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The Ideal Gas Laws Chapter 14 Expectations After this chapter, students will: Know what a “mole” is Understand and apply atomic mass, the atomic mass unit, and Avogadro’s number Understand how an ideal gas differs from real ones Use the ideal gas equation, Boyle’s Law, and Charles’ Law, to solve problems Expectations After this chapter, students will: understand the connection between the macroscopic properties of gases and the microscopic mechanics of gas molecules Preliminaries: the Mole A mole is a very large number of discrete objects, such as atoms, molecules, or sand grains. Specifically, it is Avogadro’s Number (NA) of such things: 6.022×1023 of them. The mole (“mol”) is not a dimensional unit; it is a label. Amadeo Avogadro 1776 – 1856 Native of Turin, Italy Hypothesized that equal volumes of gases at the same temperature and pressure contained equal numbers of molecules. (He was correct, too.) The Mole and Atomic Mass Mathematical definition: 12 g of C12 contains one mole of carbon-12 atoms. 12 g .012 kg Mass of one C12 atom: 1.993 10 -26 kg N A 6.022 10 23 The mass of one C12 atom is also 12 atomic mass units (amu), so: 1.993 10 -26 kg 1 amu 1.661 10 -27 kg 12 The Mole and Atomic Mass Atomic masses for the elements may be found in the periodic table of the elements, located inside the back cover of your textbook. These are often erroneously called “atomic weights.” Atomic masses may be added to calculate molecular masses for chemical compounds (or diatomic elements). The Mole: Calculations If we have N particles, how many moles is that? N number of moles n NA If we have a given mass of something, how many moles do we have? mass of sample mass of sample n mass per mole atomic or molecular mass The Ideal Gas The notion of an “ideal” gas developed from the efforts of scientists in the 18th and 19th centuries to link the macroscopic behavior of gases (volume, temperature, and pressure) to the Newtonian mechanics of the tiny particles that were increasingly seen as the microscopic constituents of gases. The Ideal Gas An ideal gas was one whose particles are well- behaved, in terms of the Newtonian theory of collisions: elastic collisions and the impulse- momentum theorem. An ideal gas is one in which the particles have no interaction, except for perfectly-elastic collisions with each other, and with the walls of their container. The Ideal Gas An ideal gas has no chemistry. That is, the particles (atoms or molecules) have no tendency to “stick” to other particles through chemical bonds. Inert gases (He, Ne, Ar, Kr, Xe, Rn) at low densities are very good approximations to the ideal gas. Our analytic model of the ideal gas gives us insights into the properties of many real gases, inert or not. The Ideal Gas Equation Observations from experience The pressure of a gas is directly proportional to the number of moles of particles in a given space. Example: blow up a balloon, and you’re adding to n, the number of moles of molecules. Conclusion: Pn The Ideal Gas Equation Observations from experience The pressure of a gas is directly proportional to its temperature. Example: toss a spray can into a fire (no, wait, really, don’t do it, just think about it). Increasing pressure will cause the can to fail catastrophically. Conclusion: P T The Ideal Gas Equation Observations from experience The pressure of a gas is inversely proportional to its volume. Example: squeeze the air in a half-filled balloon down to one end and squeeze it tighter. Increased pressure makes the balloon’s skin tight. 1 Conclusion: P V The Ideal Gas Equation Combine the observations nT P V A constant of proportionality, R, makes this an equation: nT PR or PV nRT V The Ideal Gas Equation The constant of proportionality, R, is called the universal gas constant. Its value and units depend on the units used for P, V, and T. pressure volume absolute PV nRT temperature universal gas constant number of moles Value and SI units of R: 8.31 J / (mol K) The Ideal Gas Equation We can also write the ideal gas equation in terms of the number of particles, N, instead of the number of moles, n. Since N = n·NA, we can both multiply and divide the right-hand side by NA: R PV nN A N T Boltzmann’s constant A R PV NkT where k 1.38 10-23 J/K NA Ludwig Boltzmann Austrian physicist 1844 – 1906 Boyle’s Law Suppose we hold both n and T constant: how are P and V related? PV nRT PV constant PV1 P2V2 1 This is called Boyle’s Law. Robert Boyle Irish mathematician 1627 – 1691 Charles’ Law Suppose we hold both n and P constant: how are T and V related? V nR PV nRT constant T P V1 V2 T1 T2 This is called Charles’ Law. Jacques Alexandre Cesar Charles French scientist 1746 – 1823 Built and flew the first large hydrogen-filled balloon. Kinetic Theory of the Ideal Gas Macroscopic properties of a gas: temperature, pressure, volume, density Microscopic properties of the particles making up the gas: mass, velocity, momentum, kinetic energy How are they related? Kinetic Theory of the Ideal Gas Consider a gas molecule contained in a cube having edge length L. The molecule’s mass is m, and its velocity (in the X direction only) is v. Time between collisions with the right-hand wall: 2L t v Kinetic Theory of the Ideal Gas The time between collisions with the right-hand wall is just the round-trip time: 2L t v From the impulse-momentum theorem, we can calculate the average force exerted on the particle by the wall: J F t p p f p0 m v mv Kinetic Theory of the Ideal Gas Substitute for the time and simplify: J F t p p f p0 m v mv 2L F 2mv v mv 2 F L By Newton’s third law, the average mv 2 force exerted on the wall is F L Kinetic Theory of the Ideal Gas The average force on the wall from one particle is mv 2 F L If there are N particles, and their directions are random, we could expect 1/3 of them to be moving in the X direction. 2 N mv Total force on the wall: F 3 L Kinetic Theory of the Ideal Gas Average pressure on the wall: N mv 2 F N mv 2 P 3 2L A L 3 L3 But V L 3 So: 2 N mv N P PV mv 2 NkT 3 V 3 Kinetic Theory of the Ideal Gas Substituting kinetic energy: 1 1 1 2 2 kT mv 2 mv KE 2 3 3 2 3 3 KE kT 2 So, we see that for an ideal gas, the average molecular kinetic energy is directly proportional to the absolute temperature. Kinetic Theory of the Ideal Gas 3 KE kT This result is true for any ideal gas. 2 By a similar argument, if an ideal gas is monatomic (the gas particles are single atoms), the internal energy of n moles of the gas at an absolute temperature T is 3 U nRT 2

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