{"id":262,"date":"2016-01-11T19:59:52","date_gmt":"2016-01-11T19:59:52","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistryclone\/chapter\/other-gas-laws-2\/"},"modified":"2020-05-05T17:24:41","modified_gmt":"2020-05-05T17:24:41","slug":"other-gas-laws","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistryclone\/chapter\/other-gas-laws\/","title":{"raw":"Other Gas Laws","rendered":"Other Gas Laws"},"content":{"raw":"[latexpage]\r\n
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

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  1. Review other simple gas laws.<\/li>\r\n \t
  2. Learn and apply the combined gas law.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nYou may notice in Boyle\u2019s law and Charles\u2019s law that we actually refer to four physical properties of a gas: pressure (P<\/i>), volume (V<\/i>), temperature (T<\/i>), and amount (in moles,\u00a0n<\/i>). We do this because these are the only four independent physical properties of a gas. There are other physical properties, but they are all related to one (or more) of these four properties.\r\n\r\nBoyle\u2019s law is written in terms of two of these properties, with the other two being held constant. Charles\u2019s law is written in terms of two different properties, with the other two being held constant. It may not be surprising to learn that there are other gas laws that relate other pairs of properties\u2014as long as the other two are held constant. Here we will mention a few.\r\n\r\n[pb_glossary id=\"1564\"]Gay-Lussac\u2019s law[\/pb_glossary] relates pressure with absolute temperature. In terms of two sets of data, Gay-Lussac\u2019s law is:\r\n\r\n\\[\\dfrac{P_1}{T_1}=\\dfrac{P_2}{T_2}\\text{ at constant }V\\text{ and }n\\]\r\n\r\nNote that it has a structure very similar to that of Charles\u2019s law, only with different variables \u2014 pressure instead of volume. [pb_glossary id=\"1566\"]Avogadro\u2019s law[\/pb_glossary] introduces the last variable for amount. The original statement of Avogadro\u2019s law states that equal volumes of different gases at the same temperature and pressure contain the same number of particles of gas. Because the number of particles is related to the number of moles (1 mol = 6.022 \u00d7 1023<\/sup> particles), Avogadro\u2019s law essentially states that equal volumes of different gases at the same temperature and pressure contain the same amount<\/em> (moles, particles) of gas. Put mathematically into a gas law, Avogadro\u2019s law is:\r\n\r\n\\[\\dfrac{V_1}{n_1}=\\dfrac{V_2}{n_2}\\text{ at constant }P\\text{ and }T\\]\r\n\r\n(First announced in 1811, it was Avogadro\u2019s proposal that volume is related to the number of particles that eventually led to naming the number of things in a mole as Avogadro\u2019s number.) Avogadro\u2019s law is useful because for the first time we are seeing amount, in terms of the number of moles, as a variable in a gas law.\r\n
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    Example 6.7<\/p>\r\n\r\n<\/header>\r\n

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    Problem<\/h1>\r\nA 2.45 L volume of gas contains 4.5 \u00d7 1021<\/sup> gas particles. How many gas particles are there in 3.87 L if the gas is at constant pressure and temperature?\r\n

    Solution<\/h2>\r\nWe can set up Avogadro\u2019s law as follows:\r\n\r\n\\[\\dfrac{2.45\\text{ L}}{4.5\\times 10^{21}\\text{ particles}}=\\dfrac{3.87\\text{ L}}{n_2}\\]\r\n\r\nWe algebraically rearrange to solve for n<\/i>2<\/sub>:\r\n\r\n\\[n_2=\\dfrac{(3.87\\text{ \\cancel{L}})(4.5\\times 10^{21}\\text{ particles})}{2.45\\text{ \\cancel{L}}}\\]\r\n\r\nThe L units cancel, so we solve for n<\/i>2<\/sub>:\r\n

    n<\/i>2<\/sub> = 7.1 \u00d7 1021<\/sup> particles<\/p>\r\n\r\n

    Test Yourself<\/h1>\r\nA 12.8 L volume of gas contains 3.00 \u00d7 1020<\/sup> gas particles. At constant temperature and pressure, what volume do 8.22 \u00d7 1018<\/sup> gas particles fill?\r\n

    Answer<\/h2>\r\n0.351 L\r\n\r\n<\/div>\r\n<\/div>\r\nThe variable n<\/i> in Avogadro\u2019s law can also stand for the number of moles of gas in addition to the number of particles.\r\n\r\nOne thing we notice about all the gas laws is that, collectively, volume and pressure are always in the numerator, and temperature is always in the denominator. This suggests that we can propose a gas law that combines pressure, volume, and temperature. This gas law is known as the [pb_glossary id=\"1568\"]combined gas law[\/pb_glossary], and its mathematical form is:\r\n\r\n\\[\\dfrac{P_1V_1}{T_1}=\\dfrac{P_2V_2}{T_2}\\text{ at constant }n\\]\r\n\r\nThis allows us to follow changes in all three major properties of a gas. Again, the usual warnings apply about how to solve for an unknown algebraically (isolate it on one side of the equation in the numerator), units (they must be the same for the two similar variables of each type), and units of temperature (must be in kelvins).\r\n
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    Example 6.8<\/p>\r\n\r\n<\/header>\r\n

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    Problem<\/h1>\r\nA sample of gas at an initial volume of 8.33 L, an initial pressure of 1.82 atm, and an initial temperature of 286 K simultaneously changes its temperature to 355 K and its volume to 5.72 L. What is the final pressure of the gas?\r\n

    Solution<\/h2>\r\nWe can use the combined gas law directly; all the units are consistent with each other, and the temperatures are given in Kelvin. Substituting,\r\n\r\n\\[\\dfrac{(1.82\\text{ atm})(8.33\\text{ L})}{286\\text{ K}}=\\dfrac{P_2(5.72\\text{ L})}{355\\text{ K}}\\]\r\n\r\nWe rearrange this to isolate the P<\/i>2<\/sub> variable all by itself. When we do so, certain units cancel:\r\n\r\n\\[\\dfrac{(1.82\\text{ atm})(8.33\\text{ \\cancel{L}})(355\\text{ \\cancel{K}})}{(286\\text{ \\cancel{K}})(5.72\\text{ \\cancel{L}})}=P_2\\]\r\n\r\nMultiplying and dividing all the numbers, we get:\r\n

    P<\/i>2<\/sub> = 3.29 atm<\/p>\r\nUltimately, the pressure increased, which would have been difficult to predict because two properties of the gas were changing.\r\n

    Test Yourself<\/h1>\r\nIf P<\/i>1<\/sub> = 662 torr, V<\/i>1<\/sub> = 46.7 mL, T<\/i>1<\/sub> = 266 K, P<\/i>2<\/sub> = 409 torr, and T<\/i>2<\/sub> = 371 K, what is V<\/i>2<\/sub>?\r\n

    Answer<\/h2>\r\n105 mL\r\n\r\n<\/div>\r\n<\/div>\r\nAs with other gas laws, if you need to determine the value of a variable in the denominator of the combined gas law, you can either cross-multiply all the terms or just take the reciprocal of the combined gas law. Remember, the variable you are solving for must be in the numerator and all by itself on one side of the equation.\r\n
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    Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

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