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73-505: The crystal lattice of aragonite differs from that of calcite, resulting in a different crystal shape, an orthorhombic crystal system with acicular crystal . Repeated twinning results in pseudo-hexagonal forms. Aragonite may be columnar or fibrous, occasionally in branching helictitic forms called flos-ferri ("flowers of iron") from their association with the ores at the Carinthian iron mines. The type location for aragonite

146-752: A i + ⋯ d ( T S ) − S d T = d U + d ( p V ) − V d p − ∑ i = 1 k μ i d N i + ∑ i = 1 n X i d a i + ⋯ d ( U − T S + p V ) = V d p − S d T + ∑ i = 1 k μ i d N i − ∑ i = 1 n X i d

219-1054: A i + ⋯ d G = V d p − S d T + ∑ i = 1 k μ i d N i − ∑ i = 1 n X i d a i + ⋯ {\displaystyle {\begin{aligned}T\,\mathrm {d} S&=\mathrm {d} U+p\,\mathrm {d} V-\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} (TS)-S\,\mathrm {d} T&=\mathrm {d} U+\mathrm {d} (pV)-V\,\mathrm {d} p-\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}+\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} (U-TS+pV)&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} G&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \end{aligned}}} where: This

292-495: A calcium chloride solution to a sodium carbonate solution at temperatures above 60 °C (140 °F) or in water-ethanol mixtures at ambient temperatures. Aragonite is a thermodynamically unstable phase of calcium carbonate at any pressure below about 3,000 bars (300,000 kPa) at any temperature. Aragonite nonetheless frequently forms in near-surface environments at ambient temperatures. The weak Van der Waals forces inside aragonite give an important contribution to both

365-624: A ); similarly for the reciprocal lattice. So, in this common case, the Miller indices ( ℓmn ) and [ ℓmn ] both simply denote normals/directions in Cartesian coordinates . For cubic crystals with lattice constant a , the spacing d between adjacent (ℓmn) lattice planes is (from above): Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes: For face-centered cubic (fcc) and body-centered cubic (bcc) lattices,

438-424: A ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers. One important characteristic of a crystalline structure is its atomic packing factor (APF). This is calculated by assuming that all the atoms are identical spheres, with

511-506: A closed system at constant temperature (in a heat bath), T d S ≥ δ Q {\displaystyle TdS\geq \delta Q} , and so it follows that Assuming that only mechanical work is done, this simplifies to This means that for such a system when not in equilibrium, the Gibbs energy will always be decreasing, and in equilibrium, the infinitesimal change dG will be zero. In particular, this will be true if

584-452: A critical role in determining many physical properties, such as cleavage , electronic band structure , and optical transparency . Crystal structure is described in terms of the geometry of arrangement of particles in the unit cells. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as a parallelepiped , providing six lattice parameters taken as

657-404: A crystal lattice are described by the three-value Miller index notation. This syntax uses the indices h , k , and ℓ as directional parameters. By definition, the syntax ( hkℓ ) denotes a plane that intercepts the three points a 1 / h , a 2 / k , and a 3 / ℓ , or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with

730-413: A crystal. Likewise, the crystallographic planes are geometric planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows: Some directions and planes are defined by symmetry of the crystal system. In monoclinic, trigonal, tetragonal, and hexagonal systems there is one unique axis (sometimes called

803-512: A cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Considering only ( hkℓ ) planes intersecting one or more lattice points (the lattice planes ), the distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula The crystallographic directions are geometric lines linking nodes ( atoms , ions or molecules ) of

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876-448: A given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition. The initial state of the body, according to Gibbs, is supposed to be such that "the body can be made to pass from it to states of dissipated energy by reversible processes ". In his 1876 magnum opus On

949-599: A lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystallographic point group or crystal class

1022-413: A radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows: Another important characteristic of a crystalline structure is its coordination number (CN). This is the number of nearest neighbours of a central atom in

1095-461: A reduction in G {\displaystyle G} is necessary for a reaction to be spontaneous under these conditions. The concept of Gibbs free energy, originally called available energy , was developed in the 1870s by the American scientist Josiah Willard Gibbs . In 1873, Gibbs described this "available energy" as the greatest amount of mechanical work which can be obtained from

1168-405: A single equation the necessary and sufficient condition of thermodynamic equilibrium for a substance when surrounded by a medium of constant pressure p and temperature T , this equation may be written: when δ refers to the variation produced by any variations in the state of the parts of the body, and (when different parts of the body are in different states) in the proportion in which

1241-966: A thermodynamic process. The temperature dependence of the Gibbs energy for an ideal gas is given by the Gibbs–Helmholtz equation , and its pressure dependence is given by G N = G ∘ N + k T ln ⁡ p p ∘ . {\displaystyle {\frac {G}{N}}={\frac {G^{\circ }}{N}}+kT\ln {\frac {p}{p^{\circ }}}.} or more conveniently as its chemical potential : G N = μ = μ ∘ + k T ln ⁡ p p ∘ . {\displaystyle {\frac {G}{N}}=\mu =\mu ^{\circ }+kT\ln {\frac {p}{p^{\circ }}}.} In non-ideal systems, fugacity comes into play. The Gibbs free energy total differential with respect to natural variables may be derived by Legendre transforms of

1314-882: Is Molina de Aragón in the Province of Guadalajara in Castilla-La Mancha , Spain , for which it was named in 1797. Aragonite is found in this locality as cyclic twins inside gypsum and marls of the Keuper facies of the Triassic . This type of aragonite deposit is very common in Spain, and there are also some in France. An aragonite cave, the Ochtinská Aragonite Cave , is situated in Slovakia . In

1387-422: Is enthalpy , T {\displaystyle T} is absolute temperature , and S {\displaystyle S} is entropy . According to the second law of thermodynamics , for systems reacting at fixed temperature and pressure without input of non- Pressure Volume (pV) work , there is a general natural tendency to achieve a minimum of the Gibbs free energy. A quantitative measure of

1460-465: Is a description of ordered arrangement of atoms , ions , or molecules in a crystalline material . Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in material that constitutes this repeating pattern is unit cell of the structure. The unit cell completely reflects symmetry and structure of

1533-436: Is acquired by a system at an electrical potential Ψ, the electrical work associated with this is −Ψ d e , which would be included in the infinitesimal expression. Other work terms are added on per system requirements. Each quantity in the equations above can be divided by the amount of substance, measured in moles , to form molar Gibbs free energy . The Gibbs free energy is one of the most important thermodynamic functions for

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1606-455: Is considered essential for the replication of reef conditions. Aragonite provides the materials necessary for much sea life and also keeps the pH of the water close to its natural level, to prevent the dissolution of biogenic calcium carbonate . Aragonite has been successfully tested for the removal of pollutants like zinc , cobalt and lead from contaminated wastewaters. Crystal lattice In crystallography , crystal structure

1679-446: Is expressed as G ( p , T ) = U + p V − T S = H − T S {\displaystyle G(p,T)=U+pV-TS=H-TS} Where: The Gibbs free energy change ( Δ G = Δ H − T Δ S {\displaystyle \Delta G=\Delta H-T\Delta S} , measured in joules in SI )

1752-438: Is given by where δQ is energy added as heat, and δW is energy added as work. The work done on the system may be written as δW = − pdV + δW x , where − pdV is the mechanical work of compression/expansion done on or by the system and δW x is all other forms of work, which may include electrical, magnetic, etc. Then and the infinitesimal change in G is The second law of thermodynamics states that for

1825-399: Is held at constant temperature and pressure, and is closed (no matter can come in or out). The Gibbs energy of any system is ⁠ G = U + p V − T S {\displaystyle G=U+pV-TS} ⁠ and an infinitesimal change in G , at constant temperature and pressure, yields By the first law of thermodynamics , a change in the internal energy U

1898-508: Is one form of the Gibbs fundamental equation . In the infinitesimal expression, the term involving the chemical potential accounts for changes in Gibbs free energy resulting from an influx or outflux of particles. In other words, it holds for an open system or for a closed , chemically reacting system where the N i are changing. For a closed, non-reacting system, this term may be dropped. Any number of extra terms may be added, depending on

1971-412: Is the entropy , V is the system volume, p is its pressure and T is its absolute temperature . The combination ( E {\displaystyle {\mathcal {E}}} , Q ele ) is an example of a conjugate pair of variables . At constant pressure the above equation produces a Maxwell relation that links the change in open cell voltage with temperature T (a measurable quantity) to

2044-418: Is the maximum amount of non-volume expansion work that can be extracted from a closed system (one that can exchange heat and work with its surroundings, but not matter) at fixed temperature and pressure. This maximum can be attained only in a completely reversible process . When a system transforms reversibly from an initial state to a final state under these conditions, the decrease in Gibbs free energy equals

2117-442: Is the case with internal energy. However, simply substituting the above integrated result for U into the definition of G gives a standard expression for G : This result shows that the chemical potential of a substance i {\displaystyle i} is its (partial) mol(ecul)ar Gibbs free energy. It applies to homogeneous, macroscopic systems, but not to all thermodynamic systems. The system under consideration

2190-424: Is the first to form. The presence of magnesium ions may inhibit calcite formation in favor of aragonite. Once formed, aragonite tends to alter to calcite on scales of 10 to 10 years. The mineral vaterite , also known as μ-CaCO 3 , is another phase of calcium carbonate that is metastable at ambient conditions typical of Earth's surface, and decomposes even more readily than aragonite. In aquaria , aragonite

2263-403: Is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements . There are 32 possible crystal classes. Each one can be classified into one of

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2336-424: Is the result of a 1988 IUPAC meeting to set unified terminologies for the international scientific community, in which the removal of the adjective "free" was recommended. This standard, however, has not yet been universally adopted. The name "free enthalpy " was also used for G in the past. The quantity called "free energy" is a more advanced and accurate replacement for the outdated term affinity , which

2409-428: Is the same as G ( p , T ) = H − T S , {\displaystyle G(p,T)=H-TS,} where: [REDACTED] The expression for the infinitesimal reversible change in the Gibbs free energy as a function of its "natural variables" p and T , for an open system , subjected to the operation of external forces (for instance, electrical or magnetic) X i , which cause

2482-419: Is usually a measure of the diminution of the total energy of the system ( internal energy ). Thus, G or F is the amount of energy "free" for work under the given conditions. Until this point, the general view had been such that: "all chemical reactions drive the system to a state of equilibrium in which the affinities of the reactions vanish". Over the next 60 years, the term affinity came to be replaced with

2555-495: The Gibbs free energy (or Gibbs energy as the recommended name; symbol G {\displaystyle G} ) is a thermodynamic potential that can be used to calculate the maximum amount of work , other than pressure-volume work , that may be performed by a thermodynamically closed system at constant temperature and pressure . It also provides a necessary condition for processes such as chemical reactions that may occur under these conditions. The Gibbs free energy

2628-458: The calcareous endoskeleton of warm- and cold-water corals ( Scleractinia ). Several serpulids have aragonitic tubes. Because the mineral deposition in mollusk shells is strongly biologically controlled, some crystal forms are distinctively different from those of inorganic aragonite. In some mollusks, the entire shell is aragonite; in others, aragonite forms only discrete parts of a bimineralic shell (aragonite plus calcite). The nacreous layer of

2701-439: The elimination of cyclohexanol to cyclohexene , can be seen as coupling an unfavorable reaction (elimination) to a favorable one (burning of coal or other provision of heat) such that the total entropy change of the universe is greater than or equal to zero, making the total Gibbs free energy change of the coupled reactions negative. In traditional use, the term "free" was included in "Gibbs free energy" to mean "available in

2774-458: The internal energy . The definition of G from above is Taking the total differential, we have Replacing d U with the result from the first law gives The natural variables of G are then p , T , and { N i }. Because S , V , and N i are extensive variables , an Euler relation allows easy integration of d U : Because some of the natural variables of G are intensive, d G may not be integrated using Euler relations as

2847-420: The principal axis ) which has higher rotational symmetry than the other two axes. The basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis. For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted

2920-591: The trigonal crystal system ), orthorhombic , monoclinic and triclinic . Bravais lattices , also referred to as space lattices , describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals , fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above. The crystal structure consists of

2993-621: The Equilibrium of Heterogeneous Substances , a graphical analysis of multi-phase chemical systems, he engaged his thoughts on chemical-free energy in full. If the reactants and products are all in their thermodynamic standard states , then the defining equation is written as Δ G ∘ = Δ H ∘ − T Δ S ∘ {\displaystyle \Delta G^{\circ }=\Delta H^{\circ }-T\Delta S^{\circ }} , where H {\displaystyle H}

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3066-654: The US, aragonite in the form of stalactites and "cave flowers" ( anthodite ) is known from Carlsbad Caverns and other caves. For a few years in the early 1900s, aragonite was mined at Aragonite, Utah (now a ghost town). Massive deposits of oolitic aragonite sand are found on the seabed in the Bahamas . Aragonite is the high pressure polymorph of calcium carbonate . As such, it occurs in high pressure metamorphic rocks such as those formed at subduction zones . Aragonite forms naturally in almost all mollusk shells, and as

3139-417: The affinity as the largest quantity of work which can be gained when the reaction is carried out in a reversible manner, e.g., electrical work in a reversible cell. The maximum work is thus regarded as the diminution of the free, or available, energy of the system ( Gibbs free energy G at T = constant, P = constant or Helmholtz free energy F at T = constant, V = constant), whilst the heat given out

3212-413: The amount of "free" or "useful" energy available to do non- pV work at constant temperature and pressure. The equation can be also seen from the perspective of the system taken together with its surroundings (the rest of the universe). First, one assumes that the given reaction at constant temperature and pressure is the only one that is occurring. Then the entropy released or absorbed by the system equals

3285-453: The aragonite fossil shells of some extinct ammonites forms an iridescent material called ammolite . Aragonite also forms naturally in the endocarp of Celtis occidentalis . The skeleton of some calcareous sponges is made of aragonite. Aragonite also forms in the ocean inorganic precipitates called marine cements (in the sediment ) or as free crystals (in the water column). Inorganic precipitation of aragonite in caves can occur in

3358-465: The body is divided between the different states. The condition of stable equilibrium is that the value of the expression in the parenthesis shall be a minimum. In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body... Thereafter, in 1882, the German scientist Hermann von Helmholtz characterized

3431-400: The characterization of a system. It is a factor in determining outcomes such as the voltage of an electrochemical cell , and the equilibrium constant for a reversible reaction . In isothermal, isobaric systems, Gibbs free energy can be thought of as a "dynamic" quantity, in that it is a representative measure of the competing effects of the enthalpic and entropic driving forces involved in

3504-426: The crystal lattice leaves it unchanged. All crystals have translational symmetry in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in

3577-413: The crystallographic and elastic properties of this mineral. The difference in stability between aragonite and calcite, as measured by the Gibbs free energy of formation , is small, and effects of grain size and impurities can be important. The formation of aragonite at temperatures and pressures where calcite should be the stable polymorph may be an example of Ostwald's step rule , where a less stable phase

3650-565: The entire crystal, which is built up by repetitive translation of unit cell along its principal axes. The translation vectors define the nodes of Bravais lattice . The lengths of principal axes/edges, of unit cell and angles between them are lattice constants , also called lattice parameters or cell parameters . The symmetry properties of crystal are described by the concept of space groups . All possible symmetric arrangements of particles in three-dimensional space may be described by 230 space groups. The crystal structure and symmetry play

3723-453: The entropy that the environment must absorb or release, respectively. The reaction will only be allowed if the total entropy change of the universe is zero or positive. This is reflected in a negative Δ G , and the reaction is called an exergonic process . If two chemical reactions are coupled, then an otherwise endergonic reaction (one with positive Δ G ) can be made to happen. The input of heat into an inherently endergonic reaction, such as

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3796-433: The external parameters of the system a i to change by an amount d a i , can be derived as follows from the first law for reversible processes: T d S = d U + p d V − ∑ i = 1 k μ i d N i + ∑ i = 1 n X i d

3869-431: The favorability of a given reaction under these conditions is the change Δ G (sometimes written "delta G " or "d G ") in Gibbs free energy that is (or would be) caused by the reaction. As a necessary condition for the reaction to occur at constant temperature and pressure, Δ G must be smaller than the non-pressure-volume (non- pV , e.g. electrical) work, which is often equal to zero (then Δ G must be negative). Δ G equals

3942-424: The following series: This arrangement of atoms in a crystal structure is known as hexagonal close packing (hcp) . If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises: This type of structural arrangement is known as cubic close packing (ccp) . The unit cell of

4015-517: The form of speleothems . Aragonite is common in serpentinites where magnesium-rich pore solutions apparently inhibit calcite growth and promote aragonite precipitation. Aragonite is metastable at the low pressures near the Earth's surface and is thus commonly replaced by calcite in fossils. Aragonite older than the Carboniferous is essentially unknown. Aragonite can be synthesized by adding

4088-437: The form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified. Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not

4161-513: The form of useful work". The characterization becomes more precise if we add the qualification that it is the energy available for non-pressure-volume work. (An analogous, but slightly different, meaning of "free" applies in conjunction with the Helmholtz free energy , for systems at constant temperature). However, an increasing number of books and journal articles do not include the attachment "free", referring to G as simply "Gibbs energy". This

4234-400: The interactions of homogeneous substances in contact, i.e., bodies composed of part solid, part liquid, and part vapor, and by using a three-dimensional volume - entropy - internal energy graph, Gibbs was able to determine three states of equilibrium, i.e., "necessarily stable", "neutral", and "unstable", and whether or not changes would ensue. Further, Gibbs stated: If we wish to express in

4307-461: The lengths of the cell edges ( a , b , c ) and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates ( x i , y i , z i ) along the cell edges, measured from a reference point. It is thus only necessary to report the coordinates of a smallest asymmetric subset of particles, called the crystallographic asymmetric unit. The asymmetric unit may be chosen so that it occupies

4380-418: The maximum amount of non- pV work that can be performed as a result of the chemical reaction for the case of a reversible process. If analysis indicates a positive Δ G for a reaction, then energy — in the form of electrical or other non- pV work — would have to be added to the reacting system for Δ G to be smaller than the non- pV work and make it possible for the reaction to occur. One can think of ∆G as

4453-400: The particular system being considered. Aside from mechanical work , a system may, in addition, perform numerous other types of work. For example, in the infinitesimal expression, the contractile work energy associated with a thermodynamic system that is a contractile fiber that shortens by an amount −d l under a force f would result in a term f d l being added. If a quantity of charge −d e

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4526-411: The primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions . The spacing d between adjacent ( hkℓ ) lattice planes is given by: The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on

4599-482: The same as the seven crystal systems . aP mP mS oP oS oI oF tP tI hR hP cP cI cF The most symmetric, the cubic or isometric system, has the symmetry of a cube , that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle ) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal , tetragonal , rhombohedral (often confused with

4672-525: The same group of atoms, the basis , positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group . A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to

4745-401: The seven crystal systems. In addition to the operations of the point group, the space group of the crystal structure contains translational symmetry operations. These include: There are 230 distinct space groups. By considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc., it is possible to form a general view of

4818-424: The smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the space group of the crystal structure. Vectors and planes in

4891-611: The structure. The APFs and CNs of the most common crystal structures are shown below: The 74% packing efficiency of the FCC and HCP is the maximum density possible in unit cells constructed of spheres of only one size. Interstitial sites refer to the empty spaces in between the atoms in the crystal lattice. These spaces can be filled by oppositely charged ions to form multi-element structures. They can also be filled by impurity atoms or self-interstitials to form interstitial defects . Gibbs free energy of formation In thermodynamics ,

4964-447: The structures and alternative ways of visualizing them. The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer were placed directly over plane A, this would give rise to

5037-563: The system is experiencing any number of internal chemical reactions on its path to equilibrium. When electric charge dQ ele is passed between the electrodes of an electrochemical cell generating an emf E {\displaystyle {\mathcal {E}}} , an electrical work term appears in the expression for the change in Gibbs energy: d G = − S d T + V d p + E d Q e l e , {\displaystyle dG=-SdT+Vdp+{\mathcal {E}}dQ_{ele},} where S

5110-626: The term free energy. According to chemistry historian Henry Leicester, the influential 1923 textbook Thermodynamics and the Free Energy of Chemical Substances by Gilbert N. Lewis and Merle Randall led to the replacement of the term "affinity" by the term "free energy" in much of the English-speaking world. The Gibbs free energy is defined as G ( p , T ) = U + p V − T S , {\displaystyle G(p,T)=U+pV-TS,} which

5183-493: The unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in (1 2 3). In an orthogonal coordinate system for

5256-410: The work done by the system to its surroundings, minus the work of the pressure forces. The Gibbs energy is the thermodynamic potential that is minimized when a system reaches chemical equilibrium at constant pressure and temperature when not driven by an applied electrolytic voltage. Its derivative with respect to the reaction coordinate of the system then vanishes at the equilibrium point. As such,

5329-558: Was used by chemists in the earlier years of physical chemistry to describe the force that caused chemical reactions . In 1873, Josiah Willard Gibbs published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces , in which he sketched the principles of his new equation that was able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying

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