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</head>
<body>
<div class="document">
<div class="section" id="heat-n-stuff">
<h1>Heat n' stuff</h1>
<div class="section" id="st-law-of-thermodynamics">
<h2>1st law of thermodynamics</h2>
<p>absorbed heat = change in internal energy + work done by gas</p>
<div class="line-block">
<div class="line"><span class="formula"><i>Q</i> = Δ<i>E</i><sub><i>internal</i></sub> + <i>W</i><sub><i>by</i> <i>gas</i></sub></span></div>
<div class="line"><span class="formula"><i>Q</i> = Δ<i>E</i><sub><i>internal</i></sub> − <i>W</i><sub><i>on</i> <i>gas</i></sub></span></div>
</div>
</div>
<div class="section" id="nd-law-of-thermodynamics">
<h2>2nd law of thermodynamics</h2>
<p>Heat never flows spontaneously from a colder body to a hotter body.</p>
</div>
<div class="section" id="rd-law-of-thermodynamics">
<h2>3rd law of thermodynamics</h2>
<p>It is impossible for any procedure to lead to the isotherm <cite>T = 0K</cite> in a finite number of steps.</p>
</div>
<div class="section" id="processes">
<h2>Processes</h2>
<ul class="simple">
<li>isobaric: constant pressure -&gt; work done by gas = <span class="formula"><i>p</i> × Δ<i>V</i></span></li>
<li>isochoric: constant volume -&gt; no work done</li>
<li>isothermal: constant temperature</li>
<li>adiabatic: no heat exchange</li>
</ul>
</div>
<div class="section" id="molar-heat-capacity">
<h2>Molar heat capacity</h2>
<div class="section" id="constant-volume">
<h3>Constant volume</h3>
<ul class="simple">
<li>monoatomic -&gt; <span class="formula"><i>C</i><sub><i>v</i></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator">3</span><span class="ignored">)/(</span><span class="denominator">2</span><span class="ignored">)</span></span><i>R</i></span></li>
<li>diatomic -&gt; <span class="formula"><i>C</i><sub><i>v</i></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator">5</span><span class="ignored">)/(</span><span class="denominator">2</span><span class="ignored">)</span></span><i>R</i></span></li>
<li><span class="formula"><i>Q</i> = <i>C</i><sub><i>v</i></sub><i>n</i>Δ<i>T</i></span></li>
</ul>
</div>
<div class="section" id="constant-pressure">
<h3>Constant pressure</h3>
<ul class="simple">
<li>monoatomic -&gt; <span class="formula"><i>C</i><sub><i>p</i></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator">5</span><span class="ignored">)/(</span><span class="denominator">2</span><span class="ignored">)</span></span><i>R</i></span></li>
<li>diatomic -&gt; <span class="formula"><i>C</i><sub><i>p</i></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator">7</span><span class="ignored">)/(</span><span class="denominator">2</span><span class="ignored">)</span></span><i>R</i></span></li>
<li><span class="formula"><i>Q</i> = <i>C</i><sub><i>p</i></sub><i>n</i>Δ<i>T</i></span></li>
</ul>
</div>
</div>
<div class="section" id="pv-diagram">
<h2>PV diagram</h2>
<ul class="simple">
<li>y axis: pressure</li>
<li>x axis: volume</li>
<li>work done: area under the line (negative if going left)</li>
<li>work done in a cycle: area inside (negative if counterclockwise)</li>
<li>horizontal lines are isobaric</li>
<li>vertical lines are isochoric</li>
</ul>
</div>
<div class="section" id="heat-engine">
<h2>Heat engine</h2>
<ul class="simple">
<li>takes heat from a hot reservoir, releases heat to a cold reservoir, does work</li>
<li>properties: <span class="formula"><i>Q</i><sub><i>in</i></sub>, <i>Q</i><sub><i>out</i></sub>, <i>W</i></span></li>
<li>efficiency: <span class="formula"><i>η</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>W</i></span><span class="ignored">)/(</span><span class="denominator"><i>Q</i><sub><i>in</i></sub></span><span class="ignored">)</span></span></span></li>
<li><span class="formula"><i>W</i> = <i>Q</i><sub><i>in</i></sub> − <i>Q</i><sub><i>out</i></sub></span></li>
</ul>
</div>
<div class="section" id="heat-pump">
<h2>Heat pump</h2>
<ul class="simple">
<li>we do work on it, takes heat from a cold reservoir, releases heat to a hot reservoir</li>
<li>properties: <span class="formula"><i>Q</i><sub><i>in</i></sub>, <i>Q</i><sub><i>out</i></sub>, <i>W</i></span></li>
<li>Coefficient of Performance: COP<ul>
<li><span class="formula"><i>COP</i><sub><i>heating</i></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>Q</i><sub><i>out</i></sub></span><span class="ignored">)/(</span><span class="denominator"><i>W</i></span><span class="ignored">)</span></span> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>Q</i><sub><i>out</i></sub></span><span class="ignored">)/(</span><span class="denominator"><i>Q</i><sub><i>out</i></sub> − <i>Q</i><sub><i>in</i></sub></span><span class="ignored">)</span></span></span></li>
<li><span class="formula"><i>COP</i><sub><i>cooling</i></sub> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>Q</i><sub><i>in</i></sub></span><span class="ignored">)/(</span><span class="denominator"><i>W</i></span><span class="ignored">)</span></span> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>Q</i><sub><i>in</i></sub></span><span class="ignored">)/(</span><span class="denominator"><i>Q</i><sub><i>out</i></sub> − <i>Q</i><sub><i>in</i></sub></span><span class="ignored">)</span></span></span></li>
</ul>
</li>
</ul>
</div>
</div>
<div class="section" id="gases">
<h1>Gases</h1>
<div class="section" id="kinetic-gas-theory">
<h2>Kinetic gas theory</h2>
<dl class="docutils">
<dt>Brownian motion</dt>
<dd>Observed the motion of pollen particles in water.</dd>
</dl>
</div>
<div class="section" id="kinetic-gas-model">
<h2>Kinetic gas model</h2>
<p>A sample of ideal gas consists of
- alike point like particles
- there is no interaction between the particles
- the particles undergo uniform straight line motion until they collide
- the collision is totally elastic</p>
</div>
<div class="section" id="macroscopic-properties">
<h2>Macroscopic properties</h2>
<ul class="simple">
<li>Volume: volume of container</li>
<li>mass: total mass of particles</li>
<li>pressure: comes from the force exerted on the wall when the particles collide with the wall</li>
<li>temperature: ?</li>
</ul>
</div>
<div class="section" id="a-simplified-model">
<h2>A simplified model</h2>
<p>The gas in a cube</p>
<div class="line-block">
<div class="line"><span class="formula"><i>a</i></span>: side length</div>
<div class="line"><span class="formula"><i>N</i></span>: number of particles</div>
<div class="line"><span class="formula"><i>m</i></span>: mass of a particle</div>
<div class="line"><span class="formula"><i>v</i></span>: average speed of the particles</div>
<div class="line"><br /></div>
<div class="line"><span class="formula"><i>N</i>* = <span class="fraction"><span class="ignored">(</span><span class="numerator">1</span><span class="ignored">)/(</span><span class="denominator">6</span><span class="ignored">)</span></span>*<i>N</i>*<i>v</i>*<i>dt</i> ⁄ <i>a</i></span></div>
<div class="line"><span class="formula"><i>p</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator">2</span><span class="ignored">)/(</span><span class="denominator">3</span><span class="ignored">)</span></span>*<i>N</i> ⁄ <i>V</i>*<i>E</i><sub><i>kinetic</i></sub></span></div>
<div class="line"><br /></div>
<div class="line"><span class="formula"><i>T</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator">3</span><span class="ignored">)/(</span><span class="denominator">2</span><span class="ignored">)</span></span><i>E</i><sub><i>kin</i></sub>*<i>Na</i> ⁄ <i>R</i></span></div>
</div>
</div>
</div>
</div>
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<p>2.a) (2√3)/3<br/>
2.c) 6-3√3<br/>
4.c) 3<sup>b+2</sup><br/>
6.a) 2<sup>-4</sup><br/>
6.b) 3<sup>k+4</sup><br/>
6.c) 5<sup>3a-b</sup><br/>
8.a) 2a<sup>18</sup>/8b<sup>6</sup><br/>
8.b) 25d<sup>-8</sup><br/>
8.c) 2z<sup>4</sup><br/>
10.a) 3<sup>3-2a</sup><br/>
10.b) 3<sup>6-8x</sup></p>
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<h2>3B</h2>
<p>1.h) n<sup>12</sup><br/>
1.i) 5<sup>3t</sup><br/>
1.j) 7<sup>x+2</sup><br/>
3.c) 3<sup>3</sup><br/>
3.f) 3<sup>-1</sup><br/>
3.h) 3<sup>-4</sup><br/>
5.a) 3<sup>p+2</sup><br/>
5.b) 3<sup>3a</sup><br/>
5.c) 3<sup>2n+1</sup><br/>
5.d) 3<sup>d+3</sup><br/>
5.e) 3<sup>3t+2</sup><br/>
5.f) 3<sup>y-1</sup><br/>
5.g) 3<sup>1-y</sup><br/>
5.h) 3<sup>2-3t</sup><br/>
5.i) 3<sup>3a-1</sup><br/>
5.j) 3<sup>3</sup><br/>
8.d) 9b<sup>2</sup>/a<sup>4</sup><br/>
8.e) a<sup>2</sup>/bc<sup>2</sup><br/>
8.i) 2/ad<sup>2</sup><br/>
8.j) 12am<sup>3</sup><br/>
9.b) b<sup>n</sup><br/>
9.d) a<sup>n</sup>b<sup>m</sup><br/>
10.c) 6<br/>
10.e) 9/16<br/>
10.g) 27/125</p>
<h2>3C</h2>
<p>1.g) 2<sup>3/2</sup><br/>
1.h) 2<sup>3/2</sup><br/>
1.i) 2<sup>-4/3</sup><br/>
1.j) 2<sup>-3/2</sup><br/>
2.b) 3<sup>-1/3</sup><br/>
2.e) 3<sup>-5/2</sup><br/>
3.b) 3<sup>&frac34;</sup><br/>
3.d) 2<sup>5/3</sup><br/>
3.f) 7<sup>-1/3</sup><br/>
3.h) 2<sup>-4/5</sup><br/>
3.j) 7<sup>-2/7</sup><br/>
5.a) 8<br/>
5.c) 8<br/>
5.e) 4<br/>
5.f) &frac12;<br/>
5.h) 1/16<br/>
5.j) 1/25</p>
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<h2>E1</h2>
<p>((a<sup>2</sup>)<sup>x-1</sup>(b/c)<sup>2x+1</sup>/a<sup>3-x</sup>/c(-4))<sup>3</sup>=a<sup>3x-5</sup>b<sup>2x+1</sup>c(3-2x)</p>
<h2>3B</h2>
<p>1.c) k<sup>5</sup><br/>
1.f) 3<sup>16</sup><br/>
1.k) 10<sup>3-q</sup><br/>
2.b) 2<sup>-2</sup><br/>
2.i) 2<sup>6</sup><br/>
2.j) 2<sup>-6</sup><br/>
4.a) 2<sup>a+1</sup><br/>
4.d) 2<sup>2x+2</sup><br/>
4.e) 2<sup>n-1</sup><br/>
4.g) 2<sup>2m</sup><br/>
4.h) 2<sup>1+n</sup><br/>
8.a) a/b<sup>2</sup><br/>
8.c) 4a<sup>2</sup>/b<sup>2</sup><br/>
8.f) a<sup>2</sup>c<sup>2</sup>/b<br/>
8.g) a<sup>3</sup><br/>
9.c) 3<sup>n-2</sup><br/>
9.e) a<sup>-2n-2</sup><br/>
10.b) 4/7</p>
<h2>E2</h2>
<p>a) 2
b) &frac12;
c) 25</p>
<h2>3C</h2>
<p>1.a) 2<sup>1/5</sup><br/>
1.b) 2<sup>-1/5</sup><br/>
1.c) 2<sup>3/2</sup><br/>
1.d) 2<sup>5/2</sup><br/>
1.e) 2<sup>-1/3</sup><br/>
1.f) 2<sup>4/3</sup><br/>
2.a) 3<sup>1/3</sup><br/>
2.d) 3<sup>3/2</sup><br/>
3.a) 7<sup>1/3</sup><br/>
3.c) 2<sup>4/5</sup><br/>
3.e) 7<sup>2/7</sup><br/>
3.g) 3<sup>-&frac34;</sup><br/>
3.i) 2<sup>-5/3</sup><br/>
5.b) 32<br/>
5.d) 125<br/>
5.g) 1/27<br/>
5.i) 1/81</p>
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