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fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../252/thermal.html">Thermal</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Modelling gases</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../thermal/gas-1-s-1.jpg" style="float: left; width: 150px; height: 225px;"><img class="sibico" src="../../../img/sibico/tok.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Theory if Knowledge"> Because there are no gravitational or electrostatic forces between gas molecules (elastic <a href="../244/collisions.html" title="Collisions">Collisions</a>), the gaseous state is the simplest to model. </p> <p>In this section of the course, we will be linking the microscopic concept of kinetic energy of individual molecules with the macroscopic concepts of pressure, temperature and volume. </p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>The absolute temperature of a gas (in Kelvin) is directly proportional to the average kinetic energy of the molecules.</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(\bar{E_k}={3\over2} kT\)</span></p> <p style="text-align: center;"><em>k</em> is the Boltzmann constant 1.38 x 10<sup>-23</sup> JK<sup>-1</sup></p> </div> <p>Pressure = force <span class="math-tex">\(\div\)</span> area. The force exerted by the gas molecules is due to their change of momentum as they hit the walls (due to <a href="../238/newtons-second-law.html" title="Newton's second law">Newton's second law</a>).</p> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-yellow panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Gas laws</p> </div> </div> <div class="panel-body"> <div> <p><strong>Boyle's law:</strong> The pressure of a fixed mass of gas at constant temperature is inversely proportional to its volume.</p> <p style="text-align: center;"><span class="math-tex">\(p\propto{1\over V}\)</span></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div> <p>To help you remember that temperature is constant, think that Boyle sounds like 'boil'!</p> </div> </section> <p><span class="snippet-fa fa fa-flask fa-lg"> </span> To verify that a gas obeys Boyle's Law, you can measure the pressure of gas in a syringe as the volume is reduced. This must be done slowly to keep the temperature constant. Alternatively, your school may have 'gas syringes' connected to pressure sensors.</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/images/blog/boyle.JPG"></p> <p>If <em>p</em> is proportional to 1/<em>V</em> then a graph of <em>p</em> vs <em>V</em> will be a curve.</p> <p style="text-align: center;"><img alt="" height="357" src="../../../ib/physics/activities/gaslaws1.png" width="376"></p> <hr class="hidden-separator"> <p>Note that this particular curve is for higher pressures and temperatures than would be found in the classroom!</p> <p>The curve is called an<em> isothermal</em> as it's at constant temperature. To verify the relationship, you would need to plot <em>p</em> against <span class="math-tex">\(1\over V\)</span>to yield a straight line through the origin (<a href="../189/linear-graphs.html" title="Linear graphs">Linear graphs</a>).</p> <p><strong><span avant="" garde="" style="color: inherit; font-family: ">Pressure Law:</span></strong> The pressure of a fixed mass of gas with constant volume is directly proportional to its absolute temperature (in Kelvin).</p> <p style="text-align: center;"><span class="math-tex">\(p\propto T\)</span></p> <p><span class="snippet-fa fa fa-flask fa-lg"> </span> A pressure sensor is used to measure the pressure of a fixed volume of gas as its temperature is changed. Software will then enable you to plot p vs T.</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/gaslaws2.png" style="width: 297px; height: 380px;"></p> <p>Since pressure is directly proportional to absolute temperature, a graph of <em>p</em> vs <em>T</em> will be linear and through the origin.</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/activities/gaslaws3.png" style="width: 439px; height: 355px;"></p> <hr class="hidden-separator"> <p><strong>Charles' Law: </strong>The volume of a fixed mass of gas at a constant pressure is directly proportional to its temperature in Kelvin.</p> <p style="text-align: center;"><span class="math-tex">\(V \propto T\)</span></p> <p><span class="snippet-fa fa fa-flask fa-lg"> </span> To verify Charles' law, measure the volume of a fixed mass of gas at different temperatures. This needs a piston that moves very smoothly; ask if your school Physics department has the PASCO apparatus.</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/activities/gaslaws4.png"></p> <p>The temperature of the air in the metal can is changed by placing it in a water bath. As the water is heated, the gas expands pushing the piston up.</p> <p>Since volume is proportional to temperature, a graph of <em>V</em> vs <em>T</em> will be linear and through the origin.</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/activities/gaslaws5.png" style="width: 401px; height: 328px;"></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Ideal gases</p> </div> </div> <div class="panel-body"> <div> <p>The three laws can be combined to produce a single equation. As well as pressure, volume and temperature, the <em>ideal gas equation</em> also includes the number of moles of the gas, <em>n</em>, and a constant of proportionality, <em>R</em>.</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(pV=nRT\)</span></p> <ul> <li><span class="math-tex">\(n\)</span> is the number of moles</li> <li><span class="math-tex">\(R\)</span> is the molar gas constant = 8.31 Jmol<sup>-1</sup>K<sup>-1</sup></li> </ul> </div> <p>This is represented by a <em>p-V</em> graph. Different temperatures are represented by different isothermals.</p> <p style="text-align: center;"><img alt="" src="../../../ib/physics/activities/gaslaws6.png" style="width: 434px; height: 388px;"></p> <p>Much like contours for height on a map, the isothermals are plotted at constant intervals (every 500 K).</p> <div class="magenta"> <h4>Deviations from the ideal gas model</h4> <p><img class="sibico" src="../../../img/sibico/tok.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Theory if Knowledge"> Recall that we made several assumptions about the particles of a gas:</p> <ul> <li>very small, relative to the size of the container</li> <li>perfectly elastic collisions</li> <li>time between collisions >> time of a collision</li> <li>spherical</li> <li>identical</li> <li>have no forces between them unless colliding</li> </ul> <p><img class="sibico" src="../../../img/sibico/quiz.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Quiz"> Which 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