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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 mu2 1 Kinetic theory of gases and … • Compressibility of Gases • Boyle’s Law P a collision rate with wall Collision rate a number density Number density a 1/V P a 1/V • Gay-Lussac’s Law P a collision rate with wall Collision rate a average kinetic energy of gas molecules Average kinetic energy a T PaT 2 Kinetic theory of gases and … • Compressibility of Gases • Charles’s Law T a average kinetic energy of molecules Collision rate a average kinetic energy of gas molecules P a collision rate with wall …. But! In Charles’s law, pressure stays constant… To decrease the collision rate, the volume needs to be increased VaT 3 Pressure and Volume (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 Copyright©2000 by Houghton Mifflin Company. All rights reserved. 4 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 Copyright©2000 by Houghton Mifflin Company. All rights reserved. 5 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 6 Copyright©2000 by Houghton Mifflin Company. All rights reserved. Kinetic theory of gases and … • Avogadro’s Law P a collision rate with wall Collision rate a number density Number density a n Pan • 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 Ptotal = SPi 7 Volume and Number of Moles (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. Copyright©2000 by Houghton Mifflin Company. All rights reserved. 8 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?? 9 Deviations from Ideal Behavior 1 mole of ideal gas PV = nRT PV = 1.0 n= RT Repulsive Forces Attractive Forces 10 Effect of intermolecular forces on the pressure exerted by a gas. 11 Van der Waals equation nonideal gas } corrected pressure } 2 an ( P + V2 ) (V – nb) = nRT 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?? 12 Given that 3.50 moles of NH3 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. Ideal gas law Van der Waals Equation V=5.20 L V=5.20 L 2 K T=(47+273.15) = 3.20x10 T=(47+273.15) = 3.20x102 K n=3.50 mol n=3.50 mol R=0.0821 L·atm/K·mol 2 2 a=4.17 atm·L /mol nRT P b=0.0371 L/mol 2 an2 (4.17 atmL mol 2 )(3.50m ol) 1.89atm 2 2 V (5.20L) nb (3.50m ol)(0.0371L mol ) 0.130L 2 V (3.50m ol)(0.0821Latm K mol )(320K ) P 5.20L P 17.7atm an2 ( P 2 )(V nb) nRT V ( P 1.89atm)(5.20L 0.130L) (3.50m ol)(0.081Latm K mol )(320K ) P 16.2atm 13 Homework: Problem 5.89 Calculate the pressure exerted by 2.50 moles of CO2 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=0.0427. 14 The Meaning of Temperature (KE)avg 3 RT 2 Kelvin temperature is an index of the random motions of gas particles (higher T means greater motion.) Copyright©2000 by Houghton Mifflin Company. All rights reserved. 15 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 KE 1 2 mu 2 KE = kinetic energy • Therefore NA 1 R = universal gas constant T = temperature (in K) m = mass u = speed Bar over top = root mean square average 2 m u 3 2 RT 2 3RT u 2 u urms 2 3RT where R 8.3145J Copyright©2000 by Houghton Mifflin Company. All rights reserved. K m ol Apparatus for Studying Molecular Speed Distributiona 17 The distribution of speeds of three different gases at the same temperature The distribution of speeds for nitrogen gas molecules at three different temperatures urms = M 3RT 18 Calculate the root mean squared speed of molecular chlorine in m/s at 20ºC. urms urms 3RT M 3(8.314 J K mol )(298 K ) 0.07090 kg mol 1J 1 kgm 2 s2 urms 321m s 19 Problem 5.78 The temperature in the stratosphere is 23ºC. Calculate the root mean square speeds of N2 molecules in this region. N2=____ 20 Gas diffusion is the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties. r1 r2 = M2 M1 molecular path NH4Cl NH3 17 g/mol HCl 36 g/mol 21 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. r1 r2 = t2 t1 = M2 M1 Nickel forms a gaseous compound of the formula Ni(CO)x What is the value of x given that under the same conditions methane (CH4) effuses 3.3 times faster than the compound? r1 2 x M1 = (3.3)2 x 16 = 174.2 r1 = 3.3 x r2 M2 = r2 x = 4.1 ~ 4 22 M1 = 16 g/mol 58.7 + x • 28 = 174.2 ( ) Justify the statement: A helium-filled rubber balloon deflates faster than an air-filled one. 23 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 N2 gas 12.0 min to effuse through the same barrier. Calculate the molar mass of the gas. What could this gas be? 24