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particle duality</a></label></li><li class="expanded parent selected"><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="atomic-models.html">Atomic models</a></label></li></ul></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Atomic models <a href="#" class="mark-page-favorite pull-right" data-pid="1133" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../physics.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../354/atomic-nuclear-and-particles.html">Atomic, nuclear and particles</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1132/ahl-quantum.html">AHL Quantum</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Atomic models</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="../../quantum/level.jpg" style="float: left; width: 250px; height: 166px;">Quantitative models can be used to calculate approximate electron energies. Electron energies and angular momenta in an atom are quantised.</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> <div class="panel panel-has-colored-body panel-has-border panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Bohr model</p> </div> </div> <div class="panel-body"> <div> <p>The Bohr model predicts the hydrogen spectrum by applying the equations for circular motion for electron orbits that have quantised angular momentum.</p> <p>The energy of an electron in an energy level is given by:</p> <p style="text-align: center;"><span class="math-tex">\(E=-{13.6\over n^2}\)</span></p> <ul> <li><span class="math-tex">\(E\)</span> is the energy associated with the <span class="math-tex">\(n^{th}\)</span> energy level (eV)</li> <li><span class="math-tex">\(n\)</span> is the number of the energy level</li> </ul> <p>The consequence is that, moving out from the nucleus, orbits get further apart but the energy difference gets less.</p> <p>Hence, the energy of an emitted photon when an electron moves down between energy levels is:</p> <p style="text-align: center;"><span class="math-tex">\(E = 13.6({1\over {n_f}^2} - {1\over {n_i}^2})\)</span></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/376912607"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Electron in a box model</p> </div> </div> <div class="panel-body"> <div> <p>A helpful model for understanding the probability of an electron's position is a standing wave. Just as an electron cannot move outside an atom, a wave cannot move from a string clamped at both ends. A standing wave may only oscillate at particular harmonic frequencies; an electron may only have particular discrete energies.</p> <p>Substituting into the de Broglie wavelength equation, <span class="math-tex">\(\lambda = {h\over mv}\)</span>, we can find the kinetic energy, <span class="math-tex">\(E_k={1\over 2}mv^2\)</span>:</p> <p style="text-align: center;"><span class="math-tex">\(\Rightarrow v={h\over m\lambda}=\sqrt{{2E_k\over m}}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(E_k={h^2\over 2\lambda^2m}\)</span></p> <p>The possible wavelengths for a clamped string of length <span class="math-tex">\(L\)</span> are <span class="math-tex">\(2L, {2L\over 2}, {2L\over 3}... {2L\over n}\)</span> etc. For the <span class="math-tex">\(n^{th}\)</span> harmonic:</p> <p style="text-align: center;"><span class="math-tex">\(E_k={n^2h^2\over 8L^2m}\)</span></p> <p>This 'electron in a box' model predicts discrete energy levels.</p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </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-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Quantised properties</p> </div> </div> <div class="panel-body"> <div> <h4>Angular momentum</h4> <p>In the Bohr model, angular momentum is quantised. We commence by:</p> <ol> <li>Recalling the de Broglie wavelength of an electron, <span class="math-tex">\(\lambda={h\over mv}\)</span></li> <li>Using the standing wave condition that a circumference of an orbit must be equal to a whole number of wavelengths, <span class="math-tex">\(2\pi r=n\lambda_n\)</span></li> </ol> <p>Combining these conditions:</p> <p style="text-align: center;"><span class="math-tex">\(\Rightarrow 2\pi r=n{h\over mv}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(mvr={nh\over 2\pi}\)</span></p> <ul> <li><span class="math-tex">\(mvr\)</span> is angular momentum (kgm<sup>2</sup>s<sup>-2</sup>)</li> <li><span class="math-tex">\(n\)</span> is an integer equal to 1 minus the quantum number</li> <li><span class="math-tex">\(h\)</span> is Planck's constant (Js)</li> </ul> <p>This calculation reveals the flaw in the Bohr model. We cannot know both the position and momentum of a particle.</p> <h4>Spin</h4> <p>Observations from the Stern-Gerlach experiment demonstrate that electron spin is another quantised property. Electron spin can take one of two values.</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/376909932"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Electron tunnelling</p> </div> </div> <div class="panel-body"> <div> <p>Tunnelling is the term given to a particle passing through a potential energy barrier without having sufficient energy to surmount the barrier. It is a conceptually similar phenomenon to a ball rolling through a physical tunnel in a hill, rather than being given an energetic kick to climb over. But in quantum tunnelling barriers, no such physical tunnel exists.</p> <p>Instead, the continuous nature wave function of the particle means that there is a finite probability of the particle's position being on the far side of the potential barrier.</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/376909960"></iframe></div> <p>Many factors affect the likelihood of tunnelling. Those which would reduce the likelihood of tunnelling include:</p> <ul> <li>Increasing the width of the barrier</li> <li>Increasing the mass of the particle</li> <li>Increasing the energy deficit between the particle and the barrier energy potential</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </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="#b0d3be7b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="b0d3be7b"> <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%;"> Further atomic models <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="6-367-1133" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>According to the Bohr model, the angular momentum of the hydrogen electron equals:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(nh\over2π\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(2πn\over h\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(n\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(nh\)</span></span></label> </p></div><div class="q-explanation"><p>This equation with <span class="math-tex">\(mvr\)</span> comes from the dde Broglie wavelength and the model of standing waves on the circumference of the orbit.</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>According to the Bohr model, which statement about the separation of consecutive orbit radii and energy levels is true as the orbit radius of a hydrogen electron increases?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>The orbits get further apart but the energy difference decreases.</span></label> </p><p><label class="radio"> <input type="radio"> <span>The orbits get closer and the energy difference decreases.</span></label> </p><p><label class="radio"> <input type="radio"> <span>The orbits get further apart and the energy difference increases.</span></label> </p><p><label class="radio"> <input type="radio"> <span>The orbits get closer and the energy difference increases.</span></label> </p></div><div class="q-explanation"><p>Radius is proportional to <span class="math-tex">\(n\)</span> so their separation increases.</p><p>Energy is proportional to <span class="math-tex">\(1\over n^2\)</span> to the energy differences decrease.</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>According to the Bohr model, the electron closest to the nucleus has:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>highest kinetic and lowest potential energy</span></label> </p><p><label class="radio"> <input type="radio"> <span>lowest kinetic and highest potential energy</span></label> </p><p><label class="radio"> <input type="radio"> <span>highest kinetic and highest potential energy</span></label> </p><p><label class="radio"> <input type="radio"> <span>lowest kinetic and lowest potential energy</span></label> </p></div><div class="q-explanation"><p>Both are proportional to <span class="math-tex">\(1\over r\)</span> but potential energy is negative as zero potential is defined as ionisation.</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>According to Heisenberg, the Bohr model can't be correct as it implies that electrons would have specific values of both:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>energy and momentum</span></label> </p><p><label class="radio"> <input type="radio"> <span>angular momentum and energy</span></label> </p><p><label class="radio"> <input type="radio"> <span>energy and position</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>momentum and position</span></label> </p></div><div class="q-explanation"><p>The Heisenberg uncertainty principle states that <span class="math-tex">\(ΔxΔp ≥ {h\over 4π}\)</span></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>Which of the following is not a failing of the Bohr model?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>It only worked for hydrogen</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>It did not correctly predict the electron energy levels of hydrogen</span></label> </p><p><label class="radio"> <input type="radio"> <span>It didn't predict the fine detail in the hydrogen spectrum</span></label> </p><p><label class="radio"> <input type="radio"> <span>It didn't predict the variation in intensity of the spectral lines of hydrogen</span></label> </p></div><div class="q-explanation"><p>The energy levels were correct, but even this success in the model was only due to the spherical nature of the hydrogen atom.</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 image represents the wave function of an atomic electron.</p><p style="text-align: center;"><img alt="" height="169" src="../../screenshot-2019-11-02-at-17.25.18.png" width="293"></p><p>The electron is most likely to be in position:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>C</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>A and B have equal probability</span></label> </p><p><label class="radio"> <input type="radio"> <span>A, B and C have equal probability</span></label> </p></div><div class="q-explanation"><p>The probability is the square of the amplitude. In a standing wave, the amplitude is greatest at the antinodes.</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 image represents the wave function of a free electron.</p><p style="text-align: center;"><img alt="" height="151" src="../../screenshot-2019-11-02-at-17.25.18(1).png" width="262"></p><p>The electron is most likely to be in position:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>A and B have equal probability</span></label> </p><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>A, B and C have equal probability</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p></div><div class="q-explanation"><p>The probability is the square of the amplitude. In a progressive wave, the amplitude is the same for all points.</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>If 1 kg of matter annihilated with 1 kg of antimatter the amount of energy released would be:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>9 x 10<sup>17</sup> J</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>1.8 x 10<sup>17</sup> J</span></label> </p><p><label class="radio"> <input type="radio"> <span>6 x 10<sup>8</sup> J</span></label> </p><p><label class="radio"> <input type="radio"> <span>2 J</span></label> </p></div><div class="q-explanation"><p>Mass-energy is conserved according to <span class="math-tex">\(E = mc^2\)</span>. Don't forget to add the masses!</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 <span class="scayt-misspell-word" data-scayt-word="Stern-Gerlach" data-wsc-id="k2iqidgdxon11kaog" data-wsc-lang="en_US">Stern-Gerlach</span> experiment passed silver atoms through:</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>a non-uniform magnetic field.</span></label> </p><p><label class="radio"> <input type="radio"> <span>a uniform electric field.</span></label> </p><p><label class="radio"> <input type="radio"> <span>a uniform magnetic field.</span></label> </p><p><label class="radio"> <input type="radio"> <span>a non-uniform electric field.</span></label> </p></div><div class="q-explanation"><p>A non-uniform magnetic field was used to deflect silver atoms to find out whether spin was continuous or discrete</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 images represent four arrangements of two atoms. the probability wave of an electron in the left hand atom is shown.</p><p style="text-align: center;"><img alt="" height="163" src="../../paper.journal.12.png" width="345"></p><p>Which electron is most likely to tunnel into the other atom?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>C</span></label> </p><p><label class="radio"> <input type="radio"> <span>D</span></label> </p></div><div class="q-explanation"><p>C has the narrowest barrier and the lowest energy deficit.</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="page-container panel-self-assessment" 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