1. 8.1 Maxwell Distribution of Molecular Speeds Kinetic molecular theory attempts to explain macroscopic behavior, e.g., pressure, by making the following assumptions about molecular (microscopic) behavior: atoms or molecules occupy negliable volume atoms or molecules do not exert any long range through space forces on each other, but can only interact by redistributing their kinetic energies in elastic (billard ball) type collisions Where have you seen these conditions before and what do they imply about any conclusions we might derive based on kinetic molecular theory? The theory additionally assumes a large number of atoms or molecules in random thermal motion and then proceeds to use statistical arguments to derive, the Maxwell distribution of speeds: P c = dN c / N = 4  [ m / 2  k T ] 3/2 c 2 e - m c 2 / 2 k T dc In this equation P c is the probability that an atom or molecule will have a speed in the differential interval of speeds between c and c+dc. This probability is equal to the number of atoms or molecules in this speed interval, dN c, divided by the total number of molecules, N, under consideration. In the expression on the right m is the mass of an individual atom or molecule, k is Boltzmann’s constant, T is the Kelvin temperature, and c is the atomic or molecular speed. Could you convert the Maxwell Distribution of speeds, P c, to a distribution of energies P , using the relation between kinetic energy and speed,  = 1/2 m c 2 ?

2. 8.2 College, Cambridge, he held professorships at Marischal College in Aberdeen (1856) and King's College in London (1860) and became the first Cavendish Professor of Physics at Cambridge in 1871. Maxwell's first major contribution to science was a study of the planet Saturn's rings, the nature of which was much debated. Maxwell showed that stability could be achieved only if the rings consisted of numerous small solid particles, an explanation still accepted. Maxwell next considered molecules of gases in rapid motion. By treating them statistically he was able to formulate (1866), independently of Ludwig Boltzmann, the Maxwell-Boltzmann kinetic theory of gases. This theory showed that temperatures and heat involved only molecular movement. Philosophically, this theory meant a change from a concept of certainty--heat viewed as flowing from hot to cold--to one of statistics--molecules at high temperature have only a high probability of moving toward those at low temperature. This new approach did not reject the earlier studies of thermodynamics; rather, it used a better theory of the basis of thermodynamics to explain these observations and experiments. James Clerk Maxwell was a Scottish physicist, who was born on Nov. 13 in 1831, d. Nov. 5, 1879, did revolutionary work in electromagnetism and the kinetic theory of gases. After graduating (1854) with a degree in mathematics from Trinity James Clerk Maxwell

3. 8.3 Maxwell's most important achievement was his extension and mathematical formulation of Michael Faraday's theories of electricity and magnetic lines of force. In his research, conducted between 1864 and 1873, Maxwell showed that a few relatively simple mathematical equations could express the behavior of electric and magnetic fields and their interrelated nature; that is, an oscillating electric charge produces an electromagnetic field. These four partial differential equations first appeared in fully developed form in Electricity and Magnetism (1873). Since known as Maxwell's equations they are one of the great achievements of 19th-century physics. Maxwell also calculated that the speed of propagation of an electromagnetic field is approximately that of the speed of light. He proposed that the phenomenon of light is therefore an electromagnetic phenomenon. Because charges can oscillate with any frequency, Maxwell concluded that visible light forms only a small part of the entire spectrum of possible electromagnetic radiation. Maxwell used the later-abandoned concept of the ether to explain that electromagnetic radiation did not involve action at a distance. He proposed that electromagnetic-radiation waves were carried by the ether and that magnetic lines of force were disturbances of the ether. Heinrich Hertz discovered such waves in 1888. James Clerk Maxwell died on Nov. 5, 1879. This material is taken from the WEB site http://sirius.phy.hr/~dpaar/fizicari/xmaxwell.html maintained by Dalibor Paar at the Department of Physics, Faculty of Science, University of Zagreb.

4. 8.4 The Maxwell distribution of speeds is best understood by plotting the probability density for speeds, P c / dc, versus atomic or molecular speed: Why does the distribution go through a maximum? How does the distribution change as temperature or mass are changed? Use the link Maxwell.xls to answer this question.Maxwell.xls

5. 8.5 Single numbers are often used to characterize distributions, e.g., the average grade is used to characterize the distribution of grades in a class. The Maxwell distribution of speeds is typically characterized by the most probable speed (the speed associated with the maximum in the distribution), the average or mean speed, and the root mean squared speed: What is the average speed of N 2 molecules at 25.0 o C in miles per hour? Why are the mean and most probable speeds not the same? What would the distribution look like if they were? Could you derive a formula that you could use to calculate the most probable speed? The area under the above curve between zero and infinite speeds represents the sum of the probabilites for all possible speeds. What should this area be equal to?

6. 8.6 Maxwell Distribution Bonus Problem This Bonus Problem is worth 10 points and is due, if you decide to pursue it, one week at 5:00 P.M. from the day that it is assigned and will only be graded on the answer. The fraction of molecues with speeds greater than a given speed, say c’, is equal to the area under the curve describing the Maxwell distribution from speed c’ to infinite speed: ratioed to the total area under the curve. Since the square of these speeds is propotional to their kinetic energy through  = 1/2 mc 2, this fraction is also equal to the fraction of molecules with energies greater than a given energy,  ‘ = 1/2 mc’ 2. This fraction in turn can be viewed as the fraction of molecules with sufficient energy to react upon collision and is important in theories of molecular dynamics (kinetics). Calculate the fraction of He atoms at 25.0 o C that have speeds greater than 900 meters/sec. You should find the resulting integral, which also occurs when considering normal distributions, challenging! As a test case the fraction of He atoms at 25.0 o C that have speeds greater than 500 meters/sec is 0.940.

7. 8.7 To see how to calculate the mean or average of the continuously distributed speeds consider a discrete distribution of speeds of N total molecules in which N 1 molecules have speed c 1, N 2 molecules have speed c 2, etc. The mean speed,, would be given by: = (N 1 c 1 +N 2 c 2 + …) / (N 1 + N 2 + …) =  N i c i /  N i =  N i c i / N N, which is a constant and does not depend on the summation index i, can be brought under the summation sign to give: =  (N i /N) c i =  P i c i Where P i, the probability that a molecule will have speed c i, is equal to the fraction, N i /N, of molecules with speed c i (does this make sense to you?) As the number of molecules becomes large and the difference between successive possible speeds becomes small, the distribution becomes continuous and the summation used to calculate the mean speed is replaced by an integral over all possible speeds: =   0  P c c =  0  [4  [ m / 2  k T ] 3/2 c 2 e - m c 2 / 2 k T ] c dc Derive a formula for the mean speed by evaluating this integral.

8. 8.8 The average kinetic energy per molecule of a collection of atoms or molecules that follow the assumptions of kinetic molecular theory would be given by: =  0  P   =  0  P c (1/2 mc 2 ) =  0  [4  [ m / 2  k T ] 3/2 c 2 e - m c 2 / 2 k T ] (1/2 mc 2 ) dc = 3/2 k T Since Avogadro’s number, N o, times Boltzmann’s constant is equal to the gas constant: R = N o k the average kinetic energy per mole is given by: = N o  = N o (3/2 k T) = 3/2 (N o k) T = 3/2 R T Since the assumptions that underlye kinetic moleculear theory are essentially the same assumptions that form our notion of an ideal gas, these results apply to an ideal gas. What does the average energy of an ideal gas depend on? If the pressure of a fixed amount of an ideal gas is lowered by expanding the volume isothermally does the energy change?