Download Lecture Notes on The Kinetic Theory of Gases | CH 331 and more Study notes Physical Chemistry in PDF only on Docsity! 1 Chemistry 331 Lecture 3 The Kinetic Theory of Gases NC State University Kinetic Model of Gases Assumptions: 1. A gas consists of molecules that move randomly. 2. The size of the molecules is negligible. 3. There are no interactions between the gas molecules. Because there are such large numbers of gas molecules in any system we will interested in average quantities. We have written average with an angle bracket. For example, the average speed is: <u2> = c 2 = s12 + s22 + s3 + .. . + sN N <u> = c = s1 + s2 + s3 + ... + sN N Atkins uses s for speed and c for mean speed. Velocity and Speed When we considered the derivation of pressure using a kinetic model we used the fact that the gas exchanges momentum with the wall of the container. Therefore, the vector (directional) quantity velocity was appropriate. However, in the energy expression the velocity enters as the square and so the sign of the velocity does not matter. In essence it is the average speed that is relevant for the energy. Another way to say this is the energy is a scalar. E =12m<u 2> = 12 mv 2 = 12mc 2 p = mu = mv All of these notations mean the same thing. The root-mean-square speed The ideal gas equation of state is consistent with an interpretation of temperature as proportional to the kinetic energy of a gas. If we solve for <u2> we have the mean-square speed. If we take the square root of both sides we have the r.m.s. speed. 〈u2〉 = 3RTM 1 3 M u 2 = RT u2 1/2 = 3RTM The mean speed The mean value is more commonly used than the root-mean-square of a value. The relationship between them is: or using Atkins notation: Atkins points out that the r.m.s. speed of oxygen at 25 oC (298 K) is 482 m/s. Note: M is converted to kg/mol ! 〈 u〉 = 83π〈 u 2〉 c = 83πc 2 〈 u2〉 1/2 = 3 8.31 J /mol–K 298K 0.032kg /mol = 481.8 m / s The Maxwell Distribution Not all molecules have the same speed. Maxwell assumed that the distribution of speeds was Gaussian . As temperature increases the r.m.s. speed increases and the width of the distribution increases. Moreover, the fucntions is a normalized distribution. This just means that the integral over the distribution function is equal to 1. F (s) = 4π M2πRT 3/2 s 2exp – Ms 2 RT F(s) 0 ∞ ds = 1 See the MAPLE worksheets for examples.
