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<body>
<div class="document">
<p>Constant volume
~~~~~~~~~~~~~~~
- monoatomic -> <code>\(C_v = \frac{3}{2}R\)</code>
- diatomic -> <code>\(C_v = \frac{5}{2}R\)</code>
- <code>\(Q = C_v n \Delta T\)</code></p>
<p>Constant pressure
~~~~~~~~~~~~~~~~~
- monoatomic -> <code>\(C_p = \frac{5}{2}R\)</code>
- diatomic -> <code>\(C_p = \frac{7}{2}R\)</code>
- <code>\(Q = C_p n \Delta T\)</code></p>
<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">
<ul>
<li>y axis: pressure</li>
<li>x axis: volume</li>
<li>work done: area under the line (negative if going left)</li>
@ -699,71 +52,70 @@ span.boxed, span.framebox {
<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">
<ul>
<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>
<li>properties: <code>\(Q_in, Q_out, W\)</code></li>
<li>efficiency: <code>\(η = \frac{W}{Q_in}\)</code></li>
<li><code>\(W = Q_in-Q_out\)</code></li>
</ul>
</div>
<div class="section" id="heat-pump">
<h2>Heat pump</h2>
<ul class="simple">
<ul>
<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>
<li>properties: <code>\(Q_in, Q_out, W\)</code></li>
<li><p>Coefficient of Performance: COP</p>
<ul>
<li><code>\(COP_heating = \frac{Q_out}{W} = \frac{Q_out}{Q_out-Q_in}\)</code></li>
<li><code>\(COP_cooling = \frac{Q_in}{W} = \frac{Q_in}{Q_out-Q_in}\)</code></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">
<p>Brownian motion
Observed the motion of pollen particles in water.</p>
<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">
<ul>
<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>
<p><code>\(a\)</code>: side length
<code>\(N\)</code>: number of particles
<code>\(m\)</code>: mass of a particle
<code>\(v\)</code>: average speed of the particles</p>
<p><code>\(N* = \frac{1}{6}*N*v*dt/a\)</code>
<code>\(p = \frac{2}{3}*N/V*E_{kinetic}\)</code></p>
<p><code>\(T = \frac{3}{2} E_kin * Na / R\)</code></p>
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