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<p><font face="serif">Some of you are struggling with problem 9.10
in the text (which is part of the Mastering online homework).
The key to unlocking these problems is recognizing that you are
dealing with EQUAL VOLUMES of EQUIMOLAR SOLUTIONS. Parts B and
D seem to be the hardest.</font></p>
<p><font face="serif">For part B you are told that the equation is
first order with respect to both A and B. <br>
</font></p>
<p><font face="serif"><math display="block"
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mo
stretchy="false">[</mo><mi>A</mi><mo
stretchy="false">]</mo></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo
stretchy="false">[</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation
encoding="TeX">\frac{d[A]}{dt}=k[A][B]</annotation></semantics></math></font>but
remember, these are equal volumes and equal concentrations, so
[A]=[B]</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo
stretchy="false">[</mo><mi>B</mi><mo stretchy="false">]</mo><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><msup><mo
stretchy="false">]</mo><mn>2</mn></msup></mrow><annotation
encoding="TeX">\frac{d[A]}{dt}=k[A][B]=k[A][A]=k[A]^2</annotation></semantics></math>so
you can treat this essentially using second order kinetics and use
the second order integrated rate law to determine the
concentrations as a function of time.<br>
</p>
<p>In part D you are told that the equation is first order with
respect to A and half-order with respect to B. Again, employing
the fact that [A]=[B]..</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo
stretchy="false">[</mo><mi>B</mi><msup><mo
stretchy="false">]</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo
stretchy="false">[</mo><mi>A</mi><msup><mo
stretchy="false">]</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mi>k</mi><mo
stretchy="false">[</mo><mi>A</mi><msup><mo
stretchy="false">]</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><annotation
encoding="TeX">\frac{d[A]}{dt}=k[A][B]^{1/2}=k[A][A]^{1/2}=k[A]^{3/2}</annotation></semantics></math>Use
the same separation of variable techniques that we used before but
apply it to this particular problems...</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo
stretchy="false">[</mo><mi>A</mi><msub><mo
stretchy="false">]</mo><mn>0</mn></msub></mrow><mrow><mo
stretchy="false">[</mo><mi>A</mi><msub><mo
stretchy="false">]</mo><mi>t</mi></msub></mrow></msubsup><mfrac><mrow><mi>d</mi><mo
stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><mrow><mo
stretchy="false">[</mo><mi>A</mi><msup><mo
stretchy="false">]</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mi>k</mi><mi>d</mi><mi>t</mi></mrow><annotation
encoding="TeX">\int_{[A]_0}^{[A]_t}\frac{d[A]}{[A]^{3/2}}=\int_0^tkdt</annotation></semantics></math><br>
</p>
<p>to come up with the integrated rate law that you can then use to
solve for the concenrations.</p>
<p><br>
</p>
<div class="moz-signature">-- <br>
<b><i>Andrew J. Pounds, Ph.D.</i></b><br>
<i>Professor of Chemistry and Computer Science</i><br>
<i>Director of the Computational Science Program</i><br>
<i>Mercer University, Macon, GA 31207 (478) 301-5627</i></div>
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