4/22/2008 1 GENERAL PHYSICS II Electromagnetism & Thermal Physics 4/22/2008 2 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 gases 4/22/2008 3  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 4 §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 5 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 as Boyle’s or Mariotte’s law 4/22/2008 6 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 7 Relationship between the volume V and mass or the number of moles n: Keeping pressure and temperature constant, the volume V is proportional to the number of moles n. Combining three mentioned relationships, one has a single equation : 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 m 3 → R = 8.314 J/mol.ºK. • We can expess the equation in terms of mass of gas: m tot = n.M pV RT  n pressure volume # moles gas constant temperature pV RT M m pV tot  4/22/2008 8 1.2 Kinetic-molecular model of an ideal gas: 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 GOAL: to relate state variables (temperature, pressure) to molecular motions. In other words, we want construct 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 9  For a single collision: (the x-component changes sign) A F p  v m v x x x x p (mv ) F t t       x x mvp 2   t mv F x x   2  Pressure is the outward force per unit area exerted by the gas on any wall :  The force on a wall from gas is the time-averaged momentum transfer due to collisions of the molecules off the walls:  If the time between such collisions = dt, then the average force on the wall due to this particle is: t F x <F x > means "time average"  One of the keys of the kinetic-molecular model is to relate pressure to collisions of molecules with any wall: 4/22/2008 10 Assume we have a very sparse gas (no molecule-molecule collisions!): x v d t 2   Pressure from molecular collisions proportional to the average translational kinetic energy of molecules: d mv vd mv t mv F x x xx x 2 )/2( 22    2 xx v d Nm F  22 xx x v V Nm v Ad Nm A F p   Average force: (one molecule)  Net average force: (N molecules) PRESSURE:  We can relate this to the average translational kinetic energy of each molecule:  Time between collisions with wall: round-trip time (depends on speed) d v x Area A average"time" means        2 2 22 2 3 2 1 xzyxtr vmvvvmk  tr k V N p 3 2  macroscopic variable microscopic property [...]... speed,… • Distribution of molecules according to any properties, for example: • How many per cent, or probability of molecules having the speed v ? • Probability of molecules at a height z in a gravitational field? Distribution of molecules is given by distribution functions We will consider two such distribution functions: • Distribution on the height (or potential energy) in a gravitational field • Distribution. .. only for a range of temperature – not quite low or quite high • The large values of CV for some polyatomic gases is due to the contribution of vibrational energy Beyond this limitation we must have a more accurately (using the quantum theory) 4/22/2008 29 3.3 Heat capacity of solids: The microscopic interpretation of values of heat capacity of gases can apply also for heat capacities of solids An atom... The motion of atom is seen as a combinaton of three independent modes of vibration Each mode of vibration has two degree of freedom (corresponding to kinetic and elastic potential energy), and we know the total energy of each mode is Applying the principle of equipartition of energy we have, in average, for each of degree of freedom: kT For three orthogonal directions there are six degree of freedom... Maxwell-Boltzmann distribution function depends on temperature This dependence is shown in the picture 4/22/2008 20 T2 > T > T1 At higher temperature the distribution curve is flatter, and the maximum of the curve shifts to higher speed 4/22/2008 21 2.3 Average speeds of molecules: Using the distribution function of molecular speeds one can calculate average values of molecular speeds There are three types of average... average translational kinetic energy of a single molecule is 3 ktr  kT 2 which depends only on absolute temperature The temperature can be considered as the measure of random motion of molecules 4/22/2008 12 §2 Distribution functions for molecules: In the view point of a microscopic theory an amount of ideal gas is an ensemble of molecules, in which • The number of molecules is very large • Every molecule... This formula is called “the law of atmospheres” 4/22/2008 This is the distribution function on gravitational potential energy of molecules 15 2.2 Distribution of molecular speeds in an ideal gas: • Boltzmann pointed out that the decrease in molecular density with height in a uniform gravitational field can be understood in terms of the distribution of the velocities of molecules at lower levels in... (N: the total number of molecules of the whole system) The number of molecules with speeds between v and v + Δ is the v number of loints in the sperical shell between the radius v and v +Δ : v 4/22/2008 19 Deviding by N we have the fraction of molecules in a gas at temperature T with speeds between v and v + Δ : v The function P(v) = gives the Maxwell-Boltzmann distribution function of molecular speeds... energy is proportional to the absolute temperature 3 ktr  kT 2 What is the energy of an amount of gas ? 3.1 Internal energy of ideal gas: The kinetic-molecular model states that The internal energy of an ideal gas is of the sum of the kinetic energies of all molecules For a single molecule the kinetic energy consists of two parts: the translational and rotational 4/22/2008 25 We have had the formula... speed (or kinetic energy) of molecules 4/22/2008 13 2.1 Distribution of molecules in a gravitational field: • Consider an ideal gas in a uniform gravitational fields, for example in the earth’s gravity • Assume that the temperature T is the same everywhere The equation of state gives the pressure as a function of height z : the density of the gas at the height z the number of molecules in unit volume... interval vx → vx + Δ x ; y-component of velocity lying in the interval vy → vy + Δy ; v v z-component of velocity lying in the interval vz → vz + Δ z v 4/22/2008 18 The function is known as the Maxwell-Boltzmann velocity distribution function In a velocity diagram, the velocity of a single molecule is represented by a point having coordinates (vx, vy, vz ) The number of molecules having velocities in . Physics 4/22/2008 2 Chapter XIV Kinetic-molecular theory of gases – Distribution functions §1. Kinetic–molecular model of an ideal gas §2. Distribution functions. an modelization of gas: “ideal gas”. 4/22/2008 4 §1. Kinetic–molecular model of an ideal gas: 1.1 Equations of state of an ideal gas: Conditions in which an amount of