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<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">I got a few questions from a student concerning some of the Tutorial Mastering homework.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The following two questions come from Tutorial 6e (section 6.6).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">Question 1 (Part D):<o:p></o:p></span></b></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">How would the
</span><i><span style="font-size:14.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">d</span></i><i><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">x</span></i><sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">2</span></sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">−<i>y</i><sup>2</sup>
orbital in the <i>n</i>=5 shell <span class="GramE">compare</span> to the </span>
<i><span style="font-size:14.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">d</span></i><i><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">x</span></i><sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">2</span></sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">−<i>y</i><sup>2</sup>
orbital in the <i>n</i>=3 subshell?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">Which of the following are True?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
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<li class="MsoListParagraph" style="margin-left:0in;mso-list:l1 level1 lfo3;tab-stops:list .5in">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The contour of the orbital would extend further out along the
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<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The value of ℓ would increase by 2.<o:p></o:p></span></li><li class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-list:l1 level1 lfo3;tab-stops:list .5in">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The radial probability function would include two more nodes.<o:p></o:p></span></li><li class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-list:l1 level1 lfo3;tab-stops:list .5in">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The orientation of the orbital would be rotated 45<sup>o</sup></span><span style="font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman";mso-bidi-font-family:"Cambria Math"">
</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">along the
<span class="SpellE"><i>xy</i></span> plane.<o:p></o:p></span></li><li class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-list:l1 level1 lfo3;tab-stops:list .5in">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The
<i>m_</i>ℓ value would be the same.<o:p></o:p></span></li></ol>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">A.<span style="mso-spacerun:yes">
</span>True.<span style="mso-spacerun:yes"> </span>The most probable distance (radius) for the 5d orbital is further out than the 3d.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><span style="mso-tab-count:1">
</span>Also, the </span><i><span style="font-size:14.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">d</span></i><i><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">x</span></i><sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">2</span></sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">−<i>y</i><sup>2
</sup>looks like two p orbitals, one on the x-axis and one on the y-axis.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">B. False.<span style="mso-spacerun:yes">
</span></span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">The “</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">”
value is always the same for an s orbital no matter what shell it’s in.<o:p></o:p></span></p>
<p class="MsoNormal" style="text-indent:.5in"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span>It’s the same for a p orbital (subshell) no matter what shell it’s in.<span style="mso-spacerun:yes">
</span>This applies<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:.5in"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span>for all orbitals with the same value of </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ.<span style="mso-spacerun:yes">
</span>For a d-orbital ℓ = 2, not matter what<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:.5in"><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span>shell it’s in.<span style="mso-spacerun:yes"> </span><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">C. True.<span style="mso-spacerun:yes">
</span>See below.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">D. False.<span style="mso-tab-count:1">
</span>A </span><i><span style="font-size:14.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">d</span></i><i><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">x</span></i><sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">2</span></sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">−<i>y</i><sup>2</sup>
orbital looks the same in terms of shape, no matter the shell.<span style="mso-spacerun:yes">
</span>It<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><span style="mso-tab-count:2">
</span>doesn’t turn into a </span><span class="SpellE"><i><span style="font-size:14.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">d</span></i><i><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">xy</span></i></span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">
orbital.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">E. True.<span style="mso-spacerun:yes">
</span>The ℓ value tells us the shape and the <i>m_</i>ℓ value tells us the direction (orientation)<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><span style="mso-tab-count:1">
</span><span style="mso-spacerun:yes"> </span>of the orbitals.<span style="mso-spacerun:yes">
</span>For ℓ = 2, <i>m_</i>ℓ = -2, -1, 0, +1, +2.<span style="mso-spacerun:yes">
</span>So, a </span><i><span style="font-size:14.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">d</span></i><i><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">x</span></i><sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">2</span></sup><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">−<i>y</i><sup>2</sup>
has the same <i>m_</i>ℓ <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><span style="mso-tab-count:1">
</span><span style="mso-spacerun:yes"> </span>no matter which shell it’s in.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">For C above it involves nodes.<span style="mso-spacerun:yes">
</span>So does another question later in this e-mail.<span style="mso-spacerun:yes">
</span>So, I’ve<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">included a discussion below.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">total number of nodes for an orbital = n-1 (where n is the principal quantum #).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">#
<span class="GramE">spherical</span> (radial) nodes = n – </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> – 1 , where
</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ
</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">= the angular quantum # for the orbital.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">#
<span class="GramE">angular</span> (planar) nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">So, here’s a little table dealing with nodes:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">1s has 1-1 = 0 total nodes.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">n = 2, 2-1 = 1 total node<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">2s: # spherical (radial) nodes = 2 – 0 – 1 = 1<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 0 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=0 for an s orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">2p: # spherical nodes = 2 – 1 – 1 = 0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 1 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=1 for a p orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">n = 3,<span style="mso-spacerun:yes">
</span>3 -1 = 2 total nodes<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">3s : # spherical (radial) nodes = 3 – 0 – 1 = 2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 0 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=0 for an s orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">3p:<span style="mso-spacerun:yes">
</span># spherical nodes = 3 – 1 – 1 = 1<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 1 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=1 for a p orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">3d:<span style="mso-spacerun:yes">
</span># spherical nodes = 3 – 2 – 1 = 0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 2 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=2 for a d orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span>(<span class="GramE">for</span> the d_z^2 there are two canonical nodes<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span>which are angular nodes, not planar)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">n = 4,<span style="mso-spacerun:yes">
</span>4 – 1 = 3 total nodes<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">4s: # spherical (radial) nodes = 4 – 0 – 1 = 3<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = 0 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> =0 for an s orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">4p: # spherical nodes = 4 – 1 – 1 = 2<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 1 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=1 for a p orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">4d: #spherical nodes = 4 – 2 – 1 = 1<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 2 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=2 for a d orbital)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">4f: #spherical nodes = 4 – 3 – 1 = 0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes">
</span># <span class="GramE">planar</span> nodes = </span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""> = 3 (</span><span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">ℓ</span><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">
=3 for an f orbital)<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">Question 2 (Part B):<o:p></o:p></span></b></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">The probability of finding an electron at a point in an atom is referred to as<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">the probability density (<span class="mi"><i>ψ</i></span><span class="mn">2</span>). The spatial distribution of these densities can be<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">derived from the radial wave function
<span class="mi"><i>R</i></span><span class="mo">(</span><span class="mi"><i>r</i></span><span class="mo">)</span> and angular wave function
<span class="mi">Y</span><span class="mo">(</span><span class="SpellE"><span class="mi"><i>θ</i></span><span class="mo">,</span><span class="mi"><i>ϕ</i></span></span><span class="mo">)</span>,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">then solving the Schrödinger equation for a specific set of quantum numbers.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">Which of the following statements about nodes and probability density are<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif">accurate?<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<table class="MsoNormalTable" border="0" cellpadding="0" style="mso-cellspacing:1.5pt;mso-yfti-tbllook:1184">
<tbody>
<tr style="mso-yfti-irow:0;mso-yfti-firstrow:yes">
<td style="padding:.75pt .75pt .75pt .75pt">
<p class="MsoNormal"><span style="mso-ascii-font-family:Calibri;mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri">The probability of finding an electron at the center of a
<i>d</i> orbital is greater<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-ascii-font-family:Calibri;mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri">than zero.<span style="mso-spacerun:yes">
</span><b>FALSE</b><o:p></o:p></span></p>
</td>
</tr>
<tr style="mso-yfti-irow:1">
<td style="padding:.75pt .75pt .75pt .75pt">
<p class="MsoNormal"><span style="mso-ascii-font-family:Calibri;mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri">The 2<i>s</i> orbital does not have any nodes.<span style="mso-spacerun:yes">
</span><b>FALSE</b><o:p></o:p></span></p>
</td>
</tr>
<tr style="mso-yfti-irow:2">
<td style="padding:.75pt .75pt .75pt .75pt">
<p class="MsoNormal"><span style="mso-ascii-font-family:Calibri;mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri">The 4<i>f</i> orbitals have three nodes.<span style="mso-spacerun:yes">
</span><b>TRUE</b><o:p></o:p></span></p>
</td>
</tr>
<tr style="mso-yfti-irow:3">
<td style="padding:.75pt .75pt .75pt .75pt">
<p class="MsoNormal"><span style="mso-ascii-font-family:Calibri;mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri">The probability of finding an electron at the center of a
<i>p</i> orbital is zero.<span style="mso-spacerun:yes"> </span><b>TRUE</b><o:p></o:p></span></p>
</td>
</tr>
<tr style="mso-yfti-irow:4;mso-yfti-lastrow:yes">
<td style="padding:.75pt .75pt .75pt .75pt">
<p class="MsoNormal"><span style="mso-ascii-font-family:Calibri;mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri">The 3<i>p</i> orbitals have two nodes.
<b>TRUE</b><o:p></o:p></span></p>
</td>
</tr>
</tbody>
</table>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">Some of this is answered above in question one<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">The probability of finding the electron at the center of any orbital (at the<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">nucleus) is zero.<span style="mso-spacerun:yes">
</span>This answers both the 1<sup>st</sup> and 4<sup>th</sup> statements,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">as they’re covering the same concept.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">The 3p has 1 spherical (radial) node.<span style="mso-spacerun:yes">
</span>It also has 1 planar node.<span style="mso-spacerun:yes"> </span>That’s<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">at total of 2 nodes.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">The 2s has 1 node, a spherical (radial) node.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:"Arial",sans-serif;mso-bidi-font-family:"Times New Roman"">The 3p orbital has two notes, 1 radial and 1 planar<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">The 4f orbital has 3 nodes, 0 spherical (radial) nodes and 3 angular nodes.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman""><o:p> </o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;mso-add-space:auto">
<span style="font-family:"Arial",sans-serif;mso-fareast-font-family:"Times New Roman"">Dr. Zellmer<o:p></o:p></span></p>
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