chemical potential formula derivation

The differential of the Gibbs free energy is: where is volume, is pressure, is entropy and is temperature. A potential in physics is defined as the energy stored per unit of matter (i.e., a potential describes the corresponding potential energy intensity). Published online by Cambridge University Press: where x i (= ${{N_i } \over N}$, where N = N 1 + N 2 ++ N n) are mole fractions. Using $$ \mathrm{d}G = V\mathrm{d}p-S\mathrm{d}T, $$ we can write $$ \mu = \mu^{0} + RT\ln P. $$ But for mixtures, $$ \mathrm{d}G = V\mathrm{d}p - S\mathrm{d}t + \mu_{\ce{A}} \mathrm{d}n_{\ce{A}} + \mu_{\ce{B}} \mathrm{d}n_{\ce{B}} $$ Can the original formula for . PV = nRT where n is the amount of gas in moles, and R is the gas constant. . In direct analogy to electrical potential, gravitational potential, thermal potential, and mechanical potential, the chemical potential of a chemical substance, , can be simply defined as the chemical energy (U c) possessed by 1 mol of the substance. We derive a microscopic expression for a quantity that plays the role of chemical potential of active Brownian particles (ABPs) in a steady state in the absence of vortices. 7 0 obj << The chemical potential is a measure of the magnitude of this tendency. I am now to determine the energy at the height h using the energy i.e. The derivation of the mirror formula or spherical mirror formula is one of the most common formulas in optics. As you have access to this content, full HTML content is provided on this page. By either argument, the chemical potential for a gas of photon inside a box at fixed In each case the chemical potential of a given species at equilibrium is the same in all phases of the system. [A-] is the concentration of the weak/conjugate base. In much the same fashion as the partial molar volume is defined, the partial molar Gibbs function is defined for compound \(i\) in a mixture: \[ \mu_i = \left( \dfrac{\partial G}{\partial n_i} \right) _{p,T,n_j\neq i} \label{eq1}\], This particular partial molar function is of particular importance, and is called the chemical potential. << /S /GoTo /D [6 0 R /Fit ] >> 8-2 View the article. Nernst Equation for Single Electrode Potential E cell = E 0 - [RT/nF] ln Q Where, E cell = cell potential of the cell E 0 = cell potential under standard conditions R = universal gas constant T = temperature n = number of electrons transferred in the redox reaction F = Faraday constant Q = reaction quotient Fig. This can be misleading, because chemical potential is not a form of energycalling a potential as some sort of energy adds to the confusion and difficulty in understanding the concept of chemical potential. If a system contains more than one species of particle, there is a separate chemical potential associated with each species, defined as the change in energy when the number of particles of that species . If the substance is highly compressible (such as a gas) the pressure dependence of the molar volume is needed to complete the integral. for this article. As is well known for a constant composition system, $\mu$ (symbolizing chemical potential) is equal to the molar Gibbs energy. and with this I am then to derive the above equation for the chemical potential. Within the internal circuit, chemical energy is converted to electric potential energy (i.e., the battery). The chemical potential meets the first two criteria, albeit the second one only barely. Using this expression, it is easy to show that, \[\left( \dfrac{\partial \mu}{\partial p} \right) _{T} = V\], \[ \int_{\mu^o}^{\mu} d\mu = \int_{p^o}^{p} V\,dp \label{eq5}\], So that for a substance for which the molar volume is fairly independent of pressure at constant temperature (i. e., \(\kappa_T\) is very small), therefore Equation \ref{eq5} becomes, \[ \int_{\mu^o}^{\mu} d\mu = V \int_{p^o}^{p} dp\], Where \(p^o\) is a reference pressure (generally the standard pressure of 1 atm) and \(\mu^o\) is the chemical potential at the standard pressure. The chemical potential of a particular component is the Gibbs free energy per mole of that component in the homogeneous solution. The greater , the more active or "driven" the. The Chemical Potential Authors: Stephen Whitaker University of California, Davis Abstract The traditional development of a representation for the chemical potential of species A in an ideal gas. 5`8,-XGB4Q}BsYK&j' d\ewyOJin;={ (0 u{`@@_iF;sOp $~0!F;d2@hNy FT {64g:A^XF#B#"bmhpq3, l-V"DXn;T^Fcz4D*X?OQ~]cp8o1I=Qaxsx-a]XaSC (~J58sWD#6d.!L GyRc,N E*H"a1"/8taif-"E xK8$K]>k0(lC7},v 6&t{zF{4P2xemD`(oJ*7 As the partial most Gibbs function, it is easy to show that, where \(V\) is the molar volume, and \(S\) is the molar entropy. /Parent 17 0 R /Resources 7 0 R 2.1 Example: Barometric pressure formula Elastic Potential Energy Formula F = K x PE = 0.5 k Derivation of the Formula Total loading time: 0.656 Therefore, electrical potential represents the electrical energy intensity. For light of frequency 10 15 Hz the reverse potential is 2 V. Find Planck's constant, work function and threshold frequency. (Section 9.2.6 will introduce a more general definition of chemical potential that applies also to a constituent of a mixture.) We can rewrite Equation 5 in a different form as, Equation 6 is another form of the fundamental equation showing that the Gibbs free energy, G, is the chemical energy N (Equation 4). The effect of electrostatic potential will depend on the number of charges, z, carried by the component, giving: m = m' + zFy The sum i dn i, which enters into the expression for the total differential of all thermodynamic potentials, has been called the fundamental Gibbs equation, e.g.,: where S is the entropy and V the volume. The chemical potential, , of a pure substance has as one of its definitions (Sec. /Contents 8 0 R This energy will have the potential to do work on releasing. How would one derive an equation for chemical potential? Chemical potentials are important in many aspects of multi-phase equilibrium chemistry, including melting, boiling, evaporation, solubility, osmosis, partition coefficient, liquid-liquid extraction and chromatography. In this case e e (equilibrium) In other words, the chemical potential for photons is zero. Imposing a difference in temperature between two locations or a temperature gradient leads to entropy or heat transfer from high-temperature to low-temperature regions. If the vapour pressure at temperature T1 is P1 and the vapour pressure at temperature T2 is P2, the corresponding linear equations are: l n ( P 1) = H v a p R T 1 + l n A And l n ( P 2) = H v a p R T 2 + l n A This means that the chemical potential is the (reversible) rate of change of internal energy with mole number while keeping other variables ( S, V) constant, thus since d U = T d S p d V + i i d n i, where U is the internal energy, then i = ( U n i) T, S, n j. Another reason why chemical potential is underappreciated is the surprising lack of a unique unit associated with such a quantity of central importance in the thermodynamics of materials. Derivation of the Formula PE or U = is the potential energy of the object m = refers to the mass of the object in kilogram (kg) g = is the gravitational force h = height of the object in meter (m) Besides, the unit of measure for potential energy is Joule (J). /Font << /F19 11 0 R /F20 12 0 R /F42 14 0 R /F44 15 0 R /F67 16 0 R >> It can be a group of atoms, molecules, electrons, electron holes, atomic vacancies, phonons, or photons. The chemical potential tells how the Gibbs function will change as the composition of the mixture changes. If one has the Boltzmann equation for entropy $$ S=k \ln(W) $$ where $$ W=T^{C/k}V^{N} $$ is the number of microstates, and it is assumed that all the particles are indistinguishable. [1] P. Atkins and J. de Paula, Atkins' Physical Chemistry, 8th ed., New York: Oxford University Press, 2006. Josiah Willard Gibbs formally introduced the concept of chemical potential approximately 140 years ago in his foundational article.Reference Gibbs1 Gibbs not only established the mathematical beauty of thermodynamics by formulating the fundamental equation of thermodynamics of a system but also introduced the concept of chemical potential, which he originally called the intrinsic potential. To further understand chemical potential () and establish the link between chemical potential and Gibbs free energy (G), we consider the total internal energy, U, of a simple system by adding up the thermal, mechanical, and chemical energy from Equations 2 to 4: Gibbs defined a simple system as a system without considering the surface, electric, magnetic, and non-hydrostatic mechanical energy contributions. (2) Liquid mixtures. [L1T-2]. The mirror formula can be termed as the formula in which the relationship between the distance of object represented as 'u' and the distance of the image represented as 'v', and the focal length of the mirror given as 'f'. Combining the Kubo formula with the finite-temperature time-dependent density matrix renormalization group in the grand canonical ensemble, we developed a nearly exact algorithm to calculate the thermoelectric power factor in organic materials. 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A}{\partial n_i} \right) _{V,T,n_j\neq i}\), \(\mu_i = \left( \dfrac{\partial G}{\partial n_i} \right) _{p,T,n_j\neq i}\). Chemical potential is important in many areas . Equation 7 suggests that there is no need to introduce the term molar Gibbs free energy. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. For the same reason, we should have been working mostly with chemical potentials rather than chemical energy in applying thermodynamics to materials equilibrium and processes. Therefore, the chemical potential of a homogeneous n-component system can be written in terms of chemical potentials for the n individual components, 1, 2, n. Thermodynamic Identities. . We can rewrite the integral form of the Gibbs free energy here for an n-component system. Similarly, temperature, T, which can be considered the thermal potential or thermal energy intensity, is the thermal potential energy, U T, possessed by one unit of thermal matter, or one unit of entropy S, Pressure p can be considered as the mechanical potential. The electrochemical potential is a measure of the difference between the average energy of the outer most electrons of the . It is a central concept in thermodynamics of materials because all of the thermodynamic properties of a material at a given temperature and pressure can be obtained from knowledge of its chemical potential. Copyright Materials Research Society 2019. Thermodynamics and Equations of State), The Influence of Temperature and Pressure on the Chemical Potential and Drive and Therefore the Behaviour of Substances, Chemical Potential and Gibbs Distribution, On the Gibbs Energy and Chemical Potentials of an Ideal Gas Mixture. If the substance is highly compressible (such as a gas) the pressure dependence of the molar volume is needed to complete the integral. We welcome comments and feedback on Material Matters articles via email toBulletin@mrs.org. The Gibbs-Helmholtz equation gives the temperature dependence of the Gibbs energy divided by temperature as the negative enthalpy divided by temperature squared. *TpFKccQeI>vfcD9m2(ZO.eMY Figure 1. Discusses the theoretical basis of chemical potential (by Keith Putirka) 1 - At equilibrium, the chemical potential of the gaseous form of a substance A is equal to the chemical potential of its condensed phase. Schematic illustration of the chemical potential of a binary solution AxABxB and the chemical potentials A and B of the individual species A and B as a function of composition. This disequilibrium leads to chemical and electrochemical reaction, which in turn reduces disequilibrium. The chemical potential of a particular component is the Gibbs free energy per mole of that component in the homogeneous solution. We start with the definitions of different forms of potentials in contrast to their corresponding energies. Potential energy can be defined as the energy that a body stores within itself by virtue of its position. /ProcSet [ /PDF /Text ] Potential Equation. There is a general misconception that Equation 7 holds true only for pure substances or single-component systems. of which is mgzper molecule. /D [6 0 R /XYZ 72 501.865 null] xV4_7Rqcv@. If one would like to emphasize the amount of Gibbs free energy stored in 1 mol of matter, its unit should be J rather than J/mol. U = - GMm/r. 5 0 obj Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 5.5) def = Gm = G n That is, is equal to the molar Gibbs energy of the substance at a given temperature and pressure. = U + PV - TS (note, in this equation, ,U,S,V, as well as T and P, are intensive quantities). in electrochemistry, the electrochemical potential of electrons (or any other species) is the total potential, including both the (internal, nonelectrical) chemical potential and the electric potential, and is by definition constant across a device in equilibrium, whereas the chemical potential of electrons is equal to the electrochemical Rev. Conceptually, it is fundamentally different from electrical energy, which is proportional to the system size and is an extensive quantity. T Appendix A: Derivation of the chemical CpT ln( ) + RT ln(p/p0) (A7) potential equation T0 When the temperature is T=T0, the above expression is The expression that is commonly used in planetary at- reduced to the more familiar equation: mospheres is usually written as (Kodepudi and Prigogine [1998], Eq. Under the most common thermodynamic condition of constant temperature and pressure, chemical potential determines the stability of substances, such as chemical species, compounds, and solutions, and their tendency to chemically react to form new substances, to transform to new physical states, or to migrate from one spatial location to another. 2). Render date: 2022-12-11T21:44:47.923Z Chemical potential has been underappreciated and underutilized in applications of thermodynamics to materials science and engineering. The electric potential at a place in an electric field is the amount of effort required to transport a unit positive charge from infinity to that point, whereas electric potential energy is the amount of energy required to move a charge against the electric field. Therefore, Work done = force x displacement Now, the force here is the weight of the hammer while the displacement is the lifted height of the hammer. The establishment of the fundamental equation and introduction of chemical potential marked the birth of chemical thermodynamics and made it possible to apply thermodynamics to materials science and engineering. In summary, to facilitate the understanding of and promote applications of chemical potentials, the author makes the following recommendations: Introduce a simple definition for chemical potential: chemical potential of a substance is the chemical energy per mole of the substance. In this definition, Gibbs free energy is chemical energy, and the substance can be a pure substance or a multicomponent system. The last condition, however, is not true for the chemical potential. 22 0 obj << Potential Energy Formula or Equation & Derivation Potential Energy Formula or Equation The equation for potential energy is given as: P. E= mgh M= mass of the body g= acceleration (9.8 m/s 2 at earth's surface) h= height of body Potential Energy Derivation Potential energy is the work done on a body to take it to a specific height. /Filter /FlateDecode /D [6 0 R /XYZ 72 720 null] Chemical Potential and Gibbs Distribution 1, Free Energy. This expression can be used to calculate escape velocity, orbital energy and others. 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At equilibrium free energy is at its minimum for the system, that is, d G =0. We show that consists of (i) an intrinsic chemical potential similar to passive systems, which depends on density and self-propulsion speed, but not on the external potential, (ii) the external potential, and (iii) a . /Filter /FlateDecode Schematic illustration of the chemical potential of a binary solution A xAB xB and the chemical potentials A and B of the individual species A and B as a function of composition. In order to nd the chemical potential of this volume of gas, we need to modify 4 to write Uin terms of the potential and kinetic energy. endstream The familiar Fermi level in semiconductor device physics or solid-state physics is the chemical potential of electrons.Reference Kittel3. The last condition, however, is not true for the chemical potential. Adding to the confusion is the occasional use of Gibbs potential in place of Gibbs energy or Gibbs free energy, even when it refers to the Gibbs free energy of an entire system rather than on a per mole basis. Then G 2 = G o + nRT ln P 2 --------------------------- (4) where G o is the standard state free energy of the gas. 5.3.6): (p, T0) = (p0,T0) + RT0 . The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Thermodynamics, Flame Temperature and Equilibrium, Calculation of Chemical Potential and Activity Coefficient of Two Layers of Co2 Adsorbed on a Graphite Surface, Derivation of the Chemical Potential Equation, Linking up Pressure, Chemical Potential and Thermal Gradients Franois Montel, Hai Hoang, Guillaume Galliero, Lecture 10: 10.14.05 Chemical Potentials and the Gibbs Free Energy, The Elusive Chemical Potential Ralph Baierlein, Entropy. This derivation relied on Fig. Given: Initial frequency = 1 = 2 x 10 15 Hz, initial stopping potential = V s1 = 6 V, Final frequency = 2 = 10 15 Hz, Final stopping potential = V s2 = 2 V, speed of light = c = 3 x 10 8 m/s, Charge on electron . The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. Let's take the hammer example to derive the formula of potential energy. endobj Feature Flags: { History and meaning of the term, concentration coefficient of the chemical potential, mass action formula, (c)-curves, stan-dard values of potentials, quantity pH, law of mass action, temperature and pressure dependence of the equilibrium constants. >> endobj stream 2: The chemical potential of a substance is the slope of the total Gibbs energy of a mixture with respect to the amount of substance of interest. A PDF of this content is also available in through the Save PDF action button. /Type /Page After reaching the high potential terminal, a positive test charge will go via the external circuit and perform work on the light bulb, motor, or heater coils, converting the circuit's electric potential energy into useable forms. That is, the chemical potential increases as the internal energy, U, of the phase increases, and as the entropy, S, of the phase decreases at a given temperature, T. Further, the increases as the volume, V increases for a given pressure, P. 12 July 2019. Each chemical species, be it an atom, ion or molecule, has its own chemical potential. The chemical potential meets the first two criteria, albeit the second one only barely. The definition of chemical potential based on Equation 4 should be significantly easier to comprehend for most people, particularly for beginners in thermodynamics, than using derivatives or rate of increase in an energy function with respect to the addition of a substance, as is often the case. Equation 5 is precisely the integral form of the fundamental equation of thermodynamics formulated by Gibbs by combining the first and second laws of thermodynamics,Reference Gibbs1 which connect the seven basic thermodynamic variables, U, T, S, p, V, , and N, of a simple system. Has data issue: true The chemical potential of the vapour can be written as g = l ( T) + T log ( P/P0 ), where P0 ( T) is the saturated vapour pressure; here we have used the fact that the saturated vapour is, by definition, in equilibrium with the liquid, i.e. In its simplest mathematical form, the electrical potential is defined as. Equation 7 is true regardless of whether the substance is a single component or multicomponent. [HA] is the concentration of the weak/conjugate acid. stream (The subscript n j means that other mole number are held constant.) Electrochemical Potential. A possible reason for the discrepancy is that an external potential field exists between the two ends of the molecular chain in the system of Satija and Makarov. 9 0 obj << 13 0 obj << Derivation of several thermodynamic quantities, such as specific heat capac- ities, virial coefficients, thermodynamic potential etc., pi gi e(1) i (27) and their relation to the partition function Zm is the next logical step to understand the relation between thermo- Equation 27 is valid only if is not a function of energy. This is so because if you define it relative to the particle concentration and not the number (which would be equally valid), you end up with an energy density and not an energy. Under the most common thermodynamic condition of constant temperature and pressure, chemical potential determines the stability of substances, such as chemical species, compounds, and solutions, and their tendency to chemically react to form new substances, to transform to new physical states, or to migrate from one spatial location to another. 8 0 obj << endobj Therefore, the chemical potential of a substance introduced by Gibbs is simply the Gibbs free energy or chemical energy per mole of that substance (i.e., the molar Gibbs free energy is precisely the chemical potential). we must have g = l when P = P0. The formula for gravitational potential energy is derived from the attempt to act on gravity by bringing the mass from infinity, where the potential energy is assigned the value zero. The linear equation can be formulated in a two-point format to accomplish this. When the hammer is lifted, the change in potential energy of the hammer is equal to the work done in lifting the hammer. The formula of the potential energy is given by P.E. We can draw another analogy for chemical potential to thermal and electrical potentials. The relationship among , A, and B can be illustrated using the well-known . Hostname: page-component-5959bf8d4d-bmjgf /MediaBox [0 0 612 792] Use of this equality provides the means to establish the equilibrium constant for a chemical reaction. If the substance is an ideal gas xuUKo0WVG"_y +q ne%vZ3x_\ofg,{w~yL0ZUVLI+UeIV}4) t~ .>M}V(E[&Bwr{;ofbfk;= 9(cx{0Dy+X View all Google Scholar citations derivation of the law of diffusion, migration of ions, conductivity. Another familiar example for potential is the gravitational potential or gravitational energy intensity, which is the gravitational potential energy per unit mass. R. Satija and D. E. Makarov, " Generalized Langevin equation as a model for barrier crossing dynamics in biomolecular folding," J. Phys. The formula for the Henderson-Hasselbalch equation is: p H = p K a + l o g ( [ A ] [ H A]) Where, pKa is the negative logarithm of the acid dissociation constant. >> where N is the number of moles of the substance (i.e., the chemical potential of a chemical substance represents its chemical energy intensity in a given homogeneous system or at a given location of an inhomogeneous system). 24 24. For example, a substance can be a binary solution A xAB xB of composition x A and x B, with x A + x B = 1, = Ax A + Bx B. This equation will allow us in. We write A or (A) to signify the potential of substance A. sp= fPx(, ) xxps=1 The total derivative of the chemical potential is then d P dP x s dx s Tx s p TP p p = + , , s Tx s p TP P p dP x dx p = Or, by substituting the definition for the chemical potential, and evaluating the pressure and temperature derivatives as was done in Chapter 6: But as it turns out, the chemical potential can be defined as the partial molar derivative any of the four major thermodynamic functions \(U\), \(H\), \(A\), or \(G\): The last definition, in which the chemical potential is defined as the partial molar Gibbs function is the most commonly used, and perhaps the most useful (Equation \ref{eq1}). Many references exist in the density functional theory (DFT) literature to the chemical potential of the electrons in an atom or a molecule. Chemical Potential. The formula of potential energy is: P.E.= m.g.h Since m is mass, it can be written as [M1] g is the acceleration and has a unit of m/s2, so it can be written as [L1T-2] and h being height has a fundamental formula of [L1] Substituting these quantities into the formula gives: P.E.= [M1]. hasContentIssue true, Molar Gibbs free energy and chemical potential. The chemical energy U c is the same as the Gibbs free energy G, and Equation 4 holds for each chemical component of a homogeneous multicomponent solution as well as for the entire homogeneous solution. 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chemical potential formula derivation