Chapter XIV Kinetic-molecular theory of gases – Distribution function

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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.

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Nội dung Text: Chapter XIV Kinetic-molecular theory of gases – Distribution function

GENERAL PHYSICS II Electromagnetism & Thermal Physics 4/22/2008 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 gases 4/22/2008 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 as 4/22/2008 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 proportional to 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 M 4/22/2008 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 v (the x-component changes sign) x  If the time between such collisions = dt, then the average force on the wall due to this particle is: Fx 2 mv x Fx  t 4/22/2008 t 9
Assume we have a very sparse gas (no molecule-molecule collisions!): 2d  Time between collisions with wall:  v t x round-trip time (depends on speed) Area A 2  Average force: 2mv 2mv mv (one molecule) Fx  x  x  x  (2d / vx ) d t vx d Nm  Net average force: Fx  vx2 (N molecules) d Fx Nm 2 Nm 2 PRESSURE: p  vx  vx   means   A Ad V " time average"  We can relate this to the average translational kinetic energy of each molecule: 1 2 2 2 3 ktr  m vx vy vz2  m vx 2 2  microscopic property  Pressure from molecular collisions proportional 2N to the average translational kinetic energy of molecules: p ktr 3V macroscopic variable 4/22/2008 10
Consider 1 mole of gas: 1 mole = the amount of gas which consists the number NA of molecules NA = Avogadro’s number = 6.02 x 1023 molecules/mole mass of 1 mole in gam = molecular weight (e.g, O2:32g; H2 :2g) • Applying the equation for pressure to 1 mole of gas we have 2 2 mole mole pV  NA ktr  Ktr where Ktr is the total translational 3 3 kinetic energy of 1 mole Compare this with the ideal gas equation (for 1 mole, n=1): pV RT 3  RT 3 for n moles of gas→ K tr  nRT mole K tr 2 2 We have arrived to a simple, but important result: The average total translational kinetic energy of gas is proportional to the absolute temperature 4/22/2008 11
For a single molecule the translational kinetic energy is mole Ktr 3 R 3 R ktr   T  kT where we have denoted k NA 2 NA 2 NA The constant k occures frequently in molecular physics. It is called the Boltzmann constant. It’s value is 8 .314 J / mol .0 K k  .381   J / 0K 1 10 23 6.022  23 / mol 10 So, the 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 has an independent motion So, what we can know about them: • The average properties: average kinetic energy, average 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 on the 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 the molecular mass 4/22/2008 14
•The difference in pressure between z and z + dz is given by n at z = 0 or For the pressure This is the distribution This formula is called function on gravitational “the law of atmospheres” potential energy of molecules 4/22/2008 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 the gas: • Molecules leaving the level z = 0 with the velocities less than vz in the equation will fail to reach the height z. Similarly, The number of molecules The difference Δ between the n per unit volume which have = number per unit volume at the z-component of velocity height z and that at height z+Δz between vz and vz + Δ z v 4/22/2008 16
Differentiating the distribution function with respect to z we obtain Since the number of molecules per unit volume at temperature T with z-component of velocity must be proportional with the factor Boltzmann reasoned that this proportional relation should be the same where or not the gas is in a gravitational field, and therefore we can write where A is a constant, Pz (vz) is the probability per unit interval of vz . The constant A is determined from the condition Making the replacement and applying the formula 4/22/2008 17
The fraction of molecules with z-component of velocity between vz and vz + Δ z is given by v Similarly we have for the distribution functions for vx and vy : We now can write the expression for the fraction of molecules in an ideal gas at temperature T with x-component of velocity lying in the 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 the “volume” element is (N: the total number of molecules of the whole system) The number of molecules with speeds between v and v + Δ is thev 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. Remark that the Maxwell-Boltzmann distribution function depends on temperature. This dependence is shown in the picture. 4/22/2008 20