tag:blogger.com,1999:blog-1958934670120969772014-10-02T21:05:53.680-07:00Kinetic theoryUmerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-195893467012096977.post-66804598248027930882011-07-06T02:43:00.000-07:002011-07-06T02:43:14.362-07:00Kinetic theory<h1 class="firstHeading" id="firstHeading">Kinetic theory</h1><div id="siteSub">From Wikipedia, the free encyclopedia</div><div class="separator" style="clear: both; text-align: center;"><a class="image" href="http://en.wikipedia.org/wiki/File:Translational_motion.gif" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="" class="thumbimage" height="263" src="http://upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif" width="300" /></a></div><div class="thumb tright"> <div class="thumbinner" style="width: 302px;"> <div class="thumbcaption"> <div class="magnify"></div>The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.</div></div></div><dl><dd><i>This article applies to gases; see also Kinetic theory of solids</i></dd></dl>The <b>kinetic theory</b> of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as was Isaac Newton's conjecture, but due to collisions between molecules moving at different velocities.<br />While the particles making up a gas are too small to be visible, the jittering motion of pollen grains or dust particles which can be seen under a microscope, known as Brownian motion, results directly from collisions between the particle and gas molecules. As pointed out by Albert Einstein in 1905, this experimental evidence for kinetic theory is generally seen as having confirmed the existence of atoms and molecules.Umerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.com0tag:blogger.com,1999:blog-195893467012096977.post-33080870073430169062011-07-06T02:41:00.000-07:002011-07-06T02:41:33.673-07:00History<h2><span class="mw-headline" id="History">History</span></h2>In 1738 Daniel Bernoulli published <i>Hydrodynamica</i>, which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.<br />Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),<sup class="reference" id="cite_ref-2"><span>[</span>3<span>]</span></sup> Georges-Louis Le Sage (ca. 1780, published 1818),<sup class="reference" id="cite_ref-3"><span>[</span>4<span>]</span></sup> John Herapath (1816)<sup class="reference" id="cite_ref-4"><span>[</span>5<span>]</span></sup> and John James Waterston (1843),<sup class="reference" id="cite_ref-5"><span>[</span>6<span>]</span></sup> which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles.<sup class="reference" id="cite_ref-6"><span>[</span>7<span>]</span></sup><br />In 1857 Rudolf Clausius, according to his own words independently of Krönig, developed a similar, but much more sophisticated version of the theory which included translational and contrary to Krönig also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. <sup class="reference" id="cite_ref-7"><span>[</span>8<span>]</span></sup> In 1859, after reading a paper by Clausius, James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics.<sup class="reference" id="cite_ref-8"><span>[</span>9<span>]</span></sup> In his 1873 thirteen page article 'Molecules', Maxwell states: “we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases.”<sup class="reference" id="cite_ref-9"><span>[</span>10<span>]</span></sup> In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithmic connection between entropy and probability was first stated by him.<br />In the beginning of twentieth century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905)<sup class="reference" id="cite_ref-10"><span>[</span>11<span>]</span></sup> and Marian Smoluchowski's (1906)<sup class="reference" id="cite_ref-11"><span>[</span>12<span>]</span></sup> papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.Umerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.com0tag:blogger.com,1999:blog-195893467012096977.post-37997690400826092192011-07-06T02:39:00.000-07:002011-07-06T02:39:15.306-07:00Postulates<h2><span class="editsection"></span> <span class="mw-headline" id="Postulates">Postulates</span></h2>The theory for ideal gases makes the following assumptions:<br /><ul><li>The gas consists of very small particles, all with non-zero mass.</li><li>The number of molecules is so large that statistical treatment can be applied.</li><li>These molecules are in constant, random motion. The rapidly moving particles constantly collide with the walls of the container.</li><li>The collisions of gas particles with the walls of the container holding them are perfectly elastic.</li><li>Except during collisions, the interactions among molecules are negligible (they exert no forces on one another).</li><li>The total volume of the individual gas molecules added up is negligible compared to the volume of the container. This is equivalent to stating that the average distance separating the gas particles is large compared to their size.</li><li>The molecules are perfectly spherical in shape, and elastic in nature.</li><li>The average kinetic energy of the gas particles depends only on the temperature of the system.</li><li>Relativistic effects are negligible.</li><li>Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules are treated as classical objects.</li><li>The time during collision of molecule with the container's wall is negligible as compared to the time between successive collisions.</li><li>The equations of motion of the molecules are time-reversible.</li></ul>More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions. The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.<sup class="Template-Fact" style="white-space: nowrap;" title="This claim needs references to reliable sources from November 2007">[<i>citation needed</i>]</sup> In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number.Umerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.com0tag:blogger.com,1999:blog-195893467012096977.post-64893973047872968832011-07-06T02:37:00.000-07:002011-07-06T02:37:21.342-07:00Properties<h2><span class="mw-headline" id="Properties">Properties</span></h2><h3><span class="editsection"></span> <span class="mw-headline" id="Pressure_and_Kinetic_Energy">Pressure and Kinetic Energy</span></h3>Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container. Consider a gas of <i>N</i> molecules, each of mass <i>m</i>, enclosed in a cuboidal container of volume <i>V</i>=<i>L</i><sup>3</sup>. When a gas molecule collides with the wall of the container perpendicular to the <i>x</i> coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:<br /><dl><dd><img alt="\Delta p = p_{i,x} - p_{f,x} = 2 m v_x\," class="tex" src="http://upload.wikimedia.org/math/f/2/a/f2a426a1ca4dc88de7682b41395ff702.png" /></dd></dl>where <i>v<sub>x</sub></i> is the <i>x</i>-component of the initial velocity of the particle.<br />The particle impacts one specific side wall once every<br /><dl><dd><img alt="\Delta t = \frac{2L}{v_x}" class="tex" src="http://upload.wikimedia.org/math/9/9/f/99fadb092787b03f07a0a3a2e6116f87.png" /></dd></dl>(where <i>L</i> is the distance between opposite walls).<br />The force due to this particle is:<br /><dl><dd><img alt="F = \frac{\Delta p}{\Delta t} = \frac{m v_x^2}{L}." class="tex" src="http://upload.wikimedia.org/math/f/5/f/f5fb80f615393de54e734c180855cb25.png" /></dd></dl>The total force on the wall is<br /><dl><dd><img alt="F = \frac{Nm \overline{v_x^2}}{L}" class="tex" src="http://upload.wikimedia.org/math/e/c/7/ec7525abb30a9d98106548caacb49d35.png" /></dd></dl>where the bar denotes an average over the <i>N</i> particles. Since the assumption of molecular chaos imposes <img alt=" \overline{v_x^2} = \overline{v^2}/3 " class="tex" src="http://upload.wikimedia.org/math/f/2/2/f22aada0e3210f2d02eb51d3442b58bc.png" />, we can rewrite the force as<br /><dl><dd><img alt="F = \frac{Nm\overline{v^2}}{3L}." class="tex" src="http://upload.wikimedia.org/math/3/5/7/357c31d6f7dcbb3dce27b3e1ec7dbcbb.png" /></dd></dl>This force is exerted on an area <i>L</i><sup>2</sup>. Therefore the pressure of the gas is<br /><dl><dd><img alt="P = \frac{F}{L^2} = \frac{Nm\overline{v^2}}{3V}" class="tex" src="http://upload.wikimedia.org/math/0/4/0/0408b195e2c391fbcaf253c2f15c3b72.png" /></dd></dl>where <i>V</i>=<i>L</i><sup>3</sup> is the volume of the box. The fraction <i>n</i>=<i>N</i>/<i>V</i> is the number density of the gas (the mass density <i>ρ</i>=<i>nm</i> is less convenient for theoretical derivations on atomic level). Using <i>n</i>, we can rewrite the pressure as<br /><dl><dd><img alt=" P = \frac{n m \overline{v^2}}{3}." class="tex" src="http://upload.wikimedia.org/math/1/a/8/1a8809991b747eb61b33000d0d4b2303.png" /></dd></dl>This is a first non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule <img alt="{1 \over 2} m\overline{v^2}" class="tex" src="http://upload.wikimedia.org/math/b/5/9/b59d375bc865146d875d754263b94f2b.png" /> which is a microscopic property.Umerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.com0tag:blogger.com,1999:blog-195893467012096977.post-72156976676158356902011-07-06T02:36:00.000-07:002011-07-06T02:36:04.443-07:00Temperature and kinetic energy<h3><span class="mw-headline" id="Temperature_and_kinetic_energy">Temperature and kinetic energy</span></h3>From the ideal gas law<br /><dl><dd> <table border="0" style="width: 100%;"><tbody><tr> <td style="width: 95%;"> <img alt="\displaystyle PV = N k_B T" class="tex" src="http://upload.wikimedia.org/math/9/a/6/9a65f521e9ef53d8d325b22039b1c1d1.png" /><br /></td> <td>(1)</td> </tr></tbody></table></dd></dl>where <img alt="\displaystyle k_B" class="tex" src="http://upload.wikimedia.org/math/5/2/5/525ec9a2c1ccee8006827268ef6554cf.png" /> is the Boltzmann constant, and <img alt="\displaystyle T" class="tex" src="http://upload.wikimedia.org/math/d/5/1/d51fbb484e38555f47b6179aabb71505.png" /> the absolute temperature,<br />and from the above result <img alt="P = {Nmv^2\over 3V} " class="tex" src="http://upload.wikimedia.org/math/3/7/a/37a590ea36fe7fa412cba5c54af087a2.png" /> we have <img alt="PV = {Nmv_{rms}^2 \over 3} " class="tex" src="http://upload.wikimedia.org/math/e/a/7/ea72173200d15d35234184766e021296.png" /><br />we have <img alt=" \displaystyle N k_B T = \frac {N m v_{rms}^2} {3}" class="tex" src="http://upload.wikimedia.org/math/b/f/0/bf0a2d3d63a4f9924d3312397889bdf6.png" /><br />then the temperature <img alt="\displaystyle T" class="tex" src="http://upload.wikimedia.org/math/d/5/1/d51fbb484e38555f47b6179aabb71505.png" /> takes the form<br /><dl><dd> <table border="0" style="width: 100%;"><tbody><tr> <td style="width: 95%;"> <img alt=" \displaystyle T = \frac {m v_{rms}^2} {3 k_B}" class="tex" src="http://upload.wikimedia.org/math/c/6/a/c6a34c57648e9be67caedac8e990d1c8.png" /><br /></td> <td>(2)</td> </tr></tbody></table></dd></dl>which leads to the expression of the kinetic energy of a molecule<br /><dl><dd><img alt=" \displaystyle \frac {1} {2} mv_{rms}^2 = \frac {3} {2} k_B T." class="tex" src="http://upload.wikimedia.org/math/4/1/2/412bfa3776f954d877fddd768da57882.png" /></dd></dl>The kinetic energy of the system is N times that of a molecule <img alt=" K= \frac {1} {2} N m v_{rms}^2 " class="tex" src="http://upload.wikimedia.org/math/d/7/7/d773b0a959289284f85f656156ae020e.png" /><br />The temperature becomes<br /><dl><dd> <table border="0" style="width: 100%;"><tbody><tr> <td style="width: 95%;"> <img alt=" \displaystyle T = \frac {2} {3} \frac {K} {N k_B}." class="tex" src="http://upload.wikimedia.org/math/2/4/8/248eb93215337ea4936a51a641555a33.png" /><br /></td> <td>(3)</td> </tr></tbody></table></dd></dl>Eq.(3)<sub>1</sub> is one important result of the kinetic theory: <i>The average molecular kinetic energy is proportional to</i> the absolute temperature<i>.</i> From Eq.(1) and Eq.(3)<sub>1</sub>, we have<br /><dl><dd> <table border="0" style="width: 100%;"><tbody><tr> <td style="width: 95%;"> <img alt=" \displaystyle PV = \frac {2} {3} K." class="tex" src="http://upload.wikimedia.org/math/a/0/d/a0d3f8e2f167501eb5b15d18f66dd073.png" /><br /></td> <td>(4)</td> </tr></tbody></table></dd></dl>Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy.<br />Eq.(1) and Eq.(4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see .<sup class="reference" id="cite_ref-0"><span>[</span>1<span>]</span></sup><br />Since there are <img alt="\displaystyle 3N" class="tex" src="http://upload.wikimedia.org/math/e/1/3/e134a39e59a5c3cbfedb30e1b4575808.png" /> degrees of freedom in a monoatomic-gas system with <img alt="\displaystyle N" class="tex" src="http://upload.wikimedia.org/math/e/9/e/e9e940cb78faa948d7cfba8549b23d32.png" /> particles, the kinetic energy per degree of freedom is<br /><dl><dd> <table border="0" style="width: 100%;"><tbody><tr> <td style="width: 95%;"> <img alt=" \displaystyle \frac {K} {3 N} = \frac {k_B T} {2}." class="tex" src="http://upload.wikimedia.org/math/9/d/a/9dac8c3907101325ac99944e4864ecee.png" /><br /></td> <td>(5)</td> </tr></tbody></table></dd></dl>In the kinetic energy per degree of freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant. In addition to this, the temperature will decrease when the pressure drops to a certain point. This result is related to the equipartition theorem.<br />As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter gases act as if they have only 5.<br />Thus the kinetic energy per kelvin (monatomic ideal gas) is:<br /><ul><li>per mole: 12.47 J</li><li>per molecule: 20.7 yJ = 129 μeV.</li></ul>At standard temperature (273.15 K), we get:<br /><ul><li>per mole: 3406 J</li><li>per molecule: 5.65 zJ = 35.2 meV.</li></ul>Umerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.com0tag:blogger.com,1999:blog-195893467012096977.post-17865821685045530002011-07-06T02:33:00.001-07:002011-07-06T02:33:58.873-07:00Collisions with container<h3><span class="mw-headline" id="Collisions_with_container">Collisions with container</span></h3>One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.<br />Assuming an ideal gas, a derivation<sup class="reference" id="cite_ref-1"><span>[</span>2<span>]</span></sup> results in an equation for total number of collisions per unit time per area:<br /><dl><dd><dl><dd><img alt="A = \frac{1}{4}\frac{N}{V} v_{avg} = \frac{n}{4} \sqrt{\frac{8 k_{B} T}{\pi m}} . \," class="tex" src="http://upload.wikimedia.org/math/b/1/2/b125ed262f941e579fce6b7757505edf.png" /></dd></dl></dd></dl><h3><span class="editsection"></span> <span class="mw-headline" id="Speed_of_molecules">Speed of molecules</span></h3>From the kinetic energy formula it can be shown that<br /><dl><dd><img alt="v_{rms}^2 = \frac{3RT}{\mbox{molar mass}}" class="tex" src="http://upload.wikimedia.org/math/f/5/4/f543cadf952bf542b6e1badfd627f717.png" /></dd></dl>with <i>v</i> in m/s, <i>T</i> in kelvins, and <i>R</i> is the gas constant. The molar mass is given as kg/mol. The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (distribution of speeds).Umerhttp://www.blogger.com/profile/08695541556650316904noreply@blogger.com0