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href="../../../physics.html"><i class="fa 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><a href="../258/modelling-gases.html">Modelling gases</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Ideal gas equation</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 15 minutes"><i class="fa fa-clock-o"></i> 15'</span> </ol> <article id="main-article"> <p><img alt="" src="../../thermal/gas-3-s.jpg" style="float: left; width: 250px; height: 200px;">The ideal gas equation relates pressure, temperature, volume and the amount of gas.</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"> <div> <p style="text-align: center;"><span class="math-tex">\(pV=nRT\)</span></p> <p>Since the number of moles is linked to the number of molecules...</p> <p style="text-align: center;"><span class="math-tex">\(n={N\over N_A} \Rightarrow pV={N\over N_A}RT\)</span></p> <p>In addition,</p> <p style="text-align: center;"> <span class="math-tex">\(R = N_Ak \Rightarrow pV=NkT\)</span></p> </div> </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-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Kinetic energy and temperature, <span class="math-tex">\(\bar{E_k}={3\over2}kT\)</span></p> </div> </div> <div class="panel-body"> <div> <p>This equation can often get overlooked but is really quite extraordinary. It links the microscopic concept of the average kinetic energy of individual molecules of gas with the temperature of the gas as a whole.</p> <p>It means that you could measure the temperature of the room you are in the moment and calculate the 'root mean square speed' (<span class="math-tex">\(v_{rms}\)</span>) of the air molecules (best to assume just nitrogen and oxygen as they are similar in size and mass!):</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(\Rightarrow {1\over2}m\bar{v^2}={3\over2}kT\)</span></p> <p style="text-align: center;"><span class="math-tex">\(v_{rms}=\sqrt {3kT\over m}\)</span></p> </div> <p>This equation also enables us to calculate the internal energy, <em>U</em>, of a gas (recall that we assume that potential energy is zero): </p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(U=\sum E_p+\sum E_k=\sum E_k\)</span></p> <p style="text-align: center;"><span class="math-tex">\(\Rightarrow U=N{3\over 2} kT\)</span></p> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p><em>PVT</em> graphs</p> </div> </div> <div class="panel-body"> <div> <p><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> <em>P</em>, <em>V</em> and <em>T</em> can be plotted on a 3-D graph. In practice, however, we draw just <em>p</em> and <em>V</em> with the different temperatures represented by isotherms.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/164055813"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Real Gases</p> </div> </div> <div class="panel-body"> <div> <p><img class="sibico" src="../../../img/sibico/video.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Video"> Real gas molecules take up space and are attracted to each other.</p> <p><img class="sibico" src="../../../img/sibico/exam-style.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Exam-style question"> If the gas is at low pressure, moderate temperature and low density, then the ideal gas law will hold.</p> <div class="smart-object center" data-id="131" style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="480" mozallowfullscreen="" src="https://player.vimeo.com/video/164056445" webkitallowfullscreen="" width="640"></iframe> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-purple"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><em>Use quizzes to practise application of theory.</em> </p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#1631282a"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="1631282a"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Ideal gas equation <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-66-263" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The ideal gas equation is PV = nRT.</p><p>Which of the following are not constant</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> nR</label></p><p><label class="radio"><input class="c" type="radio"> PVT </label></p><p><label class="radio"><input type="radio"> T/PV</label></p><p><label class="radio"><input type="radio"> P<sup>2</sup>V<sup>2</sup>/T<sup>2</sup></label></p></div><div class="q-explanation"><p>PV/T = constant so the reciprocal and square will also be constant.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The unit of R is</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> NK<sup>-1</sup>mol<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> Pa m<sup>2</sup>K mol<sup>-1</sup></label></p><p><label class="radio"><input type="radio"> it's a constant so has no units</label></p><p><label class="radio"><input class="c" type="radio"> JK<sup>-1</sup>mol<sup>-1</sup></label></p></div><div class="q-explanation"><p>Pa m<sup>3</sup>K<sup>-1</sup>mol<sup>-1 </sup></p><p>Pa m<sup>3</sup> is the same as Nm which is J</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The pressure of a fixed mass of gas with temperature T is doubled and its volume halved. What is its new temperature?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> T</label></p><p><label class="radio"><input type="radio"> T/2</label></p><p><label class="radio"><input type="radio"> 4T</label></p><p><label class="radio"><input type="radio"> 2T</label></p></div><div class="q-explanation"><p>PV/T = (2P x1/2 V)/T' T' = T</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The volume of a fixed mass of gas is doubled and its temperature increased by a factor 3. If its original Pressure is P what will its pressure be after the expansion?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 2P/3</label></p><p><label class="radio"><input type="radio"> 3P</label></p><p><label class="radio"><input class="c" type="radio"> 3P/2</label></p><p><label class="radio"><input type="radio"> 2P</label></p></div><div class="q-explanation"><p>PV/T = P' x 2V/3T</p><p>P' = 3P/2</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A container of air with volume V is at a pressure of 1/2 atmospheric pressure and room temperature. the container is opened until the pressure is the same as the atmosphere and the temperature is unchanged. If there were n moles before how many moles are there at the end</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> n</label></p><p><label class="radio"><input type="radio"> 4n</label></p><p><label class="radio"><input class="c" type="radio"> 2n</label></p><p><label class="radio"><input type="radio"> n/2</label></p></div><div class="q-explanation"><p>PV/T =nR</p><p>PV/nT = R = P/2V/n'T</p><p>n' = 2n</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph represents a gas expanding at constant temperature.</p><p style="text-align: center;"><img alt="" src="../../images/pvnrtq1.png" style="width: 350px; height: 256px;"></p><p>There are 1.203 x 10<sup>-3</sup> moles so nR = 0.01 JK<sup>-1</sup></p><p>What is the temperature of the gas?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 100,000 K</label></p><p><label class="radio"><input class="c" type="radio"> 1000 K</label></p><p><label class="radio"><input type="radio"> 100 K</label></p><p><label class="radio"><input type="radio"> 10,000 K</label></p></div><div class="q-explanation"><p>T = PV/nR = PV/ 0.01 = 100 x 10<sup>3</sup> x 100x 10<sup>-6</sup>/0.01 =10/0.01 = 1000 K</p><p>Must work in SI Pa and m<sup>3</sup></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div><em>Online tutorials to help you solve original problems</em></div> <div> <div class="panel panel-has-colored-body panel-has-border panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <iframe src="../../../pages/subjects/page-networks/index-45.htm?ticket=082b9c9c4ae3624d&network-id=39&sitename=physics&subject-id=6&fullscreen=false&node=start" style="width: 100%; height: 640px; max-height: 80vh" id="network-39" class="network-viewer"></iframe> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <iframe src="../../../pages/subjects/page-networks/index-46.htm?ticket=082b9c9c4ae3624d&network-id=40&sitename=physics&subject-id=6&fullscreen=false&node=start" style="width: 100%; height: 640px; max-height: 80vh" id="network-40" class="network-viewer"></iframe> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-red"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Just for Fun</p> </div> </div> <div class="panel-body"> <div> <h4>Deriving the equation (WARNING: NOT REQUIRED)</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <h4> </h4> <p><img alt="" src="../../thermal/ideal-gas-1.png" style="float: left; width: 384px; height: 250px;">The changes in momentum for <em>x</em> and <em>y</em> directions are as follows (where <em>f</em> is final and <em>i</em> is inital):</p> <ul> <li><span class="math-tex">\(\Delta p_x=mv_{x,f}-mv_{x,i}=m(-v_{x,i})-mv_{x,i}=-2mv_{x}\)</span></li> <li><span class="math-tex">\(\Delta p_y=mv_{y,f}-mv_{y,i}=0\)</span></li> </ul> <p>You will notice that we have made the left direction positive for the <em>x </em>equation.</p> <p>Since there is no change in vertical momentum against this (arbitrarily) vertical wall, we can ignore the y-component when calculating force.</p> <hr class="hidden-separator"> <p><img alt="" src="../../thermal/ideal-gas-2.png" style="float: left; width: 319px; height: 250px;">The mass of particles that will strike the wall in a given time is half of those contained within the 'blob' shaped volume shown:</p> <p><span class="math-tex">\(m={1\over2}\rho V={1\over 2} \rho (v_x A \Delta t )\)</span></p> <p>We know from Newton's Second Law that force is equal to the rate of change of momentum. Per particle:</p> <p><span class="math-tex">\(F={\Delta p\over \Delta t}= {-2mv_{x}\over \Delta t} \)</span></p> <p>Substituting for <em>m</em>:</p> <p><span class="math-tex">\(F_{wall \rightarrow particle}= {-2 {1\over 2} \rho (v_x A \Delta t )v_{x}\over \Delta t}=-\rho A {v_{x}}^2 \)</span></p> <p>Now this force is from the wall on the particle (causing the change in momentum of the particle). Due to N3: <span class="math-tex">\(F_{particle \rightarrow wall}= \rho A {v_{x}}^2\)</span></p> <hr class="hidden-separator"> <p><img alt="" src="../../thermal/ideal-gas-3.png" style="float: left; width: 298px; height: 200px;">Concerned about why we can ignore the <em>y</em>-components of speed?</p> <p>For every particle that enters the region from above or below to strike the wall, there is (on average) another that will leave without striking the wall in the prescribed area.</p> <p>The same applies for the <em>z</em> direction too as density is constant.</p> <hr class="hidden-separator"> <p>The ideal gas law contains pressure, not force:</p> <p><span class="math-tex">\(P={F\over A}={\rho A{v_x}^2\over A}=\rho {v_x}^2\)</span></p> <p>And considering the average speed of the particles: </p> <p><span class="math-tex">\(P=\rho \bar{{v_x}^2}\)</span></p> <hr class="hidden-separator"> <p><img alt="" src="../../thermal/ideal-gas-4.png" style="float: left; width: 413px; height: 240px;"><img alt="" src="../../thermal/ideal-gas-5.png" style="float: left; width: 439px; height: 240px;"></p> <hr class="hidden-separator"> <p>Since the particles are unaffected by gravity or any other non-contact forces, there is no preferred direction. Therefore the average speed in any direction is equal:</p> <p><span class="math-tex">\(\bar {{v_x}^2}=\bar {{v_y}^2}=\bar {{v_z}^2}\)</span></p> <p>Since <span class="math-tex">\(v^2={v_x}^2+{v_y}^2+{v_z}^2\)</span></p> <p><span class="math-tex">\(\Rightarrow v^2=3{v_x}^2\)</span></p> <p><span class="math-tex">\(\Rightarrow {v_x}^2= {1\over 3} v^2\)</span></p> <p><span class="math-tex">\(P={1\over 3} \rho v^2\)</span></p> <p>Multiplying this fraction by <span class="math-tex">\({2\over m} {m\over 2}\)</span>:</p> <p><span class="math-tex">\(P={1\over 3} \rho {2\over m} {m\over 2} v^2 \Rightarrow P={2\over 3} {\rho \over m}E_k\)</span></p> <p>Now, density is the ratio of mass to volume for all of the particles together:</p> <p><span class="math-tex">\(\rho={nN_Am\over V}\)</span></p> <p><span class="math-tex">\(\Rightarrow P={2\over 3}{nN_A\over V}E_k\)</span></p> <p>In addition (and independently of the above), <span class="math-tex">\(E_k={3\over2}kT\)</span>:</p> <p><span class="math-tex">\(\Rightarrow P={2\over 3}{nN_A\over V}{3\over 2}kT\)</span></p> <p>And <span class="math-tex">\(R=N_Ak\)</span>:</p> <p><span class="math-tex">\(PV=nRT\)</span></p> <p><span class="snippet-fa fa fa-smile-o fa-2x" style="color: rgb(0, 114, 60);"> </span> </p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="263"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Ideal gas equation</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer 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