240_l03 - Copy - Kinetic Model of Gases Section 1.3 of...

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Kinetic Model of Gases Section 1.3 of Atkins, 6th Ed. Section 24.1 of Atkins, 7th Ed. Section 21.1 of Atkins, 8th Ed. Basic Assumptions Molecular Speeds RMS Speed Maxwell Distribution of Speeds Relative Mean Speed Most Probable Speed Collision Frequency Mean Free Path Last updated: Sept. 15, 2006, slight modification to slides 1, 14; minor admin changes
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ρ gas « ρ liquid . ρ solid β » β , β Properties of Gases - Low density: - High compressibility: - Exert an external pressure: An external pressure must be applied to contain a gas - Different gases diffuse into one another: Different gases completely mix in a homogeneous fashion
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Kinetic Model: Basic Assumptions In the kinetic model of gases, it is assumed that the only contribution to the energy of the gas comes from the kinetic energies of the atoms or molecules in the gas - interactions between molecules ( potential energy ) make no contribution 1. Gas consists of atoms or molecules of mass m undergoing random, neverending motion 2. Molecular size is negligible: the distances over which molecules travel are much greater than molecular size 3. Molecules are treated as hard spheres : they make perfectly elastic collisions with one another and the sides of the container - this means no energy is transferred to rotational, vibrational or electronic modes, nor to the walls - all energy is conserved for translation
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pV ' 1 3 nMc 2 , c ' ¢ v 2 ¦ 1/2 Pressure, Volume and Kinetic Energy We will demonstrate that under the kinetic model of gases that the pressure and volume of the gas are related by: where the molar mass of the molecules, M , is M = mN A , and c is the root mean square speed of the molecules. We will consider a molecule of mass m with an x -component of velocity v x . Upon colliding with the wall, its linear momentum P = mv x changes to - mv x (changes by 2 mv x ) (assume no changes in y and z for now)
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P ' nN A Av x t 2 V ×2 mv x ' nMAv 2 x t V Deriving the RMS Speed Distance : v x t In a small time interval, t , a molecule with velocity v x travels v x t along x (all molecules within this distance strike the wall during t ) Volume: Av x t If wall has area A , then all molecules in the volume V = Av x t hit the wall. Density: nN A / V The number of particles n in the “container” of volume V , so in this small volume, there are nN A / V x Av x t particles Statistics: ½ At any given time, half of the particles move left, half move right, so the average number of collisions in t is ½ nN A Av x t / V Momentum: 2 mv x In the interval t , the total momentum change is given by the product of the average number of collisions, and total change in momentum
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This note was uploaded on 10/24/2009 for the course CHEM 260 taught by Professor Staff during the Spring '08 term at University of Michigan.

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240_l03 - Copy - Kinetic Model of Gases Section 1.3 of...

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