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1 Kinetic Molecular Theory of Gases 1.A gas is composed of molecules that are separated from each other by distances far greater than their own dimensions. The molecules can be considered to be points; that is, they possess mass but have negligible volume. 2.Gas molecules are in constant motion in random directions, and they frequently collide with one another. Collisions among molecules are perfectly elastic. 3.Gas molecules exert neither attractive nor repulsive forces on one another. 4.The average kinetic energy of the molecules is proportional to the temperature of the gas in kelvins. Any two gases at the same temperature will have the same average kinetic energy KE = ½ mu 2

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2 Kinetic theory of gases and … Compressibility of Gases Boyle’s Law P collision rate with wall Collision rate number density Number density 1/V P 1/V Gay-Lussac’s Law P collision rate with wall Collision rate average kinetic energy of gas molecules Average kinetic energy T P TP T

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3 Kinetic theory of gases and … Compressibility of Gases Charles’s Law T average kinetic energy of molecules Collision rate average kinetic energy of gas molecules P collision rate with wall …. But! In Charles’s law, pressure stays constant… To decrease the collision rate, the volume needs to be increased V TV T

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Copyright©2000 by Houghton Mifflin Company. All rights reserved. 4 Pressure and Volume (Boyle’s Law)Boyle’s Law Figure 5.14 The Effects of Decreasing the Volume of a Sample of Gas at Constant Temperature Decreasing the volume causes the particles to collide with the walls of the container more frequently, increasing the pressure

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Copyright©2000 by Houghton Mifflin Company. All rights reserved. 5 Pressure and Temperature (Gay-Lussac’ Law) Figure 5.15 The Effects of Increasing the Temperature of a Sample of Gas at Constant Volume Increasing the temperature causes the particles to move faster increasing the frequency of collision with the walls of the container. If the volume is constant this results in and increase in pressure

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Copyright©2000 by Houghton Mifflin Company. All rights reserved. 6 Volume and Temperature (Charles’s Law) Figure 5.16 The Effects of Increasing the Temperature of a Sample of Gas at Constant Pressure Increasing the temperature causes the particles to move faster increasing the frequency of collision with the walls of the container. If the pressure remains constant, the volume of the container will increase to compensate for the increased motion of the particles

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7 Kinetic theory of gases and … Avogadro’s Law P collision rate with wall Collision rate number density Number density n P nP n Dalton’s Law of Partial Pressures Molecules do not attract or repel one another P exerted by one type of molecule is unaffected by the presence of another gas P total = P i

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Copyright©2000 by Houghton Mifflin Company. All rights reserved. 8 Volume and Number of Moles (Avogadro’s Law)Avogadro’s Law) Figure 5.17 The Effects of Increasing the Number of Moles of Gas Particles at Constant Temperature and Pressure Increasing the number of particles at constant temperature and pressure results in an increase in collisions. The volume of the container will increase to compensate for the increased number of collision.

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9 KMT – What happens in “real” gases A gas is composed of molecules that are separated from each other by distances far greater than their own dimensions. The molecules can be considered to be points; that is, they possess mass but have negligible volume. Gas molecules are in constant motion in random directions, and they frequently collide with one another. Collisions among molecules are perfectly elastic. Gas molecules exert neither attractive nor repulsive forces on one another. Under what conditions will gases most likely exhibit nonideal behavior??

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10 Deviations from Ideal Behavior 1 mole of ideal gas PV = nRT n = PV RT = 1.0 Repulsive Forces Attractive Forces

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11 Effect of intermolecular forces on the pressure exerted by a gas.

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12 Van der Waals equation nonideal gas P + (V – nb) = nRT an 2 V2V2 () } corrected pressure } corrected volume Pressure Correction: Takes into account the probability that a molecule will end up close enough to another molecule causing an intermolecular attraction Volume Correction: Takes into account that molecules, while extremely small, do take up some amount of volume Which element/molecule has the weakest attraction to each other??

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13 Given that 3.50 moles of NH 3 occupy 5.20 L at 47˚C, calculate the pressure of the gas (in atm) using the ideal gas law, and then the van der Waals equation. V=5.20 L T=( ) = 3.20x10 2 K n=3.50 mol R= L·atm/K·mol Ideal gas law Van der Waals Equation V=5.20 L T=( ) = 3.20x10 2 K n=3.50 mol a=4.17 atm·L 2 /mol 2 b= L/mol

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Homework: Problem 5.89 Calculate the pressure exerted by 2.50 moles of CO 2 confined in a volume of 5.00 L at 450 K if the gas is behaving ideally. It has been shown that this gas exhibits non-ideal behavior. Calculate the actual pressure given a=3.59; b=

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Copyright©2000 by Houghton Mifflin Company. All rights reserved. 15 The Meaning of Temperature Kelvin temperature is an index of the random motions of gas particles (higher T means greater motion.)

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Kinetic Molecular Theory Root mean square speed is an average molecular speed For one mole of a gas KE = 3/2 RT For one molecule Therefore Copyright©2000 by Houghton Mifflin Company. All rights reserved. KE = kinetic energy R = universal gas constant T = temperature (in K) m = mass u = speed Bar over top = root mean square average

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17 Apparatus for Studying Molecular Speed Distributiona

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18 The distribution of speeds for nitrogen gas molecules at three different temperatures The distribution of speeds of three different gases at the same temperature u rms = 3RT M

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19 Calculate the root mean squared speed of molecular chlorine in m/s at 20ºC.

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Problem 5.78 The temperature in the stratosphere is - 23ºC. Calculate the root mean square speeds of N 2 molecules in this region. N 2 =____ 20

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21 Gas diffusion Gas diffusion is the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties. NH 3 17 g/mol HCl 36 g/mol NH 4 Cl r1r1 r2r2 M2M2 M1M1 = molecular path

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22 Gas effusion is the is the process by which gas under pressure escapes from one compartment of a container to another by passing through a small opening. = r1r1 r2r2 t2t2 t1t1 M2M2 M1M1 = Nickel forms a gaseous compound of the formula Ni(CO) x What is the value of x given that under the same conditions methane (CH 4 ) effuses 3.3 times faster than the compound? r 1 = 3.3 x r 2 M 1 = 16 g/mol M 2 = r1r1 r2r2 ( ) 2 x M 1 = (3.3) 2 x 16 = x 28 = x = 4.1 ~ 4

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Justify the statement: A helium-filled rubber balloon deflates faster than an air-filled one. 23

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Problem 5.83 A gas evolved from the fermentation of glucose is found to effuse through a porous barrier in 15.0 min. Under the same condition of temperature and pressure, it takes an equal volume of N 2 gas 12.0 min to effuse through the same barrier. Calculate the molar mass of the gas. What could this gas be? 24

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