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60-514: Where N i {\displaystyle N_{i}} is the number of moles of component i , d μ i {\displaystyle i,\mathrm {d} \mu _{i}} the infinitesimal increase in chemical potential for this component, S {\displaystyle S} the entropy , T {\displaystyle T} the absolute temperature , V {\displaystyle V} volume and p {\displaystyle p}

120-454: A century earlier. In chemistry, it has been known since Proust's law of definite proportions (1794) that knowledge of the mass of each of the components in a chemical system is not sufficient to define the system. Amount of substance can be described as mass divided by Proust's "definite proportions", and contains information that is missing from the measurement of mass alone. As demonstrated by Dalton's law of partial pressures (1803),

180-450: A fact that greatly aided their acceptance: It was not necessary for a chemist to subscribe to atomic theory (an unproven hypothesis at the time) to make practical use of the tables. This would lead to some confusion between atomic masses (promoted by proponents of atomic theory) and equivalent weights (promoted by its opponents and which sometimes differed from relative atomic masses by an integer factor), which would last throughout much of

240-439: A gas cylinder filled with pure nitrogen is at room temperature (298 K) and 25 MPa, we can determine the fluid density (258 kg/m), enthalpy (272 kJ/kg), entropy (5.07 kJ/kg⋅K) or any other intensive thermodynamic variable. If instead the cylinder contains a nitrogen/oxygen mixture, we require an additional piece of information, usually the ratio of oxygen-to-nitrogen. If multiple phases of matter are present,

300-544: A generic extensive property of a mixture, it will always be true that it depends on the pressure ( P {\displaystyle P} ), temperature ( T {\displaystyle T} ), and the amount of each component of the mixture (measured in moles , n ). For a mixture with q components, this is expressed as Now if temperature T and pressure P are held constant, Z = Z ( n 1 , n 2 , ⋯ ) {\displaystyle Z=Z(n_{1},n_{2},\cdots )}

360-414: A mass of exactly 12  g . The four different definitions were equivalent to within 1%. Because a dalton , a unit commonly used to measure atomic mass , is exactly 1/12 of the mass of a carbon-12 atom, this definition of the mole entailed that the mass of one mole of a compound or element in grams was numerically equal to the average mass of one molecule or atom of the substance in daltons, and that

420-449: A measurement of mass is not even necessary to measure the amount of substance (although in practice it is usual). There are many physical relationships between amount of substance and other physical quantities, the most notable one being the ideal gas law (where the relationship was first demonstrated in 1857). The term "mole" was first used in a textbook describing these colligative properties . Developments in mass spectrometry led to

480-414: A mixture vary with the composition of the mixture, because the environment of the molecules in the mixture changes with the composition. It is the changing molecular environment (and the consequent alteration of the interactions between molecules) that results in the thermodynamic properties of a mixture changing as its composition is altered. If, by Z {\displaystyle Z} , one denotes

540-562: A quantity proportional to the number of elementary entities of a substance. One mole contains exactly 6.022 140 76 × 10 elementary entities (approximately 602 sextillion or 602 billion times a trillion), which can be atoms, molecules, ions, ion pairs, or other particles . The number of particles in a mole is the Avogadro number (symbol N 0 ) and the numerical value of the Avogadro constant (symbol N A ) expressed in mol . The value

600-448: A special name derived from the mole is the katal , defined as one mole per second of catalytic activity . Like other SI units, the mole can also be modified by adding a metric prefix that multiplies it by a power of 10 : One femtomole is exactly 602,214,076 molecules; attomole and smaller quantities cannot be exactly realized. The yoctomole, equal to around 0.6 of an individual molecule, did make appearances in scientific journals in

660-512: A specified T , P and at a constant molar ratio composition (so that the chemical potential does not change as the moles are added together), i.e. The total differential of this expression is Combining the two expressions for the total differential of the Gibbs free energy gives which simplifies to the Gibbs–Duhem relation: Another way of deriving the Gibbs–Duhem equation can be found by taking

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720-483: A substance is equal to its relative atomic (or molecular) mass multiplied by the molar mass constant , which is almost exactly 1 g/mol. Like chemists, chemical engineers use the unit mole extensively, but different unit multiples may be more suitable for industrial use. For example, the SI unit for volume is the cubic metre, a much larger unit than the commonly used litre in the chemical laboratory. When amount of substance

780-509: A substance was redefined as containing "exactly 6.022 140 76 × 10 elementary entities" of that substance. Since its adoption into the International System of Units in 1971, numerous criticisms of the concept of the mole as a unit like the metre or the second have arisen: October 23, denoted 10/23 in the US, is recognized by some as Mole Day . It is an informal holiday in honor of

840-415: Is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double Z {\displaystyle Z} . More generally, for any λ {\displaystyle \lambda } : By Euler's first theorem for homogeneous functions , this implies where Z i ¯ {\displaystyle {\bar {Z_{i}}}}

900-440: Is a quantity which describes the variation of an extensive property of a solution or mixture with changes in the molar composition of the mixture at constant temperature and pressure . It is the partial derivative of the extensive property with respect to the amount (number of moles) of the component of interest. Every extensive property of a mixture has a corresponding partial molar property. The partial molar volume

960-438: Is also expressed in kmol (1000 mol) in industrial-scaled processes, the numerical value of molarity remains the same, as kmol m 3 = 1000  mol 1000  L = mol L {\textstyle {\frac {\text{kmol}}{{\text{m}}^{3}}}={\frac {1000{\text{ mol}}}{1000{\text{ L}}}}={\frac {\text{mol}}{\text{L}}}} . Chemical engineers once used

1020-539: Is an intensive property) which means that for any λ {\displaystyle \lambda } : In particular, taking λ = 1 / n T {\displaystyle \lambda =1/n_{T}} where n T = n 1 + n 2 + ⋯ {\displaystyle n_{T}=n_{1}+n_{2}+\cdots } , one has where x i = n i n T {\displaystyle x_{i}={\frac {n_{i}}{n_{T}}}}

1080-490: Is broadly understood as the contribution that a component of a mixture makes to the overall volume of the solution. However, there is more to it than this: When one mole of water is added to a large volume of water at 25 °C, the volume increases by 18 cm . The molar volume of pure water would thus be reported as 18 cm mol . However, addition of one mole of water to a large volume of pure ethanol results in an increase in volume of only 14 cm . The reason that

1140-544: Is commonly expressed by its molar concentration , defined as the amount of dissolved substance per unit volume of solution, for which the unit typically used is mole per litre (mol/L). The number of entities (symbol N ) in a one-mole sample equals the Avogadro number (symbol N 0 ), a dimensionless quantity . Historically, N 0 approximates the number of nucleons ( protons or neutrons ) in one gram of ordinary matter . The Avogadro constant (symbol N A = N 0 /mol ) has numerical multiplier given by

1200-483: Is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at

1260-418: Is the concentration expressed as the mole fraction of component i {\displaystyle i} . Since the molar fractions satisfy the relation the x i are not independent, and the partial molar property is a function of only q − 1 {\displaystyle q-1} mole fractions: The partial molar property is thus an intensive property - it does not depend on

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1320-425: Is the partial molar Z {\displaystyle Z} of component i {\displaystyle i} defined as: By Euler's second theorem for homogeneous functions , Z i ¯ {\displaystyle {\bar {Z_{i}}}} is a homogeneous function of degree 0 (i.e., Z i ¯ {\displaystyle {\bar {Z_{i}}}}

1380-408: Is widely used in chemistry as a convenient way to express amounts of reactants and amounts of products of chemical reactions . For example, the chemical equation 2 H 2 + O 2 → 2 H 2 O can be interpreted to mean that for each 2 mol molecular hydrogen (H 2 ) and 1 mol molecular oxygen (O 2 ) that react, 2 mol of water (H 2 O) form. The concentration of a solution

1440-479: The 2019 revision of the SI , which redefined the mole by fixing the value of the Avogadro constant, making it very nearly equivalent to but no longer exactly equal to the gram-mole), but whose name and symbol adopt the SI convention for standard multiples of metric units – thus, kmol means 1000 mol. This is equivalent to the use of kg instead of g. The use of kmol is not only for "magnitude convenience" but also makes

1500-486: The heat of mixing or entropy of mixing ). By definition, properties of mixing are related to those of the pure substances by: Here ∗ {\displaystyle *} denotes a pure substance, M {\displaystyle M} the mixing property, and z {\displaystyle z} corresponds to the specific property under consideration. From the definition of partial molar properties, substitution yields: So from knowledge of

1560-414: The kilogram-mole (notation kg-mol ), which is defined as the number of entities in 12 kg of C, and often referred to the mole as the gram-mole (notation g-mol ), then defined as the number of entities in 12 g of C, when dealing with laboratory data. Late 20th-century chemical engineering practice came to use the kilomole (kmol), which was numerically identical to the kilogram-mole (until

1620-608: The pressure . I {\displaystyle I} is the number of different components in the system. This equation shows that in thermodynamics intensive properties are not independent but related, making it a mathematical statement of the state postulate . When pressure and temperature are variable, only I − 1 {\displaystyle I-1} of I {\displaystyle I} components have independent values for chemical potential and Gibbs' phase rule follows. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to

1680-471: The vapor pressure of a fluid mixture from limited experimental data. Lawrence Stamper Darken has shown that the Gibbs–Duhem equation can be applied to the determination of chemical potentials of components from a multicomponent system from experimental data regarding the chemical potential G 2 ¯ {\displaystyle {\bar {G_{2}}}} of only one component (here component 2) at all compositions. He has deduced

1740-487: The volume , T {\displaystyle T} the temperature, and S {\displaystyle S} the entropy . The thermodynamic potentials also satisfy where μ i {\displaystyle \mu _{i}} is the chemical potential defined as (for constant n j with j≠i): This last partial derivative is the same as G i ¯ {\displaystyle {\bar {G_{i}}}} ,

1800-480: The 14th CGPM. Before the 2019 revision of the SI , the mole was defined as the amount of substance of a system that contains as many elementary entities as there are atoms in 12 grams of carbon-12 (the most common isotope of carbon ). The term gram-molecule was formerly used to mean one mole of molecules, and gram-atom for one mole of atoms. For example, 1 mole of MgBr 2 is 1 gram-molecule of MgBr 2 but 3 gram-atoms of MgBr 2 . In 2011,

1860-851: The 24th meeting of the General Conference on Weights and Measures (CGPM) agreed to a plan for a possible revision of the SI base unit definitions at an undetermined date. On 16 November 2018, after a meeting of scientists from more than 60 countries at the CGPM in Versailles, France, all SI base units were defined in terms of physical constants. This meant that each SI unit, including the mole, would not be defined in terms of any physical objects but rather they would be defined by physical constants that are, in their nature, exact. Such changes officially came into effect on 20 May 2019. Following such changes, "one mole" of

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1920-527: The Avogadro number with the unit reciprocal mole (mol ). The ratio n = N / N A is a measure of the amount of substance (with the unit mole). Depending on the nature of the substance, an elementary entity may be an atom , a molecule , an ion , an ion pair, or a subatomic particle such as a proton . For example, 10 moles of water (a chemical compound ) and 10 moles of mercury (a chemical element ) contain equal numbers of substance, with one atom of mercury for each molecule of water, despite

1980-420: The adoption of oxygen-16 as the standard substance, in lieu of natural oxygen. The oxygen-16 definition was replaced with one based on carbon-12 during the 1960s. The International Bureau of Weights and Measures defined the mole as "the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon-12." Thus, by that definition, one mole of pure C had

2040-400: The basic SI unit of mol/s were to be used, which would otherwise require the molar mass to be converted to kg/mol. For convenience in avoiding conversions in the imperial (or US customary units ), some engineers adopted the pound-mole (notation lb-mol or lbmol ), which is defined as the number of entities in 12 lb of C. One lb-mol is equal to 453.592 37  g‑mol , which is

2100-448: The binary systems 1_2 and 2_3. These constants can be obtained from the previous equality by putting the complementary mole fraction x 3 = 0 for x 1 and vice versa. Thus and The final expression is given by substitution of these constants into the previous equation: Mole (unit) The mole (symbol mol ) is a unit of measurement , the base unit in the International System of Units (SI) for amount of substance ,

2160-484: The chemical convenience of having oxygen as the primary atomic mass standard became ever more evident with advances in analytical chemistry and the need for ever more accurate atomic mass determinations. The name mole is an 1897 translation of the German unit Mol , coined by the chemist Wilhelm Ostwald in 1894 from the German word Molekül ( molecule ). The related concept of equivalent mass had been in use at least

2220-503: The chemical potentials across a phase boundary are equal. Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium (i.e. that the temperature and pressure is constant throughout the system), we recover the Gibbs' phase rule . One particularly useful expression arises when considering binary solutions. At constant P ( isobaric ) and T ( isothermal ) it becomes: or, normalizing by total number of moles in

2280-471: The compositions of interest one can fit a curve to the experimental data. This function will be Z ( n 1 ) {\displaystyle Z(n_{1})} . Differentiating with respect to n 1 {\displaystyle n_{1}} will give Z 1 ¯ {\displaystyle {\bar {Z_{1}}}} . Z 2 ¯ {\displaystyle {\bar {Z_{2}}}}

2340-422: The definition of the gram was not mathematically tied to that of the dalton, the number of molecules per mole N A (the Avogadro constant) had to be determined experimentally. The experimental value adopted by CODATA in 2010 is N A = 6.022 141 29 (27) × 10  mol . In 2011 the measurement was refined to 6.022 140 78 (18) × 10  mol . The mole was made the seventh SI base unit in 1971 by

2400-484: The equations used for modelling chemical engineering systems coherent . For example, the conversion of a flowrate of kg/s to kmol/s only requires dividing by the molar mass in g/mol (as kg kmol = 1000  g 1000  mol = g mol {\textstyle {\frac {\text{kg}}{\text{kmol}}}={\frac {1000{\text{ g}}}{1000{\text{ mol}}}}={\frac {\text{g}}{\text{mol}}}} ) without multiplying by 1000 unless

2460-475: The extensivity of energy into account. Extensivity implies that where X {\displaystyle \mathbf {X} } denotes all extensive variables of the internal energy U {\displaystyle U} . The internal energy is thus a first-order homogenous function . Applying Euler's homogeneous function theorem , one finds the following relation when taking only volume, number of particles, and entropy as extensive variables: Taking

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2520-627: The following relation x i , amount (mole) fractions of components. Making some rearrangements and dividing by (1 – x 2 ) gives: or or The derivative with respect to one mole fraction x 2 is taken at constant ratios of amounts (and therefore of mole fractions) of the other components of the solution representable in a diagram like ternary plot . The last equality can be integrated from x 2 = 1 {\displaystyle x_{2}=1} to x 2 {\displaystyle x_{2}} gives: Applying LHopital's rule gives: This becomes further: Express

2580-406: The increase is different is that the volume occupied by a given number of water molecules depends upon the identity of the surrounding molecules. The value 14 cm is said to be the partial molar volume of water in ethanol. In general, the partial molar volume of a substance X in a mixture is the change in volume per mole of X added to the mixture. The partial molar volumes of the components of

2640-415: The influence of surface effects and other microscopic phenomena. The equation is named after Josiah Willard Gibbs and Pierre Duhem . Deriving the Gibbs–Duhem equation from the fundamental thermodynamic equation is straightforward. The total differential of the extensive Gibbs free energy G {\displaystyle G} in terms of its natural variables is Since the Gibbs free energy

2700-567: The mixture as they completely determine the Gibbs free energy. To measure the partial molar property Z 1 ¯ {\displaystyle {\bar {Z_{1}}}} of a binary solution, one begins with the pure component denoted as 2 {\displaystyle 2} and, keeping the temperature and pressure constant during the entire process, add small quantities of component 1 {\displaystyle 1} ; measuring Z {\displaystyle Z} after each addition. After sampling

2760-515: The mole fractions of component 1 and 3 as functions of component 2 mole fraction and binary mole ratios: and the sum of partial molar quantities gives ( G 1 ¯ ) x 2 = 1 {\displaystyle ({\bar {G_{1}}})_{x_{2}=1}} and ( G 3 ¯ ) x 2 = 1 {\displaystyle ({\bar {G_{3}}})_{x_{2}=1}} are constants which can be determined from

2820-613: The nineteenth century. Jöns Jacob Berzelius (1779–1848) was instrumental in the determination of relative atomic masses to ever-increasing accuracy. He was also the first chemist to use oxygen as the standard to which other masses were referred. Oxygen is a useful standard, as, unlike hydrogen, it forms compounds with most other elements, especially metals . However, he chose to fix the atomic mass of oxygen as 100, which did not catch on. Charles Frédéric Gerhardt (1816–56), Henri Victor Regnault (1810–78) and Stanislao Cannizzaro (1826–1910) expanded on Berzelius' works, resolving many of

2880-434: The number of daltons in a gram was equal to the number of elementary entities in a mole. Because the mass of a nucleon (i.e. a proton or neutron ) is approximately 1 dalton and the nucleons in an atom's nucleus make up the overwhelming majority of its mass, this definition also entailed that the mass of one mole of a substance was roughly equivalent to the number of nucleons in one atom or molecule of that substance. Since

2940-547: The partial molar Gibbs free energy . This means that the partial molar Gibbs free energy and the chemical potential, one of the most important properties in thermodynamics and chemistry, are the same quantity. Under isobaric (constant P ) and isothermal (constant T ) conditions, knowledge of the chemical potentials, μ i ( x 1 , x 2 , ⋯ , x m ) {\displaystyle \mu _{i}(x_{1},x_{2},\cdots ,x_{m})} , yields every property of

3000-433: The partial molar properties, deviation of properties of mixing from single components can be calculated. Partial molar properties satisfy relations analogous to those of the extensive properties. For the internal energy U , enthalpy H , Helmholtz free energy A , and Gibbs free energy G , the following hold: where P {\displaystyle P} is the pressure, V {\displaystyle V}

3060-472: The problems of unknown stoichiometry of compounds, and the use of atomic masses attracted a large consensus by the time of the Karlsruhe Congress (1860). The convention had reverted to defining the atomic mass of hydrogen as 1, although at the level of precision of measurements at that time – relative uncertainties of around 1% – this was numerically equivalent to the later standard of oxygen = 16. However

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3120-416: The same numerical value as the number of grams in an international avoirdupois pound . Greenhouse and growth chamber lighting for plants is sometimes expressed in micromoles per square metre per second, where 1 mol photons ≈ 6.02 × 10 photons. The obsolete unit einstein is variously defined as the energy in one mole of photons and also as simply one mole of photons. The only SI derived unit with

3180-489: The size of the system. The partial volume is not the partial molar volume. Partial molar properties are useful because chemical mixtures are often maintained at constant temperature and pressure and under these conditions, the value of any extensive property can be obtained from its partial molar property. They are especially useful when considering specific properties of pure substances (that is, properties of one mole of pure substance) and properties of mixing (such as

3240-435: The solid is composed of a certain number of moles of such entities. In yet other cases, such as diamond , where the entire crystal is essentially a single molecule, the mole is still used to express the number of atoms bound together, rather than a count of molecules. Thus, common chemical conventions apply to the definition of the constituent entities of a substance, in other cases exact definitions may be specified. The mass of

3300-475: The system N 1 + N 2 , {\displaystyle N_{1}+N_{2},} substituting in the definition of activity coefficient γ {\displaystyle \gamma } and using the identity x 1 + x 2 = 1 {\displaystyle x_{1}+x_{2}=1} : This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for

3360-629: The total differential, one finds Finally, one can equate this expression to the definition of d U {\displaystyle \mathrm {d} U} to find the Gibbs–Duhem equation By normalizing the above equation by the extent of a system, such as the total number of moles, the Gibbs–Duhem equation provides a relationship between the intensive variables of the system. For a simple system with I {\displaystyle I} different components, there will be I + 1 {\displaystyle I+1} independent parameters or "degrees of freedom". For example, if we know

3420-489: The two quantities having different volumes and different masses. The mole corresponds to a given count of entities. Usually, the entities counted are chemically identical and individually distinct. For example, a solution may contain a certain number of dissolved molecules that are more or less independent of each other. However, the constituent entities in a solid are fixed and bound in a lattice arrangement, yet they may be separable without losing their chemical identity. Thus,

3480-423: The unit among chemists. The date is derived from the Avogadro number, which is approximately 6.022 × 10 . It starts at 6:02 a.m. and ends at 6:02 p.m. Alternatively, some chemists celebrate June 2 ( 06/02 ), June 22 ( 6/22 ), or 6 February ( 06.02 ), a reference to the 6.02 or 6.022 part of the constant. Partial molar property In thermodynamics , a partial molar property

3540-459: The year the yocto- prefix was officially implemented. The history of the mole is intertwined with that of units of molecular mass , and the Avogadro constant . The first table of standard atomic weight was published by John Dalton (1766–1844) in 1805, based on a system in which the relative atomic mass of hydrogen was defined as 1. These relative atomic masses were based on the stoichiometric proportions of chemical reaction and compounds,

3600-472: Was chosen on the basis of the historical definition of the mole as the amount of substance that corresponds to the number of atoms in 12  grams of C , which made the mass of a mole of a compound expressed in grams, numerically equal to the average molecular mass or formula mass of the compound expressed in daltons . With the 2019 revision of the SI , the numerical equivalence is now only approximate but may be assumed for all practical purposes. The mole

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