma \ mu _ {i} \ mathrm {d} n_ {i} + \ Sigma n_ {i} \ mathrm {d} \ mu _ {i} = 0}$

By comparison with the total differential of the free enthalpy one obtains:

${\ displaystyle \ Sigma \ mu _ {i} \ mathrm {d} n_ {i} + \ Sigma n_ {i} \ mathrm {d} \ mu _ {i} = - SdT + Vdp + \ Sigma \ mu _ {i } \ mathrm {d} n_ {i}}$

Assuming constant temperature, this equation is further simplified to:

${\ displaystyle \ Sigma n_ {i} \ mathrm {d} \ mu _ {i} = V \ mathrm {d} p}$

For the sake of simplicity, assume that the system under consideration is an ideal gas , the total pressure of which results from the partial pressures of the individual components. So the last equation can only be written for one component:

${\ displaystyle n_ {i} \ mathrm {d} \ mu _ {i} = {\ frac {n_ {i} RT} {p_ {i}}} \ mathrm {d} p_ {i}}$

If one now integrates this equation on both sides from standard conditions to a freely selectable state,

${\ displaystyle \ int _ {{\ mu _ {i}} ^ {o}} ^ {\ mu _ {i}} n_ {i} \ mathrm {d} \ mu _ {i} = \ int _ {{ p} ^ {o}} ^ {p} {\ frac {n_ {i} RT} {p_ {i}}} \ mathrm {d} p_ {i}}$

this results in the dependence of the chemical potential of a component

${\ displaystyle \ mu _ {i} = {\ mu _ {i}} ^ {o} + RT \ ln \ left ({\ frac {p_ {i}} {{p_ {i}} ^ {o}} } \ right)}$

The quotient in the logarithm term can be identified as the mole fraction of the ideal gas. If interactions that are not ideal, such as with real gases or electrolyte solutions, are to be taken into account, the mole fraction must be replaced by the activity . Alternatively, one could use a gas equation above that takes into account non-ideal behavior. However, this is already very complicated in the case of the Van der Waals equation for real gases, and the approach presented is therefore chosen. So you can write down the chemical potential of a component as a function of the activity : ${\ displaystyle a_ {i}}$

${\ displaystyle \ mu _ {i} = {\ mu _ {i}} ^ {o} + RT \ ln (a_ {i})}$
The standard chemical potential μ o is defined under ideal conditions for an activity a = 1. The chemical potential μ can assume positive, but also negative values.

At the beginning the knowledge was already gained that the chemical equilibrium is defined by the chemical potentials and the coefficients in the equation of the equilibrium reaction. If you insert the expression obtained, which takes into account the activity of a component, you get after forming:

${\ displaystyle - \ Delta {G_ {r}} ^ {o} = RT \ ln \ left ({\ frac {({a_ {C}}) ^ {\ nu _ {C}} ({a_ {D} }) ^ {\ nu _ {D}}} {({a_ {A}}) ^ {\ nu _ {A}} ({a_ {B}}) ^ {\ nu _ {B}}}} \ right)}$

The term on the left is the free enthalpy of reaction under standard conditions and includes all terms that describe the chemical potentials of the components under standard conditions. The free enthalpy of reaction under standard conditions is constant. This means that the quotient in the logarithmic pattern must also be constant. It is the equilibrium constant in the chemical law of mass action : ${\ displaystyle {\ mu _ {i}} ^ {o}}$

${\ displaystyle K = {\ frac {({a_ {C}}) ^ {\ nu _ {C}} ({a_ {D}}) ^ {\ nu _ {D}}} {({a_ {A. }}) ^ {\ nu _ {A}} ({a_ {B}}) ^ {\ nu _ {B}}}}}$

The chemical equilibrium is given by the chemical potentials of the components of the equilibrium reaction. The law of chemical mass action can be derived from this.

In addition to the law of mass action, the laws of diffusion for dissolved particles and the diffusion coefficient can also be derived from the concentration dependence of the chemical potential . A detailed derivation can be found in the article Diffusion .

Chemical potential in statistical thermodynamics

So far, the chemical potential has been discussed in connection with chemical reactions. The chemical potential also occurs in distribution functions that describe the energy distribution of particles. These are the Fermi-Dirac statistics and the Bose-Einstein statistics . The Fermi-Dirac statistics describe the distribution of electrons over a density of states in a solid. In terms of formula, the Fermi-Dirac statistics are defined as follows:

${\ displaystyle f (E) = {\ frac {1} {\ exp {\ left ({\ frac {E- \ mu} {k _ {\ mathrm {B}} T}} \ right)} + 1}} }$,

where the occupation probability of a state is dependent on the energy, while the Boltzmann constant is the absolute temperature and the chemical potential of the electron (in the following text only the chemical potential is abbreviated). In this case the chemical potential is the partial derivative of the free energy according to the number of particles . ${\ displaystyle f (E)}$${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle T}$${\ displaystyle \ mu}$${\ displaystyle N}$

${\ displaystyle \ mu = {\ frac {\ partial F} {\ partial N}}}$

The chemical potential characterizes the turning point of the Fermi distribution. The Fermi energy is the limit value of the chemical potential for 0  Kelvin . The term Fermi energy is often used instead of the chemical potential for higher temperatures, as the temperature dependence is not very pronounced at room temperature. A physical interpretation of the chemical potential is analogous to that given above. It characterizes the energy that is necessary to remove an electron from the solid. This example also shows that in this case the chemical potential does not have the unit joule / mol , but simply joule. Both are permissible, since the unit mol only represents a defined number of particles.

The chemical potential also appears in the Bose-Einstein statistics. It is used to describe the population probability of bosons, which, unlike electrons, have no particle limitation per energy state.

${\ displaystyle f (E) = {\ frac {1} {\ exp {\ left ({\ frac {E- \ mu} {k _ {\ mathrm {B}} T}} \ right)} - ​​1}} }$

These particles can also include particles such as phonons or photons for which particle conservation is not necessary, such as electrons in a solid. One consequence of this is that the chemical potential for these particles is zero.

Electrochemical potential

Main article Electrochemical Potential

Initially, the chemical potential was defined with the help of the internal energy, whereby no electrical work was taken into account. If you do this you get the following expression for the change in internal energy:

${\ displaystyle \ mathrm {d} U = T \, \ mathrm {d} Sp \, \ mathrm {d} V + \ Sigma \ mu _ {i} \, \ mathrm {d} n_ {i} + \ Sigma z_ {i} F \ phi \ mathrm {d} n_ {i}}$

The last term corresponds to the electrical work that is necessary to bring an infinitesimal amount of substance of a charged particle to a place where the electrical potential prevails. is the Faraday constant and the oxidation number of the particle . This abstract definition can be described as follows. The electrical work is defined by the equation , where the charge and the electrical potential difference represent. Depending on the direction of the electric field and the charge (positive or negative) of a particle, energy has to be used to move a charged particle against an electric potential difference or it is released when the particle is accelerated by the potential difference. This electrical work term has been inserted into the above equation. The charge can be written through for the infinitesimal amount of substance of a particle . Instead of a potential difference, the electrical potential occurs , with the electrical potential as a reference at an infinite distance and being arbitrarily set to zero. The interpretation of the electrical work term for a particle can now be given as follows. It corresponds to the electrical work that is necessary to bring an infinitesimal amount of substance of an electrically charged particle from infinite distance to a place where the electrical potential prevails. ${\ displaystyle i}$ ${\ displaystyle \ phi}$${\ displaystyle F}$${\ displaystyle z_ {i}}$${\ displaystyle i}$${\ displaystyle W _ {\ text {electrical}} = QU}$${\ displaystyle Q}$${\ displaystyle U}$${\ displaystyle z_ {i} F \ mathrm {d} n_ {i}}$${\ displaystyle \ phi}$${\ displaystyle i}$${\ displaystyle \ phi}$

With the additionally inserted term for the electrical work, the partial derivative of the internal energy according to the amount of substance of a component becomes:

${\ displaystyle \ left ({\ frac {\ partial U (S, V, n_ {j})} {\ partial {n_ {i}}}} \ right) _ {S, V, n_ {j \ neq i }} = \ mu _ {i} + z_ {i} F \ phi}$

This partial derivative is now defined as the electrochemical potential : ${\ displaystyle {\ overline {\ mu}} _ {i}}$

${\ displaystyle {\ overline {\ mu}} _ {i} = \ mu _ {i} + z_ {i} F \ phi}$

The electrochemical potential therefore takes the place of the chemical potential when charged particles are treated.

values

The values ​​of the chemical potential are tabulated for standard conditions ( ; ), s. u. Web links . ${\ displaystyle T = 298 {,} 15 \ \ mathrm {K}}$${\ displaystyle p = 101 {,} 325 \ \ mathrm {kPa}}$

If the chemical potential for a certain state (e.g. standard conditions) is known, it can be calculated in a linear approximation for pressures and temperatures in the vicinity of this state:

{\ displaystyle {\ begin {aligned} \ mu (T) & = \ mu (T_ {0}) + \ alpha \ cdot (T-T_ {0}) \ qquad {\ text {or}} \\\ mu (p) & = \ mu (p_ {0}) + \ beta \ cdot (p-p_ {0}) \ end {aligned}}}

With

• the temperature coefficient ${\ displaystyle \ alpha = \ left ({\ frac {\ partial \ mu} {\ partial T}} \ right) _ {p, n}}$
• the pressure coefficient ${\ displaystyle \ beta = \ left ({\ frac {\ partial \ mu} {\ partial p}} \ right) _ {T, n}}$

From the Maxwell relations it follows that

• the temperature coefficient is equal to the negative molar entropy:
${\ displaystyle \ left ({\ frac {\ partial \ mu} {\ partial T}} \ right) _ {p, n} = - \ left ({\ frac {\ partial S} {\ partial n}} \ right) _ {T, p}}$

and

${\ displaystyle \ left ({\ frac {\ partial \ mu} {\ partial p}} \ right) _ {T, n} = \ left ({\ frac {\ partial V} {\ partial n}} \ right ) _ {T, p}}$

literature

• J. Willard Gibbs: The Scientific Papers of J. Willard Gibbs: Vol. I Thermodynamics. Dover Publications, New York 1961.
• G. Job, F. Herrmann: Chemical Potential - a quantity in search of recognition. In: Eur. J. Phys. 27, 2006, pp. 353-371 ( doi : 10.1088 / 0143-0807 / 27/2/018 , PDF ).
• Ulrich Nickel: Textbook of Thermodynamics. A clear introduction. 3rd, revised edition. PhysChem, Erlangen 2019, ISBN 978-3-937744-07-0 .