# Statistics

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## Nernst's law of distribution

The basis of the separation of mixtures of substances by chromatographic methods is the different distribution of the individual substances between two phases. First of all, the distribution of a substance between two phases should be considered:

If two phases α and β, both of which contain substance B in solution, are in equilibrium with one another,

$μB.α=μB.β$

the following applies to component B at constant temperature and constant pressure:

$μB.0α+R.T⋅ln(aB.α)=μB.0β+R.T⋅ln(aB.β)$

whereby $μB.α$ and $μB.β$ the chemical potentials of substance B are in phase α or β, while $μB.0α$ and $μB.0β$ represent the standard potentials at infinite dilution as a reference value. $aB.α$ and $aB.β$ are the activities of substance B in relation to this standard state. Forming gives you

$ln(aB.αaB.β)=μB.0β−μB.0αR.T$

or

$aB.αaB.β=e(μB.0β−μB.0αR.T)=const.$

In the case of ideally dilute solutions, these activities can be replaced by the mole fractions and Nernst's distribution theorem is obtained.

$xB.αxB.β=const.=k$

In the case of ideally dilute solutions, this ratio of the mole fractions in the two phases is constant under given external conditions. It is therefore independent of the absolute amount of the dissolved substance in the respective phase. With sufficient dilution, the mole fractions can approximately be replaced by the concentrations.

$cB.αcB.β=k$

The fact that a dissolved substance is divided into two phases in a ratio that is characteristic of it, regardless of its concentration, can be used to separate substance mixtures. This is used in chromatographic separation processes and in the so-called multiple extraction according to Craig.

Your guess is correct, but you shouldn't forget the dimensions (-)))

& quotAt equilibrium 1 this is therefore 400° C and 20bar, and thus almost 40%& quot

Okay, thank you very much - but do I only have to "read" the value that "matches" the two specified dates? We only recently started on the subject - that's why I assumed that invoices would still be necessary here.

That means, for equilibrium II it would be - & quot; for equilibrium 2 this is consequently 400° C and 60bar, and thus at just under 65% & quot?

By the way, that's a so-called Characteristic diagram.
Here it is the characteristics of the Volume part from NH3 in% depending on the temperature. parameter for the individual characteristics is the pressure

Right. You go straight up at the desired temperature and then get intersections with the isobars. At the point of intersection with the specified isobar, go horizontally to the left and read off the yield ordinate.

formally correct, but with this order of magnitude of the pressure at least & quot; need to be accepted & quot (-)))

## Macroscopy (physics)

In physics, a distinction is made between microscopic and macroscopic observation.

### Further recommended specialist knowledge

#### Guide to Basic Laboratory Skills

At a macroscopic observation If one does not derive the behavior of a system from the behavior of its smallest components, one considers meaningful statistical values ​​(e.g. mean values).

An example is the gas (air) in a balloon. Describing the trajectories of the individual molecules (nitrogen, oxygen,.) Using the means of classical physics would be of little help. Given the huge number of particles, this approach is futile.

Instead, one leads z. B. the temperature as a measure of the mean kinetic energy of the particles. This is justified because the energy is statistically evenly distributed over all degrees of freedom. This assumption of uniform distribution is always better fulfilled with an increasing number of particles, in the ideal case of an infinite number of particles it is even perfect. The actual distribution of the energy (number of particles with one energy E.) gives the Maxwell-Boltzmann distribution in classical physics and the Fermi-Dirac statistics and Bose-Einstein statistics in quantum mechanics.

With this approach, the description of the system is reduced to a few macroscopic quantities such as temperature, pressure, density, etc., which we are already familiar with from everyday life. In fact, there is e.g. B. the atmosphere of the earth made up of individual molecules. Their individual interaction (which one even has to describe correctly in terms of quantum mechanics) becomes insignificant due to the large number of effects.

The very large number of particles in everyday systems, all of which are meaningfully described macroscopically, has the advantage that statistical methods can be applied with very high precision (see also statistical physics).

But you don't always have to proceed statistically. For example, a huge system like a planet or a sun can be traced back to a mass point (microscopic) in the description (a sphere with a concentric density distribution behaves like a mass point from the point of view of gravity).

Physics is very successful and generally applicable where it describes either simple, microscopic systems or very large, macroscopic systems. In contrast, generally valid statements are difficult for systems with “few” parts. So there is e.g. B. no closed solution for the movement of three mass points (three-body problem). In such cases, numerical computer models can usually help.

## Exercises

Is it a qualitative characteristic with an ordinal scale? (Grades in a class test) (interest in soccer) (! Pet) (! Desired title) (! Gender) (! Hobby) (! Favorite color) (! Occupation) (! Marital status) (! Denomination) (! Age in years) ( ! Salary) (! Shoe size) (! Number of seats)

Is it a qualitative characteristic with a nominal scale? (Pet) (Preferred title) (Gender) (Hobby) (Favorite color) (Profession) (Denomination) (! Grades in a class test) (! Interest in football) (! Age in years) (! Salary) (! Shoe size) (! Number of seats) (! Fruit juice content of apple juice)

Is it a quantitative characteristic with a metric discrete scale? (Age in years) (Salary in full EUR) (Number of seats) (! Grades in a class test) (! Interest in soccer) (! Favorite color) (! Profession) (! Marital status) (! Denomination) (! Fruit juice content of apple juice) (! Prescription in diopters) (! Cruising speed for airplanes) (! Temperature in degrees Celsius) (! Water level of the Ruhr in Hattingen)

Is it a quantitative characteristic with a metric continuous scale? (Fruit juice content of apple juice) (Eyesight in dioptres) (Airplane cruising speed) (Temperature in degrees Celsius) (Water level in the Ruhr in Hattingen) (! Grades in a class test) (! Interest in soccer) (! Pet) (! Desired title) ( ! Gender) (! Hobby) (! Favorite color) (! Profession) (! Marital status) (! Denomination) (! Age in years) (! Salary in full EUR) (! Number of seats)

## Mnemonics

Congratulations! You have successfully completed the first chapter.

Your rulebook already contains a lot of information. Make sure you wrote everything down.

One Population is the set of all possible objects about which one would like to make a statement.

- limited (e.g. all students in class HHU5 at the Hattingen vocational college), - very large (e.g. all residents of Germany in 2014) or - unlimited.

One sample is a subset of the population, its size is always limited.

Every question examined in a statistical survey is called characteristic .

The individual answers are called Observation values and are separated according to characteristics in a Original list held. The observation values ​​are given with a i < displaystyle a_> referred to.

Each element of the sample of a statistical survey is a Feature carrier based on the characteristics examined.

## Fermi-Dirac statistics

Fermi-Dirac statistics, Fermi statistics, Quantum statistics for a many-body system consisting of fermions and in equilibrium. In contrast to the classical Maxwell-Boltzmann statistics, here, due to the quantum mechanical indistinguishability of the particles, states that differ only through the exchange of two similar particles are identical and must not be counted as different. The main difference between the Fermi-Dirac statistics and the Bose-Einstein statistics is that the occupation numbers of the individual states, which are unlimited there, can only assume the values ​​0 and 1 here. This is a consequence of the Pauli principle (fermions), according to which two or more fermions can never be in the same quantum state. An ideal Fermi gas, i.e. a system of fermions that do not interact with one another, satisfies the Fermi-Dirac distribution

. It is

the mean number of particles in the state i with the energy E.i, kB. the Boltzmann constant and T the absolute temperature (Fig. 1). The chemical potential μ is determined implicitly from the equation

, i.e. the sum of the particle numbers, summed up over all states i, is equal to the total number of particles N. In the Fermi-Dirac statistics, the chemical potential can also have positive values. Gi is the statistical weight of the state i, i.e. the number of quantum states with the energy E.i. In most applications you are dealing with fermions with a spin 1/2, e.g. with electrons. It then applies Gi = 2 according to the two different spin orientations per energy state.

In the event of E.iμ

kB.T one can neglect the 1 in the denominator, and the Fermi distribution goes over into the Boltzmann distribution. If the above condition is not met, the properties of the gas deviate from the classic ideal gas, and gas degeneration occurs. The course of the occupation number of the individual levels as a function of the energy, omitting the factor Gi, so the function

shows the following behavior: At low temperatures, i.e. with severe degeneration, the fermions predominantly occupy the lower energy states, while the higher energy states are practically empty, with the transition between occupied and empty energy levels in the proximity of the chemical potential depends on the corresponding temperature. In the borderline case T = 0 are all states below the chemical potential with two each (f & # 252r Gi = 2) Fermions occupy the states above the chemical potential, the limit value of which is f & # 252r T = 0 because of this property also as Fermi energy E.F. are denoted are empty. This means that the mean energy of the fermions does not vanish even at absolute zero, it has a finite value (Fig. 2). For free particles, the Fermischen limit energy corresponds to a limit momentum pF.. All fermions in the system have momentum at absolute zero

. This inequality describes a sphere in momentum space which Fermi sphere. The number of unequal momentum states within the Fermi sphere is (4 & # 960pF. 3 / 3)V / H 3, is there V the given volume of the Fermi gas.

The application of the Fermi-Dirac statistics to the electron gas in metals is of particular importance, because this is already in a state of severe degeneration at normal temperatures. The Fermi-Dirac statistics allow practical statements about the metal properties, for example the derivation of Richardson's law of electron emission at glow cathodes. The specific heats of the electrons calculated by means of the Fermi-Dirac statistics could also be confirmed experimentally. At very low temperatures, the electrons provide most of the specific heat of the metal. The specific heats disappear for you T

0 in accordance with the third law of thermodynamics.

The Fermi-Dirac statistics were proposed for electrons by E. Fermi in 1926. Shortly afterwards, P.A.M. Dirac, who uncovered its connection with quantum theory. The first application of the Fermi-Dirac statistics to the electron gas in metals was carried out by W. Pauli and A. Sommerfeld.

Fermi-Dirac statistic 1: Fermi-Dirac distribution for & # 252r T = 0 and kB.T = (1 / 5)μ. Both curves intersect f(E.) = 1 / 2.

Fermi-Dirac statistic 2: Fermi-Dirac distribution for a gas in three dimensions, measured for different temperatures and E.F. / kB. = 50,000 K. Note that & # 223 the curves are not at f(E.) = Cut 1/2.

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Dr. Roger Erb, Kassel [RE1] (A) (33)
Dr. Angelika Fallert-Müller, Groß-Zimmer [AFM] (A) (16, 26)
Stephan Fichtner, Heidelberg [SF] (A) (31)
Dr. Thomas Filk, Freiburg [TF3] (A) (10, 15)
Natalie Fischer, Walldorf [NF] (A) (32)
Dr. Thomas Fuhrmann, Mannheim [TF1] (A) (14)
Christian Fulda, Hanover [CF] (A) (07)
Frank Gabler, Frankfurt [FG1] (A) (22)
Dr. Harald Genz, Darmstadt [HG1] (A) (18)
Prof. Dr. Henning Genz, Karlsruhe [HG2] (A) (Essays Symmetry and Vacuum)
Dr. Michael Gerding, Potsdam [MG2] (A) (13)
Andrea Greiner, Heidelberg [AG1] (A) (06)
Uwe Grigoleit, Weinheim [UG] (A) (13)
Gunther Hadwich, Munich [GH] (A) (20)
Dr. Andreas Heilmann, Halle [AH1] (A) (20, 21)
Carsten Heinisch, Kaiserslautern [CH] (A) (03)
Dr. Marc Hemberger, Heidelberg [MH2] (A) (19)
Dr. Sascha Hilgenfeldt, Cambridge, USA (A) (essay sonoluminescence)
Dr. Hermann Hinsch, Heidelberg [HH2] (A) (22)
Priv.-Doz. Dr. Dieter Hoffmann, Berlin [DH2] (A, B) (02)
Dr. Gert Jacobi, Hamburg [GJ] (B) (09)
Renate Jerecic, Heidelberg [RJ] (A) (28)
Prof. Dr. Josef Kallrath, Ludwigshafen [JK] (A) (04)
Priv.-Doz. Dr. Claus Kiefer, Freiburg [CK] (A) (14, 15)
Dr. Ulrich Kilian, Heidelberg [UK] (A) (19)
Thomas Kluge, Jülich [TK] (A) (20)
Dr. Achim Knoll, Karlsruhe [AK1] (A) (20)
Dr. Alexei Kojevnikov, College Park, USA [AK3] (A) (02)
Dr. Bernd Krause, Munich [BK1] (A) (19)
Dr. Gero Kube, Mainz [GK] (A) (18)
Ralph Kühnle, Heidelberg [RK1] (A) (05)
Volker Lauff, Magdeburg [VL] (A) (04)
Dr. Anton Lerf, Garching [AL1] (A) (23)
Dr. Detlef Lohse, Twente, NL (A) (essay sonoluminescence)
Priv.-Doz. Dr. Axel Lorke, Munich [AL] (A) (20)
Prof. Dr. Jan Louis, Halle (A) (essay string theory)
Dr. Andreas Markwitz, Lower Hutt, NZ [AM1] (A) (21)
Holger Mathiszik, Celle [HM3] (A) (29)
Dr. Dirk Metzger, Mannheim [DM] (A) (07)
Dr. Rudi Michalak, Dresden [RM1] (A) (23 essay low temperature physics)
Günter Milde, Dresden [GM1] (A) (12)
Helmut Milde, Dresden [HM1] (A) (09)
Marita Milde, Dresden [MM2] (A) (12)
Prof. Dr. Andreas Müller, Trier [AM2] (A) (33)
Prof. Dr. Karl Otto Münnich, Heidelberg (A) (Essay Environmental Physics)
Dr. Nikolaus Nestle, Leipzig [NN] (A, B) (05, 20)
Dr. Thomas Otto, Geneva [TO] (A) (06)
Priv.-Doz. Dr. Ulrich Parlitz, Göttingen [UP1] (A) (11)
Christof Pflumm, Karlsruhe [CP] (A) (06, 08)
Dr. Oliver Probst, Monterrey, Mexico [OP] (A) (30)
Dr. Roland Andreas Puntigam, Munich [RAP] (A) (14)
Dr. Gunnar Radons, Mannheim [GR1] (A) (01, 02, 32)
Dr. Max Rauner, Weinheim [MR3] (A) (15)
Robert Raussendorf, Munich [RR1] (A) (19)
Ingrid Reiser, Manhattan, USA [IR] (A) (16)
Dr. Uwe Renner, Leipzig [UR] (A) (10)
Dr. Ursula Resch-Esser, Berlin [URE] (A) (21)
Dr. Peter Oliver Roll, Ingelheim [OR1] (A, B) (15)
Hans-Jörg Rutsch, Walldorf [HJR] (A) (29)
Rolf Sauermost, Waldkirch [RS1] (A) (02)
Matthias Schemmel, Berlin [MS4] (A) (02)
Prof. Dr. Erhard Scholz, Wuppertal [ES] (A) (02)
Dr. Martin Schön, Konstanz [MS] (A) (14 essay special theory of relativity)
Dr. Erwin Schuberth, Garching [ES4] (A) (23)
Jörg Schuler, Taunusstein [JS1] (A) (06, 08)
Dr. Joachim Schüller, Dossenheim [JS2] (A) (10)
Richard Schwalbach, Mainz [RS2] (A) (17)
Prof. Dr. Klaus Stierstadt, Munich [KS] (B)
Dr. Siegmund Stintzing, Munich [SS1] (A) (22)
Dr. Berthold Suchan, Giessen [BS] (A) (Essay Philosophy of Science)
Cornelius Suchy, Brussels [CS2] (A) (20)
Dr. Volker Theileis, Munich [VT] (A) (20)
Prof. Dr. Stefan Theisen, Munich (A) (essay string theory)
Dr. Annette Vogt, Berlin [AV] (A) (02)
Dr. Thomas Volkmann, Cologne [TV] (A) (20)
Rolf vom Stein, Cologne [RVS] (A) (29)
Dr. Patrick Voss-de Haan, Mainz [PVDH] (A) (17)
Dr. Thomas Wagner, Heidelberg [TW2] (A) (29)
Manfred Weber, Frankfurt [MW1] (A) (28)
Dr. Martin Werner, Hamburg [MW] (A) (29)
Dr. Achim Wixforth, Munich [AW1] (A) (20)
Dr. Steffen Wolf, Berkeley, USA [SW] (A) (16)
Dr. Stefan L. Wolff, Munich [SW1] (A) (02)
Priv.-Doz. Dr. Jochen Wosnitza, Karlsruhe [JW] (A) (23)
Dr. Kai Zuber, Dortmund [KZ] (A) (19)
Dr. Werner Zwerger, Munich [WZ] (A) (20)

## For lecturers

### Activities submitted by instructors

title Authors level Type topic
Concept questions for physics using PhET (Inquiry Based) Trish Loeblein Uni-Vor
S II
MC physics
Games Remote Lesson ideas Trish Loeblein S II
GS
S I
Uni-Vor
HA
On-line
chemistry
physics
mathematics
Balancing Chemical Equations - Inquiry based Introduction Trish Loeblein S I
Uni-Vor
S II
Practical
On-line
HA
MC
chemistry
Balancing Chemical Equations - Guided Inquiry Activity Timothy Herzog, Yuen-ying Carpenter Uni-Vor HA
Guided
On-line
chemistry
Balancing Chemical Equations - Clicker Questions Yuen-ying Carpenter, Robert Parson, Trish Loeblein Uni-Vor MC chemistry
Balancing Chemical Equations HTML5 Elyse room S II Practical chemistry
Alignment of PhET sims with NGSS Trish Loeblein updated by Diana López S I Otherwise physics
earth sciences
biology
chemistry
PhET Sims Aligned to the Chemistry Curriculum Julia Chamberlain Uni-Vor
S II
Otherwise chemistry
How do PhET simulations fit in my middle school program? Sarah Borenstein S I Otherwise earth sciences
physics
biology
chemistry
SECUNDARIA: Alineación PhET con programas de la SEP México (2011 y 2017) Diana Lopez S I
S II
Otherwise biology
mathematics
chemistry
physics
PREPARATORIA: Alineación de PhET con programas de la DGB México (2017) Diana Lopez Uni-Vor
S II
Otherwise physics
chemistry
mathematics
Preguntas de razonamiento para todas las simulaciones HTML5 Diana Lopez University main
Uni-Vor
S I
S II
University degree
GS
discussion
HA
astronomy
mathematics
chemistry
physics
MS and HS TEK to Sim Alignment Elyse room S II
S I
Otherwise physics
biology
chemistry
Balancing Chemical Equations Christoffer Becho S II
S I
Guided chemistry
Balancing Chemical Reactions Game Jeff Drach S II Otherwise chemistry
Explore Balancing Chemical Equations Patricia Garcia S I Guided chemistry
Student Guide for PhET - Balancing Chemical Equations in html5 Brian Libby S I
S II
HA
Guided
chemistry
Balancing Equations PhET Kristine McDaniel S II Guided chemistry
High School Chemistry 1 level Balancing Equations laura peck S II Practical chemistry
Balancing Chemical Equations Inquiry Lab Tony Tracy S II Practical chemistry
Moodle et PhetColorado Claude Divoux S II MC
Practical
Guided
chemistry
physics
Bilanciamento di reazioni chimiche___ITALIANO Natasha Samousenko S II
S I
Practical chemistry
Ajust equacions químiques Miriam Pujadas S II HA chemistry
Atividade sobre o balanceamento de equações químicas a partir da Lei de Lavoisier Carina Maria Rodrigues Lima, Maria Cleide da Silva Barroso S II demo
Guided
HA
chemistry
Uma proposta de interdisciplinaridade Ana Carla Pimentel Paiva Francisco Régis Vieira Alves S I
S II
Guided chemistry
mathematics
Roteiro para Estudo sobre Balanceamento de Equações Químicas, Utilizando o Simulado. Antônia Clarycy Barros Nojosa e Caroline de Goes Sampaio S II Guided chemistry
Balanceamento de equações químicas Victor Augusto Bianchetti Rodrigues S II Guided chemistry
Balanceamento, Leis Ponderais e Modelo de Dalton DANILO JOSE F PINTO S II Guided
HA
Practical
chemistry
Rätta Kemiska Ekvationer - steg för steg Timothy Herzog, Yuen-ying Carpenter translated by Sofia Borg S I
S II
Practical chemistry
INTRODUCCIÓN AL BALANCEO DE ECUACIONES A TRAVÉS DE UN SIMULADOR PhET. Diana Mestra S II HA
On-line
Guided
Practical
demo
chemistry
Balancear ecuaciones INES VEGA FABRA S II Guided
demo
chemistry
Actividad Balanceando Ecuaciones Químicas Mary Rivas S II Guided chemistry
Balanceo de ecuaciones químicas por tanteo andis pacheco S I HA chemistry
Guía Balanceo de ecuaciones químicas Andrea Ortega Fernández S I Guided chemistry
Balanceando Ecuaciones Elvis Rivera S II Practical chemistry

## Global production volume of chemical and textile fibers by 2020

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Textile fibers are made up of the total production volume of chemical, wool and cotton fibers.

Older values ​​were partly taken from the corresponding publications from the previous year.

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Glass, ceramic & plastic industry

Plastic production worldwide and in Europe by 2019

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Top-selling chemical companies in Germany 2005-2019

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Chemical companies with the highest turnover worldwide in 2020

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Glass, ceramic & plastic industry

Use of plastic in Germany in 2019, by area of ​​application

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