Rational expectations: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>BattyBot
changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)
 
en>Werddemer
Line 1: Line 1:
Underweight people normally have a BMI below 18.5, when a general BMI is between 18.5 plus 24.9. Overweight people have a BMI between 25 and 29.9, as well as the BMI of a overweight individual is over 30. A BMI better than 40 is considered morbidly overweight. BMI refuses to account for muscle mass, that is heavier than fat, plus consequently can give questionable results in females that are expecting or breastfeeding, competitive athletes, body-builders and people with chronic illnesses.<br><br><br><br>It's certainly a big choice starting about a modern weight reduction program. Excitement, anticipation, a little of worry plus apprehension. yet by following the simple steps 1 will reach one's objectives plus gain self esteem and confidence inside the process.I hope these simple strategies can motivate we to begin on a sensible fat reduction system plus aid you inside making the appropriate decisions before beginning a fat reduction program.<br><br>Should you truly want to work through the condition for calculations of BMI, then you're recommended to use the online [http://safedietplans.com/calories-burned-walking calories burned while walking] that will help you calculate a BMI with a height and fat. It is very easy to work this tool in order to calculate the BMI. All you ought to insert the weight and height plus it will provide you the actual happen of your BMI.<br><br>Through the BMI index to look for the harshness of weight reduction, individuals begins to formulate any adverse wellness healthy diet plan incorporating regular exercise to reduce weight plus eliminate the precarious of wanting a debilitating condition resulting from weight problems.<br><br>A common trend diet is not commonly meant for lasting weight reduction because it refuses to deal with the principal problem, that might be bodily or psychological. The diets usually last from 24 hours to 1 week and serve basic purposes, like, wanting to fit into a particular dress or the favorite jeans.<br><br>Believe it or not, not all diet plans and exercise programs have the same objective. Some of them are developed to aid we lose fat while others are developed to make we gain muscles. This really is the cause why you should be well-aware of the actual requirements of your body? Do we need to gain more fat? If you try to acquire muscles after losing weight? How long should you follow the diet or a exercise regimen? We can be able to answer all of these in the event you calculate a BMI.<br><br>Like each direction of thumb, there are a few exceptions. Body builders often have a BMI above 25, or 25. In that case, an elevated Body Mass Index will reflect the fact which body builders often have more muscle, without a surplus of fat.
[[File:Boyles Law animated.gif|thumb|300px|An animation showing the relationship between pressure and volume when amount and temperature are held constant.]]
{{Continuum mechanics}}
'''Boyle's law''' (sometimes referred to as the '''Boyle–Mariotte law''', or '''Mariotte's law'''<ref>{{cite book|last=Draper|first=John William|title=A Textbook on chemistry|year=1861|page=46|url=http://books.google.co.uk/books?id=HKwS7QDh5eMC&pg=PA1&dq=draper,+john+william&source=gbs_toc_r&cad=4#v=onepage&q&f=false}}</ref>) is an experimental [[gas laws|gas law]] which describes how the [[pressure]] of a [[gas]] tends to decrease as the [[volume]] of a gas increases. A modern statement of Boyle's law is:
<blockquote>
The absolute [[pressure]] exerted by a given mass of an [[ideal gas]] is inversely proportional to the [[volume]] it occupies if the [[temperature]] and [[amount of substance|amount of gas]] remain unchanged within a [[closed system]].<ref>Levine, Ira. N (1978). "Physical Chemistry" University of Brooklyn: [[McGraw-Hill]]</ref><ref name="levine_1">Levine, Ira. N. (1978), p12 gives the original definition.</ref>
</blockquote>
Mathematically, Boyle's law can be stated as
:<math>P \propto \frac{1}{V}</math>
or
:<math> PV = k,</math>
 
 
where ''P'' is the [[pressure]] of the gas, ''V'' is the [[volume]] of the gas, and ''k'' is a [[Constant (mathematics)|constant]].
 
The equation states that product of pressure and volume is a constant for a given mass of confined gas as long as the temperature is constant. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as
:<math>P_1 V_1 = P_2 V_2.</math>
 
The equation shows that, as volume increases, the pressure of the gas decreases in proportion. Similarly, as volume decreases, the pressure of the gas increases. The law was named after [[chemistry|chemist]] and [[physicist]] [[Robert Boyle]], who published the original law in 1662.<ref>J Appl Physiol 98: 31-39, 2005. Free download at [http://jap.physiology.org/cgi/content/full/98/1/31 Jap.physiology.org]</ref>
 
==History==
{{main|History of thermodynamics}}
[[File:Legge di Boyle dati originali.jpg|thumb|A graph of Boyle's original data{{cn|date=April 2012}}]]
This relationship between pressure and volume was first noted by two new{{clarify|date=December 2013}} scientists, [[Richard Towneley]] and [[Henry Power]]. [[Robert Boyle]] confirmed their discovery through experiments and published the results. According to [[Robert Gunther]] and other authorities, it was Boyle's assistant, [[Robert Hooke]], who built the experimental apparatus.  Boyle's law is based on experiments with air, which he considered to be a fluid of particles at rest in between small invisible springs. At that time, air was still seen as one of the four elements, but Boyle disagreed. Boyle's interest was probably to understand air as an essential element of life;<ref>The Boyle Papers BP 9, fol. 75v-76r at [http://www.bbk.ac.uk/boyle/boyle_papers/bp09_docs/bp9_075v-076r.htm BBK.ac.uk]</ref> for example, he published works on the growth of plants without air.<ref>The Boyle Papers, BP 10, fol. 138v-139r at [http://www.bbk.ac.uk/boyle/boyle_papers/bp10_docs/bp10_138v-139r.htm BBK.ac.uk]</ref> The French physicist [[Edme Mariotte]] (1620–1684) discovered the same law independently of Boyle in 1676,<ref>{{cite book |last1=Greiner |first1=Walter |last2=Neise |first2=Ludwig |last3=Stöcker |first3=Horst |authorlink3=Horst Stöcker |title=Thermodynamics and Statistical Mechanics |year=1997 |publisher=Springer |isbn=0-3-87-94299-8|url=http://www.springer.com/physics/classical+continuum+physics/book/978-0-387-94299-5 |edition=Corr. 2nd printing |accessdate=22 November 2012|page=9|quote=Already in 1664, R. Boyle, and shortly later (1676) but independently from him, E. Mariotte found a general relationship between the pressure and the volume of a gas at constant temperature...}}</ref> but Boyle had already published it in 1662. Thus this law may, improperly, be referred to as Mariotte's or the Boyle–Mariotte law. Later, in 1687 in the [[Philosophiæ Naturalis Principia Mathematica]],  [[Isaac Newton|Newton]] showed mathematically that if an elastic fluid consisting of particles at rest, between which are repulsive forces inversely proportional to their distance, the density would be directly proportional to the pressure,<ref>Principia, Sec.V,prop. XXI, Theorem XVI</ref> but this mathematical treatise is not the physical explanation for the observed relationship. Instead of a static theory a kinetic theory is needed, which was provided two centuries later by [[James Clerk Maxwell|Maxwell]] and [[Ludwig Boltzmann|Boltzmann]].
 
== Definition ==
The law itself can be stated as follows:
 
{{quotation|For a fixed amount of an [[ideal gas]] kept at a fixed temperature, pressure and volume are inversely proportional.<ref name="levine_1" />}}
Or Boyle's law is a gas law, stating that the pressure and volume of a gas have an inverse relationship, when temperature is held constant. If volume increases, then pressure decreases and vice versa, when temperature is held constant.
 
Therefore when the volume is halved, the pressure is doubled; and if the volume is doubled, the pressure is halved.
 
=== Relation to kinetic theory and ideal gases===
Boyle’s law states that ''at constant temperature'' for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional.  The law can also be stated in a slightly different manner, that the product of absolute pressure and volume is always constant.
 
Most gases behave like [[ideal gas]]es at moderate pressures and temperatures.  The technology of the 17th century could not produce high pressures or low temperatures. Hence, the law was not likely to have deviations at the time of publication. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing [[real gas]] theory.<ref name="levine_2">Levine, Ira. N. (1978), p11 notes that deviations occur with high pressures and temperatures.</ref> The deviation is expressed as the [[compressibility factor]].
 
Boyle (and  Mariotte) derived the law solely on experimental grounds.  The law can also be derived theoretically based on the presumed existence of [[atom]]s and [[molecule]]s and assumptions about motion and perfectly elastic collisions (see [[kinetic theory of gases]]).  These assumptions were met with enormous resistance in the [[positivist]] scientific community at the time however, as they were seen as purely theoretical constructs for which there was not the slightest observational evidence. 
 
[[Daniel Bernoulli]] in 1737-1738 derived Boyle's law using [[Newton's laws of motion]] with application on a molecular level. It remained ignored until around 1845, when [[John Waterston]] published a paper building the main precepts of kinetic theory; this was rejected by the [[Royal Society of England]].  Later works of [[James Prescott Joule]], [[Rudolf Clausius]] and in particular Ludwig Boltzmann firmly established the [[kinetic theory of gases]] and brought attention to both the theories of Bernoulli and Waterston.<ref name="levine_3">Levine, Ira. N. (1978), p400 – Historical background of Boyle's law relation to Kinetic Theory</ref>
 
The debate between proponents of [[Energetics]] and [[Atomism]] led Boltzmann to write a book in 1898, which endured criticism up to his suicide in 1906.<ref name="levine_3" /> [[Albert Einstein]] in 1905 showed how kinetic theory applies to the [[Brownian motion]] of a fluid-suspended particle, which was confirmed in 1908 by [[Jean Perrin]].<ref name="levine_3" />
 
===Equation===
The mathematical equation for Boyle's law is:
 
:<math> PV = k </math>
 
where:
* ''P'' denotes the [[pressure]] of the system.
* ''V'' denotes the [[volume]] of the gas.
* ''k'' is a constant value representative of the pressure and volume of the system.
 
So long as [[temperature]] remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of ''k'' will remain constant. However, due to the derivation of pressure as perpendicular applied force and the probabilistic likelihood of collisions with other particles through [[collision theory]], the application of force to a surface may not be infinitely constant for such values of ''k'', but will have a [[limit (mathematics)|limit]] when [[differential calculus|differentiating]] such values over a given time.{{huh|date=December 2013}}
 
Forcing the volume ''V'' of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure ''p'' must decrease proportionally. Conversely, reducing the volume of the gas increases the pressure.  
 
Boyle's law is used to predict the result of introducing a change, in volume and pressure only, to the initial state of a fixed quantity of gas. The before and after volumes and pressures of the fixed amount of gas, where the before and after temperatures are the same (heating or cooling will be required to meet this condition), are related by the equation:
 
: <math>P_1 V_1 = P_2 V_2. \,</math>
 
Here ''P''<sub>1</sub> and ''V''<sub>1</sub> represent the original pressure and volume, respectively, and ''P''<sub>2</sub> and ''V''<sub>2</sub> represent the second pressure and volume.
 
Boyle's law, [[Charles's law]], and [[Gay-Lussac's law#Pressure-temperature law|Gay-Lussac's law]] form the [[combined gas law]]. The three gas laws in combination with [[Avogadro's law]] can be generalized by the [[ideal gas law]].
 
==See also==
{{portal|underwater diving}}
*[[Charles's law]]
*[[Gay-Lussac's law]]
*[[Dalton's law]]
*[[Combined gas law]]
*[[Ideal gas law]]
 
==References==
{{Reflist}}
 
==External links==
* [http://www.webqc.org/boyle_gas_law.html Boyle's gas law calculator]
 
{{Diving medicine, physiology and physics}}
 
{{DEFAULTSORT:Boyle's Law}}
[[Category:Gas laws]]
[[Category:Volume]]
 
[[de:Thermische Zustandsgleichung idealer Gase#Gesetz von Boyle-Mariotte]]

Revision as of 20:14, 26 December 2013

An animation showing the relationship between pressure and volume when amount and temperature are held constant.

Template:Continuum mechanics Boyle's law (sometimes referred to as the Boyle–Mariotte law, or Mariotte's law[1]) is an experimental gas law which describes how the pressure of a gas tends to decrease as the volume of a gas increases. A modern statement of Boyle's law is:

The absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain unchanged within a closed system.[2][3]

Mathematically, Boyle's law can be stated as

or


where P is the pressure of the gas, V is the volume of the gas, and k is a constant.

The equation states that product of pressure and volume is a constant for a given mass of confined gas as long as the temperature is constant. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as

The equation shows that, as volume increases, the pressure of the gas decreases in proportion. Similarly, as volume decreases, the pressure of the gas increases. The law was named after chemist and physicist Robert Boyle, who published the original law in 1662.[4]

History

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

A graph of Boyle's original dataTemplate:Cn

This relationship between pressure and volume was first noted by two newTemplate:Clarify scientists, Richard Towneley and Henry Power. Robert Boyle confirmed their discovery through experiments and published the results. According to Robert Gunther and other authorities, it was Boyle's assistant, Robert Hooke, who built the experimental apparatus. Boyle's law is based on experiments with air, which he considered to be a fluid of particles at rest in between small invisible springs. At that time, air was still seen as one of the four elements, but Boyle disagreed. Boyle's interest was probably to understand air as an essential element of life;[5] for example, he published works on the growth of plants without air.[6] The French physicist Edme Mariotte (1620–1684) discovered the same law independently of Boyle in 1676,[7] but Boyle had already published it in 1662. Thus this law may, improperly, be referred to as Mariotte's or the Boyle–Mariotte law. Later, in 1687 in the Philosophiæ Naturalis Principia Mathematica, Newton showed mathematically that if an elastic fluid consisting of particles at rest, between which are repulsive forces inversely proportional to their distance, the density would be directly proportional to the pressure,[8] but this mathematical treatise is not the physical explanation for the observed relationship. Instead of a static theory a kinetic theory is needed, which was provided two centuries later by Maxwell and Boltzmann.

Definition

The law itself can be stated as follows:

36 year-old Diving Instructor (Open water ) Vancamp from Kuujjuaq, spends time with pursuits for instance gardening, public listed property developers in singapore developers in singapore and cigar smoking. Of late took some time to go China Danxia. Or Boyle's law is a gas law, stating that the pressure and volume of a gas have an inverse relationship, when temperature is held constant. If volume increases, then pressure decreases and vice versa, when temperature is held constant.

Therefore when the volume is halved, the pressure is doubled; and if the volume is doubled, the pressure is halved.

Relation to kinetic theory and ideal gases

Boyle’s law states that at constant temperature for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional. The law can also be stated in a slightly different manner, that the product of absolute pressure and volume is always constant.

Most gases behave like ideal gases at moderate pressures and temperatures. The technology of the 17th century could not produce high pressures or low temperatures. Hence, the law was not likely to have deviations at the time of publication. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory.[9] The deviation is expressed as the compressibility factor.

Boyle (and Mariotte) derived the law solely on experimental grounds. The law can also be derived theoretically based on the presumed existence of atoms and molecules and assumptions about motion and perfectly elastic collisions (see kinetic theory of gases). These assumptions were met with enormous resistance in the positivist scientific community at the time however, as they were seen as purely theoretical constructs for which there was not the slightest observational evidence.

Daniel Bernoulli in 1737-1738 derived Boyle's law using Newton's laws of motion with application on a molecular level. It remained ignored until around 1845, when John Waterston published a paper building the main precepts of kinetic theory; this was rejected by the Royal Society of England. Later works of James Prescott Joule, Rudolf Clausius and in particular Ludwig Boltzmann firmly established the kinetic theory of gases and brought attention to both the theories of Bernoulli and Waterston.[10]

The debate between proponents of Energetics and Atomism led Boltzmann to write a book in 1898, which endured criticism up to his suicide in 1906.[10] Albert Einstein in 1905 showed how kinetic theory applies to the Brownian motion of a fluid-suspended particle, which was confirmed in 1908 by Jean Perrin.[10]

Equation

The mathematical equation for Boyle's law is:

where:

  • P denotes the pressure of the system.
  • V denotes the volume of the gas.
  • k is a constant value representative of the pressure and volume of the system.

So long as temperature remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of k will remain constant. However, due to the derivation of pressure as perpendicular applied force and the probabilistic likelihood of collisions with other particles through collision theory, the application of force to a surface may not be infinitely constant for such values of k, but will have a limit when differentiating such values over a given time.Template:Huh

Forcing the volume V of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure p must decrease proportionally. Conversely, reducing the volume of the gas increases the pressure.

Boyle's law is used to predict the result of introducing a change, in volume and pressure only, to the initial state of a fixed quantity of gas. The before and after volumes and pressures of the fixed amount of gas, where the before and after temperatures are the same (heating or cooling will be required to meet this condition), are related by the equation:

Here P1 and V1 represent the original pressure and volume, respectively, and P2 and V2 represent the second pressure and volume.

Boyle's law, Charles's law, and Gay-Lussac's law form the combined gas law. The three gas laws in combination with Avogadro's law can be generalized by the ideal gas law.

See also

Sportspersons Hyslop from Nicolet, usually spends time with pastimes for example martial arts, property developers condominium in singapore singapore and hot rods. Maintains a trip site and has lots to write about after touring Gulf of Porto: Calanche of Piana.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Template:Diving medicine, physiology and physics

de:Thermische Zustandsgleichung idealer Gase#Gesetz von Boyle-Mariotte

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ Levine, Ira. N (1978). "Physical Chemistry" University of Brooklyn: McGraw-Hill
  3. ↑ Levine, Ira. N. (1978), p12 gives the original definition.
  4. ↑ J Appl Physiol 98: 31-39, 2005. Free download at Jap.physiology.org
  5. ↑ The Boyle Papers BP 9, fol. 75v-76r at BBK.ac.uk
  6. ↑ The Boyle Papers, BP 10, fol. 138v-139r at BBK.ac.uk
  7. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  8. ↑ Principia, Sec.V,prop. XXI, Theorem XVI
  9. ↑ Levine, Ira. N. (1978), p11 notes that deviations occur with high pressures and temperatures.
  10. ↑ 10.0 10.1 10.2 Levine, Ira. N. (1978), p400 – Historical background of Boyle's law relation to Kinetic Theory
Retrieved from "https://en.formulasearchengine.com/index.php?title=Rational_expectations&oldid=1916"