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Complex analytic space

2012-10-20T01:31:35Z

75.36.201.51: /* Definition */

A '''gas pycnometer''' is a laboratory device used for measuring the [[density]] — or more accurately the [[volume]] — of solids, be they regularly shaped, [[Porous medium|porous]] or non-porous, [[Single crystal|monolithic]], [[Powder (substance)|powdered]], [[Granular material|granular]] or in some way [[Comminution|comminuted]], employing some method of gas displacement<ref>{{US patent|2667782}} Shea, "Apparatus for measuring volumes of solid materials".</ref><ref>{{US patent|4083228}} Turner ''et al.'', "Gas comparison pycnometer".</ref><ref>{{US patent|4888718}} Furuse, "Volume measuring apparatus and method"</ref> and the volume:pressure relationship known as [[Boyle's Law]]. A gas pycnometer is also sometimes referred to as a helium pycnometer.

==Types of gas pycnometer==
===Gas expansion pycnometer===
Gas expansion pycnometer is also known as constant volume gas pycnometer. The simplest type of gas pycnometer (due to its relative lack of moving parts) consists of two chambers, one (with a removable gas-tight lid) to hold the sample and a second chamber of fixed, known (via [[calibration]]) internal volume – referred to as the reference volume or added volume. The device additionally comprises a [[valve]] to admit a gas under pressure to one of the chambers, a pressure measuring device – usually a [[Pressure sensor|transducer]] – connected to the first chamber, a valved pathway connecting the two chambers, and a valved vent from the second of the chambers. In practice the sample may occupy either chamber, that is gas pycnometers can be constructed such that the sample chamber is pressurized first, or such that it is the reference chamber that starts at the higher pressure. Various design parameters have been analyzed by Tamari.<ref>S. Tamari (2004) ''Meas. Sci. Technol''. '''15''' 549–558 "Optimum design of the constant-volume gas pycnometer for determining the volume of solid particles" {{DOI|10.1088/0957-0233/15/3/007}}</ref> The working equation of a gas pycnometer wherein the sample chamber is pressurized first is as follows:

::<math>V_{s} = V_{c} + \frac{ V_{r}} {1-\frac{P_{1}}{P_{2}}}</math>

where ''Vs'' is the sample volume, ''Vc'' is the volume of the empty sample chamber (known from a prior calibration step), ''Vr'' is the volume of the reference volume (again known from a prior calibration step), ''P''1 is the first pressure (i.e. in the sample chamber only) and ''P''2 is the second (lower) pressure after expansion of the gas into the combined volumes of sample chamber and reference chamber.

Derivation of the "working equation" and a schematic illustration of such a gas expansion pycnometer is given by Lowell ''et al.''.<ref>S. Lowell, J.E. Shields, M.A. Thomas and M. Thommes "Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density", Springer (originally by Kluwer Academic Publishers), 2004 ISBN 978-1-4020-2302-6 p. 327</ref>

===Variable volume pycnometer===
Variable volume pycnometer (or gas comparison pycnometer) consists of either a single or two variable volume chambers. The volume of the chamber(s) can be varied by either a fixed amount by a simple mechanical piston of fixed travel, or continuously and gradually by means of a graduated piston. Resulting changes in pressure can be read by means of a transducer, or nullified by adjustment of a third ancillary, graduated variable-volume chamber. This type of pycnometer is commercially obsolete; in 2006 [[ASTM]] withdrew its standard test method D2856<ref>ASTM D2856-94(1998) Standard Test Method for Open-Cell Content of Rigid Cellular Plastics by the Air Pycnometer (withdrawn in 2006).</ref> for the open-cell content of rigid cellular plastics by the air pycnometer, which relied upon the use of a variable volume pycnometer, and was replaced by test method D6226<ref name=astmd6226>ASTM D6226-05 Standard Test Method for Open Cell Content of Rigid Cellular Plastics.</ref> which describes a gas expansion pycnometer.

==Practical use==
===Volume vs density===
While pycnometers (of any type) are recognized as [[density]] measuring devices they are in fact devices for measuring volume only. Density is merely calculated as the ratio of [[mass]] to volume; mass being invariably measured on a discrete device, usually by [[Weight|weighing]]. The volume measured in a gas pycnometer is that amount of three-dimensional space which is inaccessible to the gas used, i.e. that volume within the sample chamber from which the gas is excluded. Therefore the volume measured considering the finest scale of [[Rugosity|surface roughness]] will depend on the atomic or molecular size of the gas. [[Helium]] therefore is most often prescribed as the measurement gas, not only is it of small size, it is also inert and the most [[Ideal gas law|ideal gas]].

Closed pores, i.e. those that do not communicate with the surface of the solid, are included in the measured volume. Helium may however demonstrate some measurable [[Permeation|permeability]] through low density solids ([[polymer]]s and [[Cellulose|cellulosic]] materials predominantly) thus interfering with the measurement of solid volume. In such cases larger molecule gases such as [[nitrogen]] or [[sulfur hexafluoride]] are beneficial.

[[Adsorption]] of the measuring gas should be avoided, as should excessive [[vapor pressure]] from moisture or other liquids present in the otherwise solid sample.

===Applications===
Gas pycnometers are used extensively for characterizing a wide variety of solids such as [[Heterogeneous catalysis|heterogeneous catalysts]], [[carbon]]s,<ref>
DIN 51913 Testing of carbon materials – Determination of density by gas pycnometer (volumetric) using helium as the measuring gas</ref> [[Powder metallurgy|metal powders]],<ref>ASTM B923-02(2008)Standard Test Method for Metal Powder Skeletal Density by Helium or Nitrogen Pycnometry</ref><ref>MPIF Standard 63: Method for Determination of MIM Components (Gas Pycnometer)</ref> soils,<ref>ASTM D5550 -06 Standard Test Method for Specific Gravity of Soil Solids by Gas Pycnometer</ref> [[ceramic]]s,<ref>ASTM C604 Standard Test Method for True Specific Gravity of Refractory Materials by Gas-Comparison Pycnometer</ref> [[Active ingredient|active pharmaceutical ingredients]] (API's) and [[excipient]]s,<ref>USP<699> "Density of Solids"</ref> [[petroleum coke]],<ref>ASTM D2638 – 06 Standard Test Method for Real Density of Calcined Petroleum Coke by Helium Pycnometer</ref> [[cement]] and other construction materials,<ref>C. Hall "Water Transport in Brick, Stone and Concrete", Taylor & Francis, 2002, ISBN 978-0-419-22890-5 p. 13 </ref> [[cenospheres]]/[[Glass microsphere|glass microballoons]] and [[Foam|solid foams]].<ref name=astmd6226/>

==Notes==
*Pycnometer is the preferred spelling in modern [[American English]] usage. Pyknometer is to be found in older texts, and is used interchangeably with pycnometer in [[British English]]. Pyknometer is preferred in [[European English (disambiguation)|European English]]. The term has its origins in the Greek word πυκνός, meaning "dense".
*The density calculated from the volume measured by a gas pycnometer is often referred to as ''skeletal'' density,<ref>D. Sangeeta and J. R. Lagraff "Inorganic materials chemistry desk reference", CRC Press, 2005, ISBN 978-0-8493-0910-6 p. 103</ref><ref name=water>N. P. Cheremisinoff "Handbook of Water and Wastewater Treatment Technologies", Butterworth-Heinemann, 2001, ISBN 978-0-7506-7498-0 p. 144 </ref> ''true'' density <ref>P. J. Sinko and A. N. Martin "Martin's Physical Pharmacy and Pharmaceutical Sciences, 5th Edition, Lippincott Williams & Wilkins, 2005, ISBN 978-0-7817-5027-1 p. 544</ref><ref name=speight>J. G. Speight "The Chemistry and Technology of Coal" CRC Press, 1994, ISBN 978-0-8247-9200-8 p. 202</ref> or ''helium'' density<ref name=speight/> and sometimes as [[particle density]] for non-porous solids.<ref name=water/>
*One of the most extreme examples of the gas displacement principle for volume measurement is described in {{US patent|5231873}} (Lindberg, 1993) wherein a chamber large enough to hold a [[flatbed truck]] full of [[Lumber|timber]] is described for the purpose of measuring the volume of the load.

==References==
{{reflist|35em}}

==External links==
*[http://www.astm.org ASTM International], formerly known as the American Society for Testing and Materials.
*[http://www.uspto.gov United States Patent and Trademark Office].
*[http://www.mpif.org/ MPIF]: Metal Powder Industries Federation.
*[http://www.usp.org/ USP]: United States Pharmacopeia.
*[http://www.din.de/cmd?lang=en&level=tpl-home&contextid=din&languageid=en DIN] Deutsches Institut für Normung e.V. (German Institute for Standardization).

{{DEFAULTSORT:Gas Pycnometer}}
[[Category:Laboratory equipment]]

Selmer group

2012-10-12T04:50:16Z

75.36.201.51:

[[Image:Spurious Correlation (Greyscale).svg|thumb|300px|right|An illustration of spurious correlation, this figure shows 500 observations of x/z plotted against y/z. The sample correlation is 0.53, even though x, y, and z are statistically independent of each other (i.e., the pairwise correlations between each of them are zero).]]
[[Image:Spurious Correlation (Colour).svg|thumb|300px|right|This figure shows the 500 observations of y/z plotted against x/z from above, this time with the z-values on a colour scale to highlight how dividing through by z induces spurious correlation.]]

'''Spurious correlation''' is a term coined by [[Karl Pearson]] to describe the [[correlation]] between ''ratios'' of absolute measurements that arises as a consequence of using ratios, rather than because of any actual correlations between the measurements.<ref name="Pearson">{{cite journal|last=Pearson|first=Karl|title=Mathematical Contributions to the Theory of Evolution—On a Form of Spurious Correlation Which May Arise When Indices Are Used in the Measurement of Organs|journal=Proceedings of the Royal Society of London|year=1897|volume=60|pages=489–498|url=http://www.jstor.org/stable/115879}}</ref>

The phenomenon of spurious correlation is one of the main motives for the field of [[Compositional data|compositional data analysis]] which deals with the analysis of variables that carry only ''relative'' information, such as proportions, percentages and parts-per-million.<ref name="Aitchison">{{cite book|last=Aitchison|first=John|title=The statistical analysis of compositional data|year=1986|publisher=Chapman & Hall|isbn=0-412-28060-4}}</ref><ref>{{cite book|title=Compositional Data Analysis: Theory and Applications|year=2011|publisher=Wiley|isbn=9780470711354|url=http://onlinelibrary.wiley.com/book/10.1002/9781119976462|editor1-first=Vera|editor1-last=Pawlowsky-Glahn|editor2-first=Antonella |editor2-last=Buccianti}}</ref>

Pearson's definition of spurious correlation is distinct from (and should not be confused with) misconceptions about [[Correlation and dependence#Correlation and causality|correlation and causality]], or the term [[spurious relationship]].

==Illustration of spurious correlation==

Pearson states a simple example of spurious correlation:<ref name="Pearson" />
{{Quote|Select three numbers within certain ranges at random, say ''x'', ''y'', ''z'', these will be pair and pair uncorrelated. Form the proper fractions ''x''/''y'' and ''z''/''y'' for each triplet, and correlation will be found between these indices.}}

The upper scatter plot on the right illustrates this example using 500 observations of ''x'', ''y'', and ''z''. Variables ''x'', ''y'' and ''z'' are drawn from normal distributions with means 10, 10 and 30, respectively, and standard deviation 10, i.e.,

: <math>\begin{align}
x,y & \sim N(10,\sqrt{10}) \\
z & \sim N(30,\sqrt{10}) \\
\end{align}</math>

Even though ''x'', ''y'', and ''z'' are statistically independent (i.e., pairwise uncorrelated), the ratios x/z and y/z have a sample correlation of 0.53. This is because of the common divisor (''z'') and can be better understood if we colour the points in the scatter plot by the ''z''-value. Trios of (''x'', ''y'', ''z'') with relatively large ''z'' values tend to appear in the bottom left of the plot; trios with relatively small ''z'' values tend to appear in the top right.

==Approximate amount of spurious correlation==
Pearson derived an approximation of the correlation that would be observed between two indices (<math>x_1/x_3</math> and <math>x_2/x_4</math>), i.e., ratios of the absolute measurements <math>x_1, x_2, x_3, x_4</math>:

: <math>
\rho = \frac{r_{12} v_1 v_2 - r_{14} v_1 v_4 - r_{23} v_2 v_3 + r_{24} v_2 v_4}{\sqrt{v_1^2 + v_3^2 - 2 r_{13} v_1 v_3} \sqrt{v_2^2 + v_4^2 - 2 r_{24} v_2 v_4}}
</math>

where <math>v_i</math> is the [[coefficient of variation]] of <math>x_i</math>, and <math>r_{ij}</math> the [[Correlation#Pearson.27s product-moment coefficient|Pearson correlation]] between <math>x_i</math> and <math>x_j</math>.

This expression can be simplified for situations where there is a common divisor by setting <math>x_3=x_4</math>, and <math>x_1, x_2, x_3</math> are uncorrelated, giving the spurious correlation:

: <math>
\rho_0 = \frac{v_3^2}{\sqrt{v_1^2 + v_3^2} \sqrt{v_2^2 + v_3^2}}.
</math>

For the special case in which all coefficients of variation are equal (as is the case in the illustrations at right), <math>\rho_0 = 0.5 </math>

==Relevance to biology and other sciences==
Pearson was joined by [[Galton|Sir Francis Galton]] and [[Walter Frank Raphael Weldon]] in cautioning scientists to be wary of spurious correlation, especially in biology where it is common to scale or [[Normalization (statistics)|normalize]] measurements by dividing them by a particular variable or total. The danger he saw was that conclusions would be drawn from correlations that are artifacts of the analysis method, rather than actual “organic” relationships.

However, it would appear that spurious correlation (and its potential to mislead) is not yet widely understood. In 1986 [[John Aitchison]], who pioneered the log-ratio approach to [[Compositional data|compositional data analysis]] wrote:<ref name="Pearson" />
{{Quote|It seems surprising that the warnings of three such eminent statistician-scientists as Pearson, Galton and Weldon should have largely gone unheeded for so long: even today uncritical applications of inappropriate statistical methods to compositional data with consequent dubious inferences are regularly reported.}}
More recent publications suggest that this lack of awareness prevails, at least in molecular bioscience.<ref>{{cite book|first1=David|last1=Lovell|first2=Warren|last2=Müller|first3=Jen|last3=Taylor|first4=Alec|last4=Zwart|first5=Chris|last5=Helliwell|chapter=Chapter 14: Proportions, Percentages, PPM: Do the Molecular Biosciences Treat Compositional Data Right?|title=Compositional Data Analysis: Theory and Applications|year=2011|publisher=Wiley|isbn=9780470711354|url=http://onlinelibrary.wiley.com/book/10.1002/9781119976462|editor1-first=Vera|editor1-last=Pawlowsky-Glahn|editor2-first=Antonella |editor2-last=Buccianti}}</ref>

==References==
{{reflist}}

[[Category:Covariance and correlation]]