Chapter XIV Kinetic-molecular theory of gases – Distribution function

From this Chapter we will study thermal properties of matter, that iswhat means the terms “hot” or “cold”, what is the difference between“heat” and “temparature”, and the laws relative to these concepts.We will know that the thermal phenomena are determined by internalmotions of molecules inside a matter. There exists a form of energywhich is called thermal energy, or “heat”, which is the total energy ofall molecular motions, or internal energy.
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Chapter XIV Kinetic-molecular theory of gases – Distribution function GENERAL PHYSICS II Electromagnetism & Thermal Physics4/22/2008 1 Chapter XIV Kinetic-molecular theory of gases – Distribution functions §1. Kinetic–molecular model of an ideal gas §2. Distribution functions for molecules §3. Internal energy and heat capacity of ideal gases §4. State equation for real gases4/22/2008 2 From this Chapter we will study thermal properties of matter, that is what means the terms “hot” or “cold”, what is the difference between “heat” and “temparature”, and the laws relative to these concepts. We will know that the thermal phenomena are determined by internal motions of molecules inside a matter. There exists a form of energy which is called thermal energy, or “heat”, which is the total energy of all molecular motions, or internal energy. To find thermal laws one must connect the properties of molecular motions (microscopic properties) with the macroscopic thermal properties of matter (temperature, pressure,…). First we consider an modelization of gas: “ideal gas”.4/22/2008 3§1. Kinetic–molecular model of an ideal gas: 1.1 Equations of state of an ideal gas: Conditions in which an amount of matter exists are descrbied by the following variables:  Pressure ( p )  Volume ( V )  Temperature ( T )  Amount of substance ( m or number of moles n, m = n.M) These variables are called state variables molar mass There exist relationships between these variables. By experiment measurements one could find these relationship.4/22/2008 4 Relationship between p and V at a constant temperarure: The perssure of the gas is given by where F is the force applied to the piston. By varying the force one can determine how the volume of the gas varies with the pressure. Experiment showed that where C is a constant This relation is known as4/22/2008 Boyle’s or Mariotte’s law 5 Relationship between p and T while a fixed amount of gas is confined to a closed container which has rigid wall (that means V is fixed). Experiment showed that with a appropriate temperature scale the pressure p is proportional to T, and we can write where A is a constant. This relation is applicable for temperatures in ºK (Kelvin). Temperatures in this units are called absolute temperature. The instrument shown in the picture can use as a type of thermometer called constant volume gas thermometer.4/22/2008 6 Relationship between the volume V and mass or the number of moles n: Keeping pressure and temperature constant, the volume V is proportionalto the number of moles n. Combining three mentioned relationships, one has a single equation : # moles temperature pV n RT pressure volume gas constant pV This equation is called “equation of state of an ideal gas ”. • The constant R has the same value for all gases at sufficiently high temperature and low pressure → it called the gas constant (or ideal-gas constant). In SI units: p in Pa (1Pa = 1 N/m 2); V in m3 → R = 8.314 J/mol.ºK. • We can expess the equation in terms of mass of gas: mtot = n.M mtot pV  RT M4/22/2008 7 1.2 Kinetic-molecular model of an ideal gas: GOAL: to relate state variables (temperature, pressure) to molecular motions. In other words, we want construct a “microscopic model of gas”:  Gas is a collection of molecules or atoms which move around without touching much each other  Molecular velocities are random (every direction equally likely) but there is a distribution of speeds From the microscopic view point we have the IDEAL Gas definition:  molecules occupy only a small fraction of the volume  molecules interact so little that the energy is just the sum of the separate energies of the molecules (i.e. no potential energy from interactions) Examples: The atmosphere is nearly ideal, but a gas under high pressures and low temperatures (near liquidized state) is far from ideal.4/22/2008 8 One of the keys of the kinetic-molecular model is to relate pressure to collisions of molecules with any wall:  Pressure is the outward force per unit area F p exerted by the gas on any wall : A  The force on a wall from gas is the time-averaged momentum transfer due to collisions of the molecules off the walls:   x) p (mv v  means  Fx  x time average  t t  For a single collision:  x  mvx p 2 m ...