formulasearchengine - User contributions [en]

Welding defect

2015-01-04T17:01:20Z

123.236.181.218: /* Transverse crack */

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Nucleophile

2014-02-02T09:28:46Z

123.236.90.87: /* Oxygen */

[[Image:Heat of Vaporization (Benzene+Acetone+Methanol+Water).png|thumb|280px|Temperature-dependency of the heats of vaporization for water, methanol, benzene, and acetone. ]]

The '''enthalpy of vaporization''', (symbol <math>\Delta{}H_{\mathrm{vap}}</math>), also known as the '''(latent) heat of vaporization''' or '''heat of evaporation''', is the [[enthalpy]] change required to transform a given quantity of a substance from a [[liquid]] into a [[gas]] at a given [[pressure]] (often [[atmospheric pressure]], as in [[Standard conditions for temperature and pressure|STP]]).

It is often measured at the [[normal boiling point]] of a substance; although tabulated values are usually corrected to 298 [[Kelvin|K]], the correction is often smaller than the [[Standard deviation|uncertainty]] in the measured value.

The heat of vaporization is temperature-dependent, though a constant heat of vaporization can be assumed for small temperature ranges and for reduced temperature [[Reduced temperature|Tr]]<<1.0. The heat of vaporization diminishes with increasing temperature and it vanishes completely at the critical temperature (Tr=1) because above the [[critical temperature]] the [[liquid]] and [[vapor]] phases no longer exist, since the substance is a [[supercritical fluid]].

== Units ==
Values are usually quoted in [[Joule|J]]/[[Mole (unit)|mol]] or kJ/mol (molar enthalpy of vaporization), although kJ/kg or J/g (specific heat of vaporization), and older units like [[Calorie|kcal]]/mol, cal/g and [[British thermal unit|Btu]]/lb are sometimes still used, among others.

== Physical model for vaporization==

[[Image:Physical model for vaporization.jpg|thumb|right|350px|
Fig. 1 Schematic cross section of the proposed vaporization model for monatomic liquids with one atomic surface layer.]]

A simple physical model for the liquid-gas phase transformation was proposed in 2009.<ref>{{cite doi|10.1016/j.fluid.2009.06.005 }}</ref> It is suggested that the energy required to free an atom from the liquid is equivalent to the energy needed to overcome the surface resistance of the liquid. The model allows calculating the latent heat by multiplying the maximum surface area covering an atom (Fig. 1) with the surface tension and the number of atoms in the liquid. The calculated latent heat of vaporization values for the investigated 45 elements agrees well with experiments.

== Enthalpy of condensation ==
The '''enthalpy of condensation''' (or '''heat of condensation''') is by definition equal to the enthalpy of vaporization with the opposite sign: enthalpy changes of vaporization are always positive ([[heat]] is absorbed by the substance), whereas enthalpy changes of condensation are always negative (heat is released by the substance).

== Thermodynamic background ==
[[Image:Heat Content of Zn(c,l,g).PNG|tlhumb|right|350px|'''Molar enthalpy of zinc''' above 298.15 K and at 1 atm pressure, showing discontinuities at the melting and boiling points. The enthalpy of melting (Δ''H''°m) of zinc is 7323 J/mol, and the enthalpy of vaporization (Δ''H''°v) is 115 330 J/mol.]]
The enthalpy of vaporization can be written as
:<math>\Delta{}H_{\mathrm{vap}} = \Delta{}U_{\mathrm{vap}} + p\Delta\,V</math>
It is equal to the increased [[internal energy]] of the vapor phase compared with the liquid phase, plus the work done against ambient pressure. The increase in the internal energy can be viewed as the energy required to overcome the [[Chemical bond#Intermolecular interactions|intermolecular interactions]] in the liquid (or solid, in the case of [[Sublimation (chemistry)|sublimation]]). Hence [[helium]] has a particularly low enthalpy of vaporization, 0.0845 kJ/mol, as the [[van der Waals force]]s between helium [[atom]]s are particularly weak. On the other hand, the [[molecule]]s in liquid [[Water (molecule)|water]] are held together by relatively strong [[hydrogen bond]]s, and its enthalpy of vaporization, 40.65 kJ/mol, is more than five times the energy required to heat the same quantity of water from 0 °C to 100 °C ([[Heat capacity|''c''p]] = 75.3 J K−1 mol−1). Care must be taken, however, when using enthalpies of vaporization to ''measure'' the strength of intermolecular forces, as these forces may persist to an extent in the gas phase (as is the case with [[hydrogen fluoride]]), and so the calculated value of the [[bond strength]] will be too low. This is particularly true of metals, which often form [[Covalent bond|covalently bonded]] molecules in the gas phase: in these cases, the [[enthalpy of atomization]] must be used to obtain a true value of the [[bond energy]].

An alternative description is to view the enthalpy of condensation as the heat which must be released to the surroundings to compensate for the drop in [[entropy]] when a gas condenses to a liquid. As the liquid and gas are in [[Chemical equilibrium|equilibrium]] at the boiling point (''T''b), [[Gibbs free energy|Δv''G'']] = 0, which leads to:
:<math>\Delta\,_v S = S_{gas} - S_{liquid} = \Delta\,_v H/T_b</math>
As neither entropy nor [[enthalpy]] vary greatly with [[temperature]], it is normal to use the tabulated standard values without any correction for the difference in temperature from 298 K. A correction must be made if the [[pressure]] is different from 100 [[Pascal (unit)|kPa]], as the entropy of a gas is proportional to its pressure (or, more precisely, to its [[fugacity]]): the entropies of liquids vary little with pressure, as the [[compressibility]] of a liquid is small.

These two definitions are equivalent: the boiling point is the temperature at which the increased entropy of the gas phase overcomes the intermolecular forces. As a given quantity of matter always has a higher entropy in the gas phase than in a condensed phase (<math>\Delta\,_v S</math> is always positive), and from
:<math>\Delta\,G = \Delta\,H - T\Delta\,S</math>,
the [[Gibbs free energy]] change falls with increasing temperature: gases are favored at higher temperatures, as is observed in practice.

==Vaporization enthalpy of electrolyte solutions==

Estimation of the enthalpy of vaporization of electrolyte solutions can be simply carried out using equations based on the chemical thermodynamic models, such as Pitzer model<ref>X. Ge, X. Wang. Estimation of Freezing Point Depression, Boiling Point Elevation and Vaporization enthalpies of electrolyte solutions. Ind. Eng. Chem. Res. 48(2009)2229-2235. http://pubs.acs.org/doi/abs/10.1021/ie801348c (Correction: 2009, 48, 5123)http://pubs.acs.org/doi/abs/10.1021/ie900434h</ref> or TCPC model.<ref>X. Ge, X. Wang. Calculations of Freezing Point Depression, Boiling Point Elevation, Vapor Pressure and Enthalpies of Vaporization of Electrolyte Solutions by a Modified Three-Characteristic Parameter Correlation Model. J. Sol. Chem. 38(2009)1097-1117.http://www.springerlink.com/content/21670685448p5145/</ref>

== Selected values ==
=== Elements ===
{{periodic table (enthalpy of vaporisation)}}

===Other common substances===
Enthalpies of vaporization of common substances, measured at their respective standard boiling points:

{| class="wikitable sortable"
!Compound
!Boiling Point at normal pressure
!Heat of vaporization ([[Kilojoule per mole|kJ mol-1]])
!Heat of vaporization (kJ kg−1)
|-
|[[Acetone]]
|329-330 K, 56-57 °C, 133-134 °F
|31.3
|538.9
|-
|[[Aluminium]]
|2792 K, 2519 °C, 4566 °F
|294.0
|10500
|-
|[[Ammonia]]
|240 K, −33.34 °C, -28 °F
|23.35
|1371
|-
|[[Butane]]
|272-274 K, -1°C, 30-34 °F
|21.0
|320
|-
|[[Diethyl ether]]
|307.8 K, 34.6 °C, 94.3 °F
|26.17
|353.1
|-
|[[Ethanol]]
|352 K, 78.37 °C, 173 °F
|38.6
|841
|-
|[[Hydrogen]]
|20.271 K, -252.879 °C, -423.182 °F
|0.46
|451.9
|-
|[[Iron]]
|3134 K, 2862 °C, 5182 °F
|340
|6090
|-
|[[Isopropyl alcohol]]
|356 K, 82.6 °C, 181 °F
|44.0
|732.2
|-
|[[Methane]]
|109-113 K, -164--160 °C, -263--256 °F
|8.17
|480.6
|-
|[[Methanol]]
|338 K, 64.7 °C, 148 °F
|35.3
|1104
|-
|[[Propane]]
|230.9-231.11 K,-42--42 °C, -44--44 °F
|15.7
|356
|-
|[[Phosphine]]
|185 K, -87.7 °C, -126 °F
|14.6
|429.4
|-
|[[Water]]
|373.15 K, 100 °C, 212 °F
|40.68
|2260
|}

==See also==
*[[Enthalpy of fusion]]
*[[Enthalpy of sublimation]]
*[[Joback method]] (Estimation of the heat of vaporization at the normal boiling point from molecular structures)

==References==
{{reflist}}
*[http://www.codata.org/resources/databases/key1.html CODATA Key Values for Thermodynamics]
*Kugler HK & Keller C (eds) 1985, ''Gmelin handbook of inorganic and organometallic chemistry,'' 8th ed., 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, ISBN 3-540-93516-9, pp. 116–117
*[http://webbook.nist.gov/chemistry/ NIST Chemistry WebBook]
*Sears, Zemansky et al., ''University Physics'', Addison-Wesley Publishing Company, Sixth ed., 1982, ISBN 0-201-07199-1

{{States of matter}}

{{DEFAULTSORT:Enthalpy Of Vaporization}}
[[Category:Thermodynamic properties]]
[[Category:Thermodynamics]]
[[Category:Enthalpy]]

Performance prediction

2013-11-30T17:32:18Z

123.236.52.136:

In [[geometry]], a '''complex polytope''' is a generalization of a [[polytope]] in [[real space]] to an analogous structure in a [[Complex number|complex]] [[Hilbert space]], where each real dimension is accompanied by an [[imaginary number|imaginary]] one.

On a [[real line]], two points bound a segment. This defines an edge with two bounding vertices. For a real polytope it is not possible to have a third vertex associated with an edge because one of them would then lie between the other two. On the complex line, which may be represented as an [[Argand diagram]], points are not ordered and there is no idea of "between", so more than two vertex points may be associated with a given edge.

Also, a real polygon has just two sides at each vertex, such that the boundary forms a closed loop. A real polyhedron has two faces at each edge such that the boundary forms a closed surface. A [[polychoron]] has two cells at each wall, and so on. These loops and surfaces have no analogy in complex spaces, for example a set of complex lines and points may form a closed chain of connections, but this chain does not bound a polygon. Thus, more than two elements meeting in one place may be allowed.

Since bounding does not occur, we cannot think of a complex edge as a line segment, but as the whole line. Similarly, we cannot think of a bounded polygonal face but must accept the whole plane.

Thus, a complex polytope may be understood as an [[arrangement of lines|arrangement]] of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on.

==Regular complex polytopes==
[[Image:ComplexOctagon.svg|frame|right|Two representations of a regular complex octagon 4{4}2]]
The only complex polytopes to have been systematically studied are the [[regular polytope|regular]] ones. Shephard (1952) discovered them, and Coxeter (1974) developed the idea extensively. Shephard treated his figures as [[configuration (geometry)|configurations]] from the start, while Coxeter only found it necessary to do so from Chapter 12 onwards.

In the Argand diagram, of the edge of a regular complex polytope, the vertex points lie at the vertices of a [[regular polygon]] centered on the origin. Given the general point ''x'' + ''iy'' in the complex plane, for an edge having ''p'' vertices, these lie at the ''p'' roots of the equation:

:<math>x^p -1 = 0</math>

(For ''p'' = 2 these are the real points +1 and − 1, and the edge is real).

Two real projections of the same regular complex octagon with edges ''a,b,c,d,e,f,g,h'' are illustrated. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices at which it meets another edge, hence each edge meets four other edges. In the first diagram, each edge is represented by a square. The sides of the square are ''not'' parts of the polygon - this is important to understand - but are drawn in purely to help visually relate the four vertices. The edges are laid out symmetrically (coincidentally the diagram looks the same as a common projection of the [[hypercube]], but in the case of the complex octagon the diamond shapes which can be traced are not parts of the structure). The second diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a line, and each meeting point on the line is a vertex on that edge. The connectivity between the various edges is clear to see.

===Modified Schläfli notation===

;Shephard's notation

Shephard originally devised a modified form of [[Schläfli symbol|Schläfli's notation]] for regular polytopes. For a polygon bounded by ''p''1-edges, with a ''p''2-set as vertex figure and overall symmetry group of order ''g'', we denote the polygon as ''p''1(''g'')''p''2.

The number of vertices ''V'' is then ''g''/''p''2 and the number of edges ''E'' is ''g''/''p''1.

The complex octagon illustrated has eight 4-edges (''p''1=4) and sixteen 2-vertices (''p''2=2). From this we can work out that ''g'' = 32, giving the modified Schläfli symbol 4(32)2.

;Coxeter's notation

The modern notation ''p''1{''q''}''p''2 is due to Coxeter, and is based on group theory. The nodes ''p''1 and ''p''2 represent mirrors producing ''p''1 and ''p''2 images in the plane. In group theory, this might be represented (for the example left) as AAAA = BB = 1. ''q'' represents the number of alternate reflections in the two mirrors that become equal to its opposite, i.e. for ''q''=4, ABAB = BABA. When ''q'' is odd, then ''p''1 = ''p''2, e.g. 3{5}3 means AAA = BBB = 1; ABABA = BABAB.

The example octagon is represented as 4{4}2, which belongs to symmetry group AAAA = BB = 1, ABAB = BABA.

==Real conjugates==
In the ordinary, or ''real'' plane, we can construct a visible figure as the ''real conjugate'' of some complex polygon. Likewise in ordinary space, we can construct a visible figure as the ''real conjugate'' of some complex polyhedron.

To obtain the real conjugate, we discard the imaginary part of any coordinate. For example the complex point (''a'' + ''ib'') has real conjugate ''a''.

The real conjugate of a complex edge is a line with the vertex points distributed along it (not generally evenly spaced). The second of the two octagon projections above shows the real conjugates of the sides.

== References ==
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]] and Moser, W. O. J.; ''Generators and Relations for Discrete Groups'' (1965), esp pp 67–80.
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]]; ''Regular Complex Polytopes'', Cambridge University Press, (1974).
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]] and Shephard, G.C.; Portraits of a family of complex polytopes, ''Leonardo'' Vol 25, No 3/4, (1992), pp 239–244,
* Shephard, G.C.; Regular complex polytopes, ''Proc. London math. Soc.'' Series 3, Vol 2, (1952), pp 82–97.

[[Category:Polytopes]]
[[Category:Complex analysis]]

Accounting rate of return

2012-06-24T04:28:11Z

123.236.58.206: /* Basic formulae */

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