2Expectations After this chapter, students will: Know what a “mole” is Understand and apply atomic mass, the atomic mass unit, and Avogadro’s numberUnderstand how an ideal gas differs from real onesUse the ideal gas equation, Boyle’s Law, and Charles’ Law, to solve problems
3Expectations After this chapter, students will: understand the connection between the macroscopic properties of gases and the microscopic mechanics of gas molecules
4Preliminaries: 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.
5Amadeo Avogadro 1776 – 1856 Native of Turin, Italy Hypothesized that equal volumesof gases at the same temperatureand pressure contained equalnumbers of molecules.(He was correct, too.)
6The Mole and Atomic Mass Mathematical definition: 12 g of C12 contains one mole of carbon-12 atoms.Mass of one C12 atom:The mass of one C12 atom is also 12 atomic mass units (amu), so:
7The 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).
8The Mole: Calculations If we have N particles, how many moles is that?If we have a given mass of something, how many moles do we have?number of moles
9The Ideal GasThe 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.
10The Ideal GasAn 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.
11The Ideal GasAn 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.
12The 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:
13The 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:
14The 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.Conclusion:
15The Ideal Gas Equation Combine the observations A constant of proportionality, R, makes this an equation:
16The Ideal Gas EquationThe constant of proportionality, R, is called the universal gas constant. Its value and units depend on the units used for P, V, and T.Value and SI units of R: 8.31 J / (mol K)pressurevolumeabsolute temperatureuniversal gas constantnumber of moles
17The Ideal Gas EquationWe 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:Boltzmann’s constant
21Charles’ LawSuppose we hold both n and P constant: how are T and V related?This is called Charles’ Law.
22Jacques Alexandre Cesar Charles French scientist1746 – 1823Built and flew the firstlarge hydrogen-filledballoon.
23Kinetic Theory of the Ideal Gas Macroscopic properties of a gas: temperature, pressure, volume, densityMicroscopic properties of the particles making up the gas: mass, velocity, momentum, kinetic energyHow are they related?
24Kinetic Theory of the Ideal Gas Consider a gas molecule contained in a cube having edge length L.The molecule’s mass is m, andits velocity (in the X directiononly) is v.Time between collisions with theright-hand wall:
25Kinetic Theory of the Ideal Gas The time between collisions with the right-hand wall is just the round-trip time:From the impulse-momentumtheorem, we can calculate theaverage force exerted on theparticle by the wall:
26Kinetic Theory of the Ideal Gas Substitute for the time and simplify:By Newton’s third law, the averageforce exerted on the wall is
27Kinetic Theory of the Ideal Gas The average force on the wall from one particle isIf there are N particles, andtheir directions are random, wecould expect 1/3 of them to bemoving in the X direction.Total force on the wall:
28Kinetic Theory of the Ideal Gas Average pressure on the wall:But So:
29Kinetic Theory of the Ideal Gas Substituting kinetic energy:So, we see that for an ideal gas,the average molecular kinetic energyis directly proportional to theabsolute temperature.
30Kinetic Theory of the Ideal Gas This result is true for any ideal gas.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