<html><head><title>Math Addendum 6: Gas Laws</title><style type="text/css"> p { margin: 0 0 0 0; font-family: Arial, sans-serif; font-size:10pt; font-weight: normal; font-style: normal; vertical-align: baseline; color: black; text-decoration: none; } div { border-width:0px; border-style: solid; border-color: red; } body { background-color: #fff; } p img { vertical-align: middle; } p b, li b { font-weight : bold; } p i, li i { font-style : italic; } p u, li u { text-decoration : underline; } p so, li so { text-decoration : line-through; } p sub, p sub { font-size : 70%; vertical-align: sub; } p sup, p sup { font-size : 70%; vertical-align: super; } ul { padding-left : 2em; /* for mozilla list marker box */ margin-left : 0; /* for IE list marker box */ } ol { padding-left : 2em; /* for mozilla list marker box */ margin-right : -0.5em; margin-left : 0; /* for IE list marker box */ } p.Style1, li.Style1 /*Normal*/ { margin-left: 0pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; 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margin-right: 0pt; text-indent: -14.25pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style7, li.Style7 /*Indent*/ { margin-left: 108pt; margin-right: 0pt; text-align: left; font-family: 'Arial', sans-serif;; font-size: 10pt;font-weight: normal; font-style: normal; vertical-align: baseline; } p.Style8, li.Style8 /*Title*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 24pt;font-weight: bold; font-style: normal; vertical-align: baseline; } p.Style9, li.Style9 /*Subtitle*/ { margin-left: 0pt; margin-right: 0pt; text-align: center; font-family: 'Times New Roman', sans-serif;; font-size: 18pt;font-weight: normal; font-style: normal; vertical-align: baseline; } </style></head><body><div id="mc-region-498" style="position: absolute; top: 18pt; left: 42pt; width: 273.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0002_208630390.PNG" id="generatedImage498" path=".\addendum 6 gas laws_images/IMG0002_208630390.PNG" style="width: 273.75pt;height: 41.25pt;"></img></div><div id="mc-region-6" style="position: absolute; top: 72.75pt; left: 30pt; width: 295.25pt; "><a name="" /><p class="Style1"><span style="font-size: 12pt; font-family: 'Arial', sans-serif;"><b><span>Addendum 6: Gas laws for Planetary Atmospheres</span></b></span></p></div><div id="mc-region-385" style="position: absolute; top: 104.25pt; left: 30pt; width: 286.25pt; "><a name="" /><p class="Style1"><span style="font-family: 'Arial', sans-serif;"><i><span>First define some constants and dimensional units needed below</span></i></span></p></div><div id="mc-region-640" style="position: absolute; top: 116.25pt; left: 30pt; width: 45pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0003_208630406.PNG" id="generatedImage640" path=".\addendum 6 gas laws_images/IMG0003_208630406.PNG" style="width: 45pt;height: 33.75pt;"></img></div><div id="mc-region-635" style="position: absolute; top: 128.25pt; left: 156pt; width: 179pt; "><a name="" /><p class="Style1">Acceleration of gravity at Earth's surface</p></div><div id="mc-region-639" style="position: absolute; top: 153pt; left: 30pt; width: 82.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0004_208630406.PNG" id="generatedImage639" path=".\addendum 6 gas laws_images/IMG0004_208630406.PNG" style="width: 82.5pt;height: 22.5pt;"></img></div><div id="mc-region-636" style="position: absolute; top: 158.25pt; left: 156pt; width: 114.5pt; "><a name="" /><p class="Style1">Mass of atomic hydrogen</p></div><div id="mc-region-638" style="position: absolute; top: 182.25pt; left: 30pt; width: 116.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0005_208630421.PNG" id="generatedImage638" path=".\addendum 6 gas laws_images/IMG0005_208630421.PNG" style="width: 116.25pt;height: 27.75pt;"></img></div><div id="mc-region-637" style="position: absolute; top: 188.25pt; left: 156pt; width: 98.75pt; "><a name="" /><p class="Style1">Boltzmann's constant</p></div><div id="mc-region-19" style="position: absolute; top: 218.25pt; left: 24pt; width: 408.5pt; "><a name="" /><p class="Style1">1. <b>Perfect gas law</b><span style="font-family: 'Arial', sans-serif;"><b><i><span>.</span></i></b></span><span style="font-family: 'Arial', sans-serif;"><b><i><span> </span></i></b></span><span style="font-family: 'Arial', sans-serif;"><span style="font-weight: normal; font-style: normal;">T</span></span>he 'perfect gas law relates the temperature of a 'perfect' (idealized binary collision) gas to the pressure.</p></div><div id="mc-region-501" style="position: absolute; top: 255pt; left: 192pt; width: 44.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0006_208630421.PNG" id="generatedImage501" path=".\addendum 6 gas laws_images/IMG0006_208630421.PNG" style="width: 44.25pt;height: 16.5pt;"></img></div><div id="mc-region-500" style="position: absolute; top: 284.25pt; left: 48pt; width: 335.75pt; "><a name="" /><p class="Style1">where <i>n</i> is the number density and <i>k</i><i><sub>b</sub></i><sub> </sub>is Boltzmann's constant (given above).</p></div><div id="mc-region-24" style="position: absolute; top: 305.25pt; left: 36pt; width: 376.25pt; "><a name="" /><p class="Style1"><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;">Example: At 'STP' ('standard temperature and pressure, 0</span></i></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;"><sup>o </sup></span></i></span><span style="font-family: 'Arial', sans-serif;"><i><span style="font-weight: normal;">C (=273 K) and P = 1 stmosphere), what is the number density of gas moleocules? </span></i></span></p></div><div id="mc-region-506" style="position: absolute; top: 335.25pt; left: 42pt; width: 353pt; "><a name="" /><p class="Style1"><i>The MKS unit of pressure is a Pascal (N/m</i><i><sup>2</sup></i><i>). MathCAD knows the pressure unit atmosphere (1 atm = 1.01*10</i><i><sup>5</sup></i><i> Pascals):</i></p></div><div id="mc-region-647" style="position: absolute; top: 381pt; left: 42pt; width: 57pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0007_208630421.PNG" id="generatedImage647" path=".\addendum 6 gas laws_images/IMG0007_208630421.PNG" style="width: 57pt;height: 16.5pt;"></img></div><div id="mc-region-646" style="position: absolute; top: 375pt; left: 114pt; width: 91.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0008_208630437.PNG" id="generatedImage646" path=".\addendum 6 gas laws_images/IMG0008_208630437.PNG" style="width: 91.5pt;height: 22.5pt;"></img></div><div id="mc-region-645" style="position: absolute; top: 381pt; left: 222pt; width: 59.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0009_208630437.PNG" id="generatedImage645" path=".\addendum 6 gas laws_images/IMG0009_208630437.PNG" style="width: 59.25pt;height: 16.5pt;"></img></div><div id="mc-region-641" style="position: absolute; top: 400.5pt; left: 48pt; width: 73.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0010_208630437.PNG" id="generatedImage641" path=".\addendum 6 gas laws_images/IMG0010_208630437.PNG" style="width: 73.5pt;height: 35.25pt;"></img></div><div id="mc-region-644" style="position: absolute; top: 411pt; left: 150pt; width: 99pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0011_208630437.PNG" id="generatedImage644" path=".\addendum 6 gas laws_images/IMG0011_208630437.PNG" style="width: 99pt;height: 22.5pt;"></img></div><div id="mc-region-522" style="position: absolute; top: 452.25pt; left: 30pt; width: 408.5pt; "><a name="" /><p class="Style1">2. <b>Molecular masses</b>. Atomic hydrogen has a mass = m<sub>h</sub> (given above in constants). The mass of molecules can can determined by multipying the atomic mass Z by the number of atoms per molecule times the mass of hydogren (since the mass of protons and neutrons are nearly identiucal and much more massive than the electrons). </p></div><div id="mc-region-523" style="position: absolute; top: 512.25pt; left: 54pt; width: 265.25pt; "><a name="" /><p class="Style1"><i>Example: What is the mass of molecular nitrogen (N</i><i><sub>2</sub></i><i>)? O</i><i><sub>2</sub></i><i>?</i></p></div><div id="mc-region-524" style="position: absolute; top: 537pt; left: 60pt; width: 39.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0012_208630453.PNG" id="generatedImage524" path=".\addendum 6 gas laws_images/IMG0012_208630453.PNG" style="width: 39.75pt;height: 16.5pt;"></img></div><div id="mc-region-525" style="position: absolute; top: 537pt; left: 114pt; width: 71.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0013_208630453.PNG" id="generatedImage525" path=".\addendum 6 gas laws_images/IMG0013_208630453.PNG" style="width: 71.25pt;height: 16.5pt;"></img></div><div id="mc-region-527" style="position: absolute; top: 531pt; left: 198pt; width: 99.75pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0014_208630453.PNG" id="generatedImage527" path=".\addendum 6 gas laws_images/IMG0014_208630453.PNG" style="width: 99.75pt;height: 22.5pt;"></img></div><div id="mc-region-535" style="position: absolute; top: 567pt; left: 66pt; width: 39.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0015_208630453.PNG" id="generatedImage535" path=".\addendum 6 gas laws_images/IMG0015_208630453.PNG" style="width: 39.75pt;height: 16.5pt;"></img></div><div id="mc-region-536" style="position: absolute; top: 567pt; left: 120pt; width: 71.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0016_208630453.PNG" id="generatedImage536" path=".\addendum 6 gas laws_images/IMG0016_208630453.PNG" style="width: 71.25pt;height: 16.5pt;"></img></div><div id="mc-region-537" style="position: absolute; top: 561pt; left: 204pt; width: 99.75pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0017_208630468.PNG" id="generatedImage537" path=".\addendum 6 gas laws_images/IMG0017_208630468.PNG" style="width: 99.75pt;height: 22.5pt;"></img></div><div id="mc-region-520" style="position: absolute; top: 626.25pt; left: 24pt; width: 403.25pt; "><a name="" /><p class="Style1">3. <b>Pressure and mass density of atmosphere</b>. Air 'weighs' a surprising amount. Since the Earth's atmosphere is 80% molecular nitrogen (N<sub>2</sub>) and nearly 20% molecular oxygen (O<sub>2</sub>). Hence the mean molecular mass of sn air moleculae can be estimated as</p></div><div id="mc-region-539" style="position: absolute; top: 681pt; left: 66pt; width: 106.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0018_208630468.PNG" id="generatedImage539" path=".\addendum 6 gas laws_images/IMG0018_208630468.PNG" style="width: 106.5pt;height: 16.5pt;"></img></div><div id="mc-region-540" style="position: absolute; top: 675pt; left: 198pt; width: 98.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0019_208630468.PNG" id="generatedImage540" path=".\addendum 6 gas laws_images/IMG0019_208630468.PNG" style="width: 98.25pt;height: 22.5pt;"></img></div><div id="mc-region-554" style="position: absolute; top: 704.25pt; left: 30pt; width: 236pt; "><a name="" /><p class="Style1">What is the mass density of the atmosphere at STP?</p></div><div id="mc-region-557" style="position: absolute; top: 728.25pt; left: 72pt; width: 78pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0020_208630468.PNG" id="generatedImage557" path=".\addendum 6 gas laws_images/IMG0020_208630468.PNG" style="width: 78pt;height: 17.25pt;"></img></div><div id="mc-region-558" style="position: absolute; top: 722.25pt; left: 162pt; width: 72.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0021_208630468.PNG" id="generatedImage558" path=".\addendum 6 gas laws_images/IMG0021_208630468.PNG" style="width: 72.75pt;height: 33.75pt;"></img></div><div id="mc-region-541" style="position: absolute; top: 764.25pt; left: 36pt; width: 177.5pt; "><a name="" /><p class="Style1">This means a cubic meter of air weighs:</p></div><div id="mc-region-559" style="position: absolute; top: 783pt; left: 120pt; width: 72pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0022_208630468.PNG" id="generatedImage559" path=".\addendum 6 gas laws_images/IMG0022_208630468.PNG" style="width: 72pt;height: 22.5pt;"></img></div><div id="mc-region-560" style="position: absolute; top: 789pt; left: 204pt; width: 45.75pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0023_208630484.PNG" id="generatedImage560" path=".\addendum 6 gas laws_images/IMG0023_208630484.PNG" style="width: 45.75pt;height: 12.75pt;"></img></div><div id="mc-region-30" style="position: absolute; top: 818.25pt; left: 24pt; width: 408.5pt; "><a name="" /><p class="Style1" style="margin-left: 0pt; margin-right: 0pt; text-align: left;">4. <b>Scale Height. </b><span style="font-weight: normal;">The scale height of an atmosphere is the characteristic height that the pressure decreases by a factor 1/e = 0.367. We can determine the change in pressure over a height difference dz by considering a cylinder with cross section A and differencing the pressure on top and bottom surfaces: </span></p></div><div id="mc-region-600" style="position: absolute; top: 878.25pt; left: 60pt; width: 54.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0024_208630484.PNG" id="generatedImage600" path=".\addendum 6 gas laws_images/IMG0024_208630484.PNG" style="width: 54.75pt;height: 13.5pt;"></img></div><div id="mc-region-601" style="position: absolute; top: 878.25pt; left: 138pt; width: 193.25pt; "><a name="" /><p class="Style1">see lecture notes for diagram and derivation</p></div><div id="mc-region-568" style="position: absolute; top: 902.25pt; left: 48pt; width: 50pt; "><a name="" /><p class="Style1">and since </p></div><div id="mc-region-569" style="position: absolute; top: 903pt; left: 108pt; width: 44.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0025_208630484.PNG" id="generatedImage569" path=".\addendum 6 gas laws_images/IMG0025_208630484.PNG" style="width: 44.25pt;height: 16.5pt;"></img></div><div id="mc-region-571" style="position: absolute; top: 902.25pt; left: 168pt; width: 104pt; "><a name="" /><p class="Style1">we can divide dP by P:</p></div><div id="mc-region-594" style="position: absolute; top: 927.75pt; left: 54pt; width: 84pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0026_208630484.PNG" id="generatedImage594" path=".\addendum 6 gas laws_images/IMG0026_208630484.PNG" style="width: 84pt;height: 36pt;"></img></div><div id="mc-region-589" style="position: absolute; top: 986.25pt; left: 18pt; width: 47.75pt; "><a name="" /><p class="Style1">but since:</p></div><div id="mc-region-599" style="position: absolute; top: 975.75pt; left: 72pt; width: 54.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0027_208630484.PNG" id="generatedImage599" path=".\addendum 6 gas laws_images/IMG0027_208630484.PNG" style="width: 54.75pt;height: 36pt;"></img></div><div id="mc-region-591" style="position: absolute; top: 986.25pt; left: 156pt; width: 49.25pt; "><a name="" /><p class="Style1">we obtain:</p></div><div id="mc-region-592" style="position: absolute; top: 976.5pt; left: 228pt; width: 74.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0028_208630484.PNG" id="generatedImage592" path=".\addendum 6 gas laws_images/IMG0028_208630484.PNG" style="width: 74.25pt;height: 35.25pt;"></img></div><div id="mc-region-593" style="position: absolute; top: 986.25pt; left: 330pt; width: 13.25pt; "><a name="" /><p class="Style1">or</p></div><div id="mc-region-595" style="position: absolute; top: 980.25pt; left: 354pt; width: 37.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0029_208630500.PNG" id="generatedImage595" path=".\addendum 6 gas laws_images/IMG0029_208630500.PNG" style="width: 37.5pt;height: 27.75pt;"></img></div><div id="mc-region-605" style="position: absolute; top: 1028.25pt; left: 24pt; width: 33.5pt; "><a name="" /><p class="Style1">where </p></div><div id="mc-region-604" style="position: absolute; top: 1018.5pt; left: 78pt; width: 61.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0030_208630500.PNG" id="generatedImage604" path=".\addendum 6 gas laws_images/IMG0030_208630500.PNG" style="width: 61.5pt;height: 35.25pt;"></img></div><div id="mc-region-648" style="position: absolute; top: 1029pt; left: 180pt; width: 73.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0031_208630500.PNG" id="generatedImage648" path=".\addendum 6 gas laws_images/IMG0031_208630500.PNG" style="width: 73.5pt;height: 16.5pt;"></img></div><div id="mc-region-649" style="position: absolute; top: 1029pt; left: 288pt; width: 52.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0032_208630500.PNG" id="generatedImage649" path=".\addendum 6 gas laws_images/IMG0032_208630500.PNG" style="width: 52.5pt;height: 16.5pt;"></img></div><div id="mc-region-608" style="position: absolute; top: 1058.25pt; left: 24pt; width: 364.25pt; "><a name="" /><p class="Style1">If we integrate the differential equation we get (assuming constant T =&gt; constant H)</p></div><div id="mc-region-610" style="position: absolute; top: 1082.25pt; left: 72pt; width: 92.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0033_208630500.PNG" id="generatedImage610" path=".\addendum 6 gas laws_images/IMG0033_208630500.PNG" style="width: 92.25pt;height: 41.25pt;"></img></div><div id="mc-region-651" style="position: absolute; top: 1142.25pt; left: 30pt; width: 402.5pt; "><a name="" /><p class="Style1"><i><span>Example: What is the ratio of the atmospheric pressure in Leadville, Colorado (altitude 10,000') compared with sea level? (Assume temperature is 0 C = 273 K in both locations)</span></i></p></div><div id="mc-region-652" style="position: absolute; top: 1173pt; left: 66pt; width: 41.25pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0034_208630500.PNG" id="generatedImage652" path=".\addendum 6 gas laws_images/IMG0034_208630500.PNG" style="width: 41.25pt;height: 18.75pt;"></img></div><div id="mc-region-654" style="position: absolute; top: 1179pt; left: 138pt; width: 69pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0035_208630515.PNG" id="generatedImage654" path=".\addendum 6 gas laws_images/IMG0035_208630515.PNG" style="width: 69pt;height: 16.5pt;"></img></div><div id="mc-region-656" style="position: absolute; top: 1244.25pt; left: 18pt; width: 410pt; "><a name="" /><p class="Style1">5. <b>Average speed of a gas particle</b> The average speed of a gas particle is only a function of gas temperature and the particel mass. It is calculated given by</p></div><div id="mc-region-657" style="position: absolute; top: 1281pt; left: 132pt; width: 115.5pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0036_208630515.PNG" id="generatedImage657" path=".\addendum 6 gas laws_images/IMG0036_208630515.PNG" style="width: 115.5pt;height: 36.75pt;"></img></div><div id="mc-region-658" style="position: absolute; top: 1328.25pt; left: 42pt; width: 323.75pt; "><a name="" /><p class="Style1"><i>Example: What is the average speed of an oxygen molecule in this room?</i></p></div><div id="mc-region-661" style="position: absolute; top: 1353pt; left: 72pt; width: 63pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0037_208630515.PNG" id="generatedImage661" path=".\addendum 6 gas laws_images/IMG0037_208630515.PNG" style="width: 63pt;height: 16.5pt;"></img></div><div id="mc-region-664" style="position: absolute; top: 1353pt; left: 150pt; width: 108.75pt; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0038_208630515.PNG" id="generatedImage664" path=".\addendum 6 gas laws_images/IMG0038_208630515.PNG" style="width: 108.75pt;height: 16.5pt;"></img></div><div id="mc-region-665" style="position: absolute; top: 1346.25pt; left: 276pt; width: 60.75pt; background-color: inherit; "><a name="" /><img border="0" src=".\addendum 6 gas laws_images/IMG0039_208630531.PNG" id="generatedImage665" path=".\addendum 6 gas laws_images/IMG0039_208630531.PNG" style="width: 60.75pt;height: 27.75pt;"></img></div><div></div></body></html>