Latent heat is thermal energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — a first-order phase transition. Latent heat can be understood as heat energy in hidden form, supplied or extracted to change the state of a substance without changing its temperature. Examples are latent heat of fusion and latent heat of vaporization involved in phase changes, i.e. a substance condensing or vaporizing at a specified temperature and pressure. The term was introduced around 1762 by British chemist Joseph Black, it is derived from the Latin latere. Black used the term in the context of calorimetry where a heat transfer caused a volume change in a body while its temperature was constant. In contrast to latent heat, sensible heat is a heat transfer that results in a temperature change in a body; the terms ″sensible heat″ and ″latent heat″ refer to types of heat transfer between a body and its surroundings. ″ Sensible heat ″ felt in a process as a change in the body's temperature.
″Latent heat″ is heat transferred in a process without change of the body's temperature, for example, in a phase change. Both sensible and latent heats are observed in many processes of transfer of energy in nature. Latent heat is associated with the change of phase of atmospheric or ocean water, condensation, freezing or melting, whereas sensible heat is energy transferred, evident in change of the temperature of the atmosphere or ocean, or ice, without those phase changes, though it is associated with changes of pressure and volume; the original usage of the term, as introduced by Black, was applied to systems that were intentionally held at constant temperature. Such usage referred to latent heat of expansion and several other related latent heats; these latent heats. When a body is heated at constant temperature by thermal radiation in a microwave field for example, it may expand by an amount described by its latent heat with respect to volume or latent heat of expansion, or increase its pressure by an amount described by its latent heat with respect to pressure.
Latent heat is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process. Two common forms of latent heat are latent heat of latent heat of vaporization; these names describe the direction of energy flow when changing from one phase to the next: from solid to liquid, liquid to gas. In both cases the change is endothermic, meaning. For example, when water evaporates, energy is required for the water molecules to overcome the forces of attraction between them, the transition from water to vapor requires an input of energy. If the vapor condenses to a liquid on a surface the vapor's latent energy absorbed during evaporation is released as the liquid's sensible heat onto the surface; the large value of the enthalpy of condensation of water vapor is the reason that steam is a far more effective heating medium than boiling water, is more hazardous. In meteorology, latent heat flux is the flux of heat from the Earth's surface to the atmosphere, associated with evaporation or transpiration of water at the surface and subsequent condensation of water vapor in the troposphere.
It is an important component of Earth's surface energy budget. Latent heat flux has been measured with the Bowen ratio technique, or more since the mid-1900s by the Jonathan Beaver method; the English word latent comes from Latin latēns. The term latent heat was introduced into calorimetry around 1750 when Joseph Black, commissioned by producers of Scotch whisky in search of ideal quantities of fuel and water for their distilling process, to studying system changes, such as of volume and pressure, when the thermodynamic system was held at constant temperature in a thermal bath. James Prescott Joule characterised latent energy as the energy of interaction in a given configuration of particles, i.e. a form of potential energy, the sensible heat as an energy, indicated by the thermometer, relating the latter to thermal energy. A specific latent heat expresses the amount of energy in the form of heat required to effect a phase change of a unit of mass 1kg, of a substance as an intensive property: L = Q m.
Intensive properties are material characteristics and are not dependent on the size or extent of the sample. Quoted and tabulated in the literature are the specific latent heat of fusion and the specific latent heat of vaporization for many substances. From this definition, the latent heat for a given mass of a substance is calculated by Q = m L where: Q is the amount of energy released or absorbed during the change of phase of the substance, m is the mass of the substance, L is the specific latent heat for a particular substance, either Lf for fusion, or Lv for vaporization; the following table shows the specific latent heats and change of phase temperatures of some common fluids and gases. The specific latent heat of condensation of water in the temperature range from −25 °C to 40 °C is approximated by the following empirical cubic function: L water =
Solubility is the property of a solid, liquid or gaseous chemical substance called solute to dissolve in a solid, liquid or gaseous solvent. The solubility of a substance fundamentally depends on the physical and chemical properties of the solute and solvent as well as on temperature and presence of other chemicals of the solution; the extent of the solubility of a substance in a specific solvent is measured as the saturation concentration, where adding more solute does not increase the concentration of the solution and begins to precipitate the excess amount of solute. Insolubility is the inability to dissolve in a liquid or gaseous solvent. Most the solvent is a liquid, which can be a pure substance or a mixture. One may speak of solid solution, but of solution in a gas. Under certain conditions, the equilibrium solubility can be exceeded to give a so-called supersaturated solution, metastable. Metastability of crystals can lead to apparent differences in the amount of a chemical that dissolves depending on its crystalline form or particle size.
A supersaturated solution crystallises when'seed' crystals are introduced and rapid equilibration occurs. Phenylsalicylate is one such simple observable substance when melted and cooled below its fusion point. Solubility is not to be confused with the ability to'dissolve' a substance, because the solution might occur because of a chemical reaction. For example, zinc'dissolves' in hydrochloric acid as a result of a chemical reaction releasing hydrogen gas in a displacement reaction; the zinc ions are soluble in the acid. The solubility of a substance is an different property from the rate of solution, how fast it dissolves; the smaller a particle is, the faster it dissolves although there are many factors to add to this generalization. Crucially solubility applies to all areas of chemistry, inorganic, physical and biochemistry. In all cases it will depend on the physical conditions and the enthalpy and entropy directly relating to the solvents and solutes concerned. By far the most common solvent in chemistry is water, a solvent for most ionic compounds as well as a wide range of organic substances.
This is a crucial factor in much environmental and geochemical work. According to the IUPAC definition, solubility is the analytical composition of a saturated solution expressed as a proportion of a designated solute in a designated solvent. Solubility may be stated in various units of concentration such as molarity, mole fraction, mole ratio, mass per volume and other units; the extent of solubility ranges from infinitely soluble such as ethanol in water, to poorly soluble, such as silver chloride in water. The term insoluble is applied to poorly or poorly soluble compounds. A number of other descriptive terms are used to qualify the extent of solubility for a given application. For example, U. S. Pharmacopoeia gives the following terms: The thresholds to describe something as insoluble, or similar terms, may depend on the application. For example, one source states that substances are described as "insoluble" when their solubility is less than 0.1 g per 100 mL of solvent. Solubility occurs under dynamic equilibrium, which means that solubility results from the simultaneous and opposing processes of dissolution and phase joining.
The solubility equilibrium occurs. The term solubility is used in some fields where the solute is altered by solvolysis. For example, many metals and their oxides are said to be "soluble in hydrochloric acid", although in fact the aqueous acid irreversibly degrades the solid to give soluble products, it is true that most ionic solids are dissolved by polar solvents, but such processes are reversible. In those cases where the solute is not recovered upon evaporation of the solvent, the process is referred to as solvolysis; the thermodynamic concept of solubility does not apply straightforwardly to solvolysis. When a solute dissolves, it may form several species in the solution. For example, an aqueous suspension of ferrous hydroxide, Fe2, will contain the series + as well as other species. Furthermore, the solubility of ferrous hydroxide and the composition of its soluble components depend on pH. In general, solubility in the solvent phase can be given only for a specific solute, thermodynamically stable, the value of the solubility will include all the species in the solution.
Solubility is defined for specific phases. For example, the solubility of aragonite and calcite in water are expected to differ though they are both polymorphs of calcium carbonate and have the same chemical formula; the solubility of one substance in another is determined by the balance of intermolecular forces between the solvent and solute, the entropy change that accompanies the solvation. Factors such as temperature and pressure will alter this balance. Solubility may strongly depend on the presence of other species dissolved in the solvent, for example, complex-forming anions in liquids. Solubility will depend on the excess or deficiency of a common ion in the solution, a phenomenon known as the common-ion effect. To a lesser extent, solubility will depend on the ionic strength of solutions; the last two effects can be quantified using the equation for solubility equilibrium. For a solid that dissolves in a redox reaction, solubility is expe
Myristic acid is a common saturated fatty acid with the molecular formula CH312COOH. Its salts and esters are referred to as myristates, it is named after the binomial name for nutmeg, from which it was first isolated in 1841 by Lyon Playfair. Nutmeg butter has the triglyceride of myristic acid. Besides nutmeg, myristic acid is found in palm kernel oil, coconut oil, butterfat, 8–14% of bovine milk, 8.6% of breast milk as well as being a minor component of many other animal fats. It is found in spermaceti, the crystallized fraction of oil from the sperm whale, it is found in the rhizomes of the Iris, including Orris root. It comprises 14.49% of the fats from the fruit of the Durian species Durio graveolens. Myristic acid is added co-translationally to the penultimate, nitrogen-terminus, glycine in receptor-associated kinases to confer the membrane localization of the enzyme; the myristic acid has a sufficiently high hydrophobicity to become incorporated into the fatty acyl core of the phospholipid bilayer of the plasma membrane of the eukaryotic cell.
In this way, myristic acid acts. Various "human epidemiological studies have shown that myristic acid and lauric acid were the saturated fatty acids most related to the average serum cholesterol concentrations in humans", meaning they were positively correlated with higher cholesterol levels as well as raising triglycerides in plasma by some 20% increasing the risk for cardiovascular disease, although some research points to myristic acid's positive effects on HDL cholesterol and hence improving HDL to total cholesterol ratio. Reduction of myristic acid yields myristyl myristyl alcohol. 1-Tetradecanol – the corresponding alcohol Myristyl aldehyde – the corresponding aldehyde
Thermodynamic databases for pure substances
Thermodynamic databases contain information about thermodynamic properties for substances, the most important being enthalpy and Gibbs free energy. Numerical values of these thermodynamic properties are collected as tables or are calculated from thermodynamic datafiles. Data is expressed as temperature-dependent values for one mole of substance at the standard pressure of 101.325 kPa, or 100 kPa. Both of these definitions for the standard condition for pressure are in use. Thermodynamic data is presented as a table or chart of function values for one mole of a substance. A thermodynamic datafile is a set of equation parameters from which the numerical data values can be calculated. Tables and datafiles are presented at a standard pressure of 1 bar or 1 atm, but in the case of steam and other industrially important gases, pressure may be included as a variable. Function values depend on the state of aggregation of the substance, which must be defined for the value to have any meaning; the state of aggregation for thermodynamic purposes is the standard state, sometimes called the reference state, defined by specifying certain conditions.
The normal standard state is defined as the most stable physical form of the substance at the specified temperature and a pressure of 1 bar or 1 atm. However, since any non-normal condition could be chosen as a standard state, it must be defined in the context of use. A physical standard state is one that exists for a time sufficient to allow measurements of its properties; the most common physical standard state is one, stable thermodynamically. It has no tendency to transform into any other physical state. If a substance can exist but is not thermodynamically stable, it is called a metastable state. A non-physical standard state is one whose properties are obtained by extrapolation from a physical state. Metastable liquids and solids are important because some substances can persist and be used in that state indefinitely. Thermodynamic functions that refer to conditions in the normal standard state are designated with a small superscript °; the relationship between certain physical and thermodynamic properties may be described by an equation of state.
It is difficult to measure the absolute amount of any thermodynamic quantity involving the internal energy, since the internal energy of a substance can take many forms, each of which has its own typical temperature at which it begins to become important in thermodynamic reactions. It is therefore the change in these functions, of most interest; the isobaric change in enthalpy H above the common reference temperature of 298.15 K is called the high temperature heat content, the sensible heat, or the relative high-temperature enthalpy, called henceforth the heat content. Different databases designate this term in different ways. All of these terms mean the molar heat content for a substance in its normal standard state above a reference temperature of 298.15 K. Data for gases is for the hypothetical ideal gas at the designated standard pressure; the SI unit for enthalpy is J/mol, is a positive number above the reference temperature. The heat content has been measured and tabulated for all known substances, is expressed as a polynomial function of temperature.
The heat content of an ideal gas is independent of pressure, but the heat content of real gases varies with pressure, hence the need to define the state for the gas and the pressure. Note that for some thermodynamic databases such as for steam, the reference temperature is 273.15 K. The heat capacity C is the ratio of heat added to the temperature increase. For an incremental isobaric addition of heat: Cp is therefore the slope of a plot of temperature vs. isobaric heat content. The SI units for heat capacity are J/; when heat is added to a condensed-phase substance, its temperature increases until a phase change temperature is reached. With further addition of heat, the temperature remains constant while the phase transition takes place; the amount of substance that transforms is a function of the amount of heat added. After the transition is complete, adding more heat increases the temperature. In other words, the enthalpy of a substance changes isothermally; the enthalpy change resulting from a phase transition is designated ΔH.
There are four types of enthalpy changes resulting from a phase transition. To wit: Enthalpy of transformation; this applies to the transformations from one solid phase to another, such as the transformation from α-Fe to γ -Fe. The transformation is designated ΔHtr. Enthalpy of fusion or melting; this is designated ΔHm. Enthalpy of vaporization; this is designated ΔHv. Enthalpy of sublimation; this is designated ΔHs. Cp is infinite at phase transition temperatures. At the Curie temperature, Cp shows a sharp discontinuity. Values of ΔH are given for the transition at the normal standard state temperature for the two states, if so, are designa
Enthalpy, a property of a thermodynamic system, is equal to the system's internal energy plus the product of its pressure and volume. In a system enclosed so as to prevent matter transfer, for processes at constant pressure, the heat absorbed or released equals the change in enthalpy; the unit of measurement for enthalpy in the International System of Units is the joule. Other historical conventional units still in use include the calorie. Enthalpy comprises a system's internal energy, the energy required to create the system, plus the amount of work required to make room for it by displacing its environment and establishing its volume and pressure. Enthalpy is defined as a state function that depends only on the prevailing equilibrium state identified by the system's internal energy and volume, it is an extensive quantity. Enthalpy is the preferred expression of system energy changes in many chemical and physical measurements at constant pressure, because it simplifies the description of energy transfer.
In a system enclosed so as to prevent matter transfer, at constant pressure, the enthalpy change equals the energy transferred from the environment through heat transfer or work other than expansion work. The total enthalpy, H, of a system cannot be measured directly; the same situation exists in classical mechanics: only a change or difference in energy carries physical meaning. Enthalpy itself is a thermodynamic potential, so in order to measure the enthalpy of a system, we must refer to a defined reference point; the ΔH is a positive change in endothermic reactions, negative in heat-releasing exothermic processes. For processes under constant pressure, ΔH is equal to the change in the internal energy of the system, plus the pressure-volume work p ΔV done by the system on its surroundings; this means that the change in enthalpy under such conditions is the heat absorbed or released by the system through a chemical reaction or by external heat transfer. Enthalpies for chemical substances at constant pressure refer to standard state: most 1 bar pressure.
Standard state does not speaking, specify a temperature, but expressions for enthalpy reference the standard heat of formation at 25 °C. Enthalpy of ideal gases and incompressible solids and liquids does not depend on pressure, unlike entropy and Gibbs energy. Real materials at common temperatures and pressures closely approximate this behavior, which simplifies enthalpy calculation and use in practical designs and analyses; the word enthalpy was coined late, in the early 20th century, in analogy with the 19th-century terms energy and entropy. Where energy uses the root of the Greek word ἔργον "work" to express the idea of "work-content" and where entropy uses the Greek word τροπή "transformation" to express the idea of "transformation-content", so by analogy, enthalpy uses the root of the Greek word θάλπος "warmth, heat" to express the idea of "heat-content"; the term does in fact stand in for the older term "heat content", a term, now deprecated as misleading, as dH refers to the amount of heat absorbed in a process at constant pressure only, but not in the general case.
Josiah Willard Gibbs used the term "a heat function for constant pressure" for clarity. Introduction of the concept of "heat content" H is associated with Benoît Paul Émile Clapeyron and Rudolf Clausius; the term enthalpy first appeared in print in 1909. It is attributed to Heike Kamerlingh Onnes, who most introduced it orally the year before, at the first meeting of the Institute of Refrigeration in Paris, it gained currency only in the 1920s, notably with the Mollier Steam Tables and Diagrams, published in 1927. Until the 1920s, the symbol H was used, somewhat inconsistently, for "heat" in general; the definition of H as limited to enthalpy or "heat content at constant pressure" was formally proposed by Alfred W. Porter in 1922; the enthalpy of a thermodynamic system is defined as H = U + p V, where H is enthalpy U is the internal energy of the system p is pressure V is the volume of the systemEnthalpy is an extensive property. This means, it is convenient to introduce the specific enthalpy h = H m, where m is the mass of the system, or the molar enthalpy H m = H n, where n is the number of moles.
For inhomogeneous systems the enthalpy is the sum of the enthalpies of the composing subsystems: H = ∑ k H k, where H is the total enthalpy of all the subsystems k refers to the various subsystems H k refers to the enthalpy of each subsystem ∑ k
Glycerol is a simple polyol compound. It is a colorless, viscous liquid, sweet-tasting and non-toxic; the glycerol backbone is found in many lipids which are known as glycerides. It is used in the food industry as a sweetener and humectant in pharmaceutical formulations. Glycerol has three hydroxyl groups that are responsible for its solubility in water and its hygroscopic nature. Although achiral, glycerol is prochiral with respect to reactions of one of the two primary alcohols. Thus, in substituted derivatives, the stereospecific numbering labels each carbon as either sn-1, sn-2, or sn-3. Glycerol is obtained from plant and animal sources where it occurs in triglycerides, esters of glycerol with long-chain carboxylic acids; the hydrolysis, saponification, or transesterification of these triglycerides produces glycerol as well as the fatty acid derivative: Triglycerides can be saponified with sodium hydroxide to give glycerol and fatty sodium salt or soap. Typical plant sources include soybeans or palm.
Animal-derived tallow is another source. 950,000 tons per year are produced in the United States and Europe. The EU directive 2003/30/EC set a requirement that 5.75% of petroleum fuels are to be replaced with biofuel sources across all member states by 2010. It was projected in 2006 that by the year 2020, production would be six times more than demand, creating an excess of glycerol. Glycerol from triglycerides is produced on a large scale, but the crude product is of variable quality, with a low selling price of as low as 2-5 U. S. cents per kilogram in 2011. It can be purified, but the process is expensive; some glycerol is burned for energy, but its heat value is low. Crude glycerol from the hydrolysis of triglycerides can be purified by treatment with activated carbon to remove organic impurities, alkali to remove unreacted glycerol esters, ion exchange to remove salts. High purity glycerol is obtained by multi-step distillation. Although not cost-effective, glycerol can be produced by various routes from propylene.
The epichlorohydrin process is the most important. This epichlorohydrin is hydrolyzed to give glycerol. Chlorine-free processes from propylene include the synthesis of glycerol from acrolein and propylene oxide; because of the large-scale production of biodiesel from fats, where glycerol is a waste product, the market for glycerol is depressed. Thus, synthetic processes are not economical. Owing to oversupply, efforts are being made to convert glycerol to synthetic precursors, such as acrolein and epichlorohydrin. (See the Chemical intermediate section of this article. In food and beverages, glycerol serves as a humectant and sweetener, may help preserve foods, it is used as filler in commercially prepared low-fat foods, as a thickening agent in liqueurs. Glycerol and water are used to preserve certain types of plant leaves; as a sugar substitute, it has 27 kilocalories per teaspoon and is 60% as sweet as sucrose. It does not feed the bacteria that form plaques and cause dental cavities; as a food additive, glycerol is labeled as E number E422.
It is added to icing to prevent it from setting too hard. As used in foods, glycerol is categorized by the Academy of Nutrition and Dietetics as a carbohydrate; the U. S. Food and Drug Administration carbohydrate designation includes all caloric macronutrients excluding protein and fat. Glycerol has a caloric density similar to table sugar, but a lower glycemic index and different metabolic pathway within the body, so some dietary advocates accept glycerol as a sweetener compatible with low-carbohydrate diets, it is recommended as an additive when using polyol sweeteners such as erythritol and xylitol which have a cooling effect, due to its heating effect in the mouth, if the cooling effect is not wanted. Glycerol is used in medical and personal care preparations as a means of improving smoothness, providing lubrication, as a humectant. Ichthyosis and xerosis have been relieved by the topical use glycerin, it is found in allergen immunotherapies, cough syrups and expectorants, mouthwashes, skin care products, shaving cream, hair care products and water-based personal lubricants.
In solid dosage forms like tablets, glycerol is used as a tablet holding agent. For human consumption, glycerol is classified by the U. S. FDA among the sugar alcohols as a caloric macronutrient. Glycerol is used in blood banking to preserve red blood cells prior to freezing. Glycerol is a component of glycerin soap. Essential oils are added for fragrance; this kind of soap is used by people with sensitive irritated skin because it prevents skin dryness with its moisturizing properties. It draws moisture up through skin layers and slows or prevents excessive drying and evaporation. Taken rectally, glycerol functions as a laxative by irritating the anal mucosa and inducing a hyperosmotic effect, it may be administered undiluted either as a suppository or as a small-volume enema. Alternatively, it may be administered in a dilute solution, e.g. 5%, as a high volume enema. Taken orally, glycerol can cause a rapid, temporary decrease in the internal pressure of the eye; this can be useful for the initial emergency treatment of elevated eye pressure.
In thermodynamics, chemical potential of a species is energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of a free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant; the molar chemical potential is known as partial molar free energy. When both temperature and pressure are held constant, chemical potential is the partial molar Gibbs free energy. At chemical equilibrium or in phase equilibrium the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum. In semiconductor physics, the chemical potential of a system of electrons at a temperature of zero Kelvin is known as the Fermi energy.
Particles tend to move from higher chemical potential to lower chemical potential. In this way, chemical potential is a generalization of "potentials" in physics such as gravitational potential; when a ball rolls down a hill, it is moving from a higher gravitational potential to a lower gravitational potential. In the same way, as molecules move, dissolve, etc. they will always tend to go from a higher chemical potential to a lower one, changing the particle number, conjugate variable to chemical potential. A simple example is a system of dilute molecules diffusing in a homogeneous environment. In this system, the molecules tend to move from areas with high concentration to low concentration, until the concentration is the same everywhere; the microscopic explanation for this is based in the random motion of molecules. However, it is simpler to describe the process in terms of chemical potentials: For a given temperature, a molecule has a higher chemical potential in a higher-concentration area, a lower chemical potential in a low concentration area.
Movement of molecules from higher chemical potential to lower chemical potential is accompanied by a release of free energy. Therefore, it is a spontaneous process. Another example, not based on concentration but on phase, is a glass of liquid water with ice cubes in it. Above 0 °C, an H2O molecule, in the liquid phase has a lower chemical potential than a water molecule, in the solid phase; when some of the ice melts, H2O molecules convert from solid to liquid where their chemical potential is lower, so the ice cubes shrink. Below 0 °C, the molecules in the ice phase have the lower chemical potential, so the ice cubes grow. At the temperature of the melting point, 0 °C, the chemical potentials in water and ice are the same. A third example is illustrated by the chemical reaction of dissociation of a weak acid HA: HA ⇌ H+ + A−Vinegar contains acetic acid; when acid molecules dissociate, the concentration of the undissociated acid molecules decreases and the concentrations of the product ions increase.
Thus the chemical potential of HA decreases and the sum of the chemical potentials of H+ and A− increases. When the sums of chemical potential of reactants and products are equal the system is at equilibrium and there is no tendency for the reaction to proceed in either the forward or backward direction; this explains why vinegar is acidic, because acetic acid dissociates to some extent, releasing hydrogen ions into the solution. Chemical potentials are important in many aspects of equilibrium chemistry, including melting, evaporation, osmosis, partition coefficient, liquid-liquid extraction and chromatography. In each case there is a characteristic constant, a function of the chemical potentials of the species at equilibrium. In electrochemistry, ions do not always tend to go from higher to lower chemical potential, but they do always go from higher to lower electrochemical potential; the electrochemical potential characterizes all of the influences on an ion's motion, while the chemical potential includes everything except the electric force.
The chemical potential μi of species i is defined, as all intensive quantities are, by the phenomenological fundamental equation of thermodynamics expressed in the form, which holds for both reversible and irreversible processes d U = T d S − P d V + ∑ i = 1 n μ i d N i,where dU is the infinitesimal change of internal energy U, dS the infinitesimal change of entropy S, dV is the infinitesimal change of volume V for a thermodynamic system in thermal equilibrium, dNi is the infinitesimal change of particle number Ni of species i as particles are added or subtracted. T is absolute temperature, S is entropy, P is pressure, V is volume. Other work terms, such as those involving electric, magnetic or gravitational fields may be added. From the above equation the chemical potential is given by μ i =