Tài liệu Chapter XIV Kinetic-molecular theory of gases – Distribution function doc

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 thefollowing variables:  Pressure p  V

GENERAL PHYSICS II

Electromagnetism

&

Thermal Physics

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

 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 internalmotions 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 anmodelization of gas: “ideal gas”

§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 thefollowing 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 massThere exist relationships between these variables By experiment

measurements one could find these relationship

Relationship between p and V at a constant

temperarure:

The perssure of the gas is given by

where F is the force applied to thepiston

By varying the force one candetermine how the volume of thegas varies with the pressure

Experiment showed that

where C is a constant

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 thermometercalled constant volume gas thermometer

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-gasconstant)

In SI units: p in Pa (1Pa = 1 N/m2); V in m3 → R = 8.314 J/mol.ºK

• We can expess the equation in terms of mass of gas: mtot = n.M

pV n RT

pressure volume

# moles

gas constant temperature

pV

RT m

pV  tot

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) isfar from ideal

 For a single collision:

(the x-component changes sign)

 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 onthe wall due to this particle is:

One of the keys of the kinetic-molecular model is to relate pressure

to collisions of molecules with any wall:

Assume we have a very sparse gas (no molecule-molecule collisions!):

 Pressure from molecular collisions proportional

to the average translational kinetic energy of molecules:

d

mv v

d

mv t

mv

x

x x

x

2

)/2(

22

V

Nm v

Ad

Nm A

 Time between collisions with wall:

round-trip time (depends on speed)

1

x z

y x

tr

k V

N p

3

2

macroscopic variable

microscopicproperty

NA = Avogadro’s number = 6.02 x 1023 molecules/mole

mass of 1 mole in gam = molecular weight (e.g, O2:32g; H2:2g)

Consider 1 mole of gas:

1 mole = the amount of gas which consists the number NA of molecules

• Applying the equation for pressure to 1 mole of gas we have

mole tr tr

 where Ktr mole is the total translational

kinetic energy of 1 moleCompare this with the ideal gas equation (for 1 mole, n=1):

We have arrived to a simple, but important result:

The average total translational kinetic energy of gas

is proportional to the absolute temperature

T N

R N

K k

A A

mole tr tr

2

3 2

For a single molecule the translational kinetic energy is

where we have denoted

The constant k occures frequently in molecular physics It is called the Boltzmann constant It’s value is

K

J mol

K mol J

0

/ 10

381

1 /

10 022

6

/

314

So, the average translational kinetic energy of a single molecule is

which depends only on absolute temperature

The temperature can be considered asthe measure of random motion of molecules

§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

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 number of molecules

in unit volume the molecular mass

the density of the gas at the height z

n at z = 0

•The difference in pressure between z and z + dz is given by

For the pressure

or

This formula is called

“the law of atmospheres”

This is the distribution function on gravitational

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

per unit volume which have

the z-component of velocity

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 formula

The fraction of molecules with z-component of velocity between vz and

vz + Δvz is given by

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 + Δvx ; y-component of velocity lying in the interval vy → vy+ Δvy ;z-component of velocity lying in the interval vz → vz + Δvz

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 + Δv is the

number of loints in the sperical shell between the radius v and v +Δv:

Deviding by N we have the fraction of molecules in a gas at temperature T

with speeds between v and v + Δv :

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

T2 > T > T1

At higher temperature the distribution curve is flatter,

and the maximum of the curve shifts to higher speed

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

v

The auxilary integral:

m

kT v



8



2.3.2 The root-mean-square (rms) speed:

First we take the average value of the square of the speed:

and hence the rms speed is given by

which is, obviously, not the same as the value of the average speed

The auxilary integral:

2.3.3 The most probable speed vm:

This is the speed which corresponds to the maximum of the speed

distribution curve, that is

§3 Internal energy and heat capacity of ideal gases:

We have had before the result:

Molecules in their motion have a kinetic energy

The molecular kinetic energy is proportional tothe absolute temperature

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

Diatomic molecules  translational +

rotational kinetic energy

< ½ mv x 2 > + < ½ mv y 2 > + < ½ mv z 2 >

+ < ½ I2 > + < ½ I2 > = 5(½ kT)

 free point particles  translational kinetic energy (x, y, z components)

< ½ mv x2 > + < ½ mv y2 > + < ½ mv z2 > = 3(½ kT

It is the case of monatomic molecules

We have had the formula for translational kinetic energy, how can find

that for rotational ?

average, is uniformly distributed over the degrees of freedom of motion

 Polyatomic molecules → translational +

rotational kinetic energy

< ½ mv x 2 > + < ½ mv y 2 > + < ½ mv z 2 >

+<½ I1 2 > + < ½ I2 2 >+ < ½ I3 2 > = 6(½ kT)

RT T

k N

polyatomic molecules (H2O, NH3, …)

3 translational modes (x, y, z)+ 3 rotational modes (x , y , z)

RT

U 25

RT

U 3 

Therefore we can derive the expressions for the inertial energy of

1 mole of ideal gas:

For n moles of gas the internal energy is by n times larger !

3.2 Constant-volume heat capacity of an ideal gas:

Consider an amount of ideal gas with the volume held constant In thiscaes, the internal energy of gas has a direct relation with heat capacity

The molar heat capacity at constant volume is denoted by CV :

It means

“per 1 mole”

By definition: dQ = CV dT

the heat input needed

for the change dT

(per 1 mole)

the heat input neededfor the change 1 unit oftemperature (per 1 mole)

 CV can be measured by experiment

 But we can also relate CV to the internal energy U

For 1 mole of monatomic gas: U = 3/2 RT → dU = 3/2 RdT

A constant-volume process have not a work done, therefore by theconservation of energy (or by the 1st principle we will say later):

The kinetic molecular model predicts this value of the molar

heat capacity at constant volume for monatomic molecules

For diatomic gases, CV has the value CV = 5/2 R

These predictions basically agree with measured values (see thetable 18.1 p 702, text book) We say “basically”, since:

• The above predictions is true only for a range of temperature –not quite low or quite high

• The large values of CV for some polyatomic gases is due tothe contribution of vibrational energy

Beyond this limitation we must have a more accurately (using

3.3 Heat capacity of solids:

The microscopic interpretation of values of heat capacity of gases canapply also for heat capacities of solids

An atom in a solid is bound to its six nearest

neighbours by elastic forces 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 three orthogonal directions there are six degree of freedom

For each molecule the energy of vibration is (6/2) kT = 3 kT

Therefore, for 1 mole of solid, the internal energy U = 3 RT

and the molar capacity of solid is predicted to be CV = 3 R

This result is called Dulong–Petit law)

This law is obeyed quite

well by solids at high

temperatures But at lower

§4 State equation for real gases:

In the model of an ideal gas, we ignored:

The volume of molecules themselves

The interaction (attractive) forces between them

By making approximate corrections for these two omissions,

van der Waals deduced an state equation for real gases:

number of moles

in the amount of gas

The constant a and b are empirical constants, they are different fordifferent gases.

b is the constant which represents the sum of private volumes

of the molecules in 1 mole → the free net volume for molecular

motions is only V – nb

When the interactions between molecules is taken into account,The pressure on the walls of container decreases The decrease inthe pressure is proportional to the number of molecules per unit

volume in a layer next the wall and also that in the next layer →

this decrease is proportional to (n2 / V2) The proportional

coefficient is a.

The p-V diagram for a real gas is shown in the figure

But for T < TC the curvesare very different whichrepresent the transitionsfrom gas phase to liquidphase or solid phase atlow temperature.

 Equation of state of an ideal gas pV  nRT

 The kinetic-molecular model for an ideal gas gives the relation:

tr

k V

N p

3

2



The translational kinetic energy of a single molecule:

The translational kinetic energy of 1 mole of ideal gas:



 The internal energy of ideal gas (per mole) and molar heat capacity:

For a monatomic gas:

For a diatomic gas:

 The Maxwell-Boltzmann distribution function of molecular speeds

P(v) =

 The average speed of molecules:

 The root-mean-square speed:

 The most probable speed:

m

kT v



8

