chapter 7 dong hoc chat khi revised

The Kinetic Theory of Gases7.1 Molecular Model of an Ideal Gas7.2 Molar Specific Heat of an Ideal Gas7.3 Adiabatic Processes for an Ideal Gas7.4 Degrees of Freedom and Molar Specific Hea

TẬP ĐOÀN DẦU KHÍ VIỆT NAM

TRƯỜNG ĐẠI HỌC DẦU KHÍ VIỆT NAM

Lecturer : Assoc Prof Pham Hong Quang

General Physics I

Chapter 7 The Kinetic Theory of Gases

7.1 Molecular Model of an Ideal Gas 7.2 Molar Specific Heat of an Ideal Gas 7.3 Adiabatic Processes for an Ideal Gas 7.4 Degrees of Freedom and Molar Specific Heats 7.5 The Boltzmann Distribution Law

7.6 Distribution of Molecular Speeds

Learning outcome

The student should be able to:

•Identify Avogadro’s number N A

•Identify why an ideal gas is said to be ideal.

•Apply either of the two forms of the ideal gas law, written in terms

of the number of moles n or the number of molecules N.

•Relate the ideal gas constant R and the Boltzmann constant k.

•Identify that the pressure on the interior walls of a gas container is due to the molecular collisions with the walls.

•For the molecules of an ideal gas, relate the rootmean-square

speed v rms and the average speed v avg .

•Relate the pressure of an ideal gas to the rms speed v rms of the

molecules.

•For an ideal gas, apply the relationship between the gas

temperature T and the rms speed v rms and molar mass M of the

•Identify that a measurement of a gas temperature is effectively a

measurement of the average kinetic energy of the gas molecules.

•Identify what is meant by mean free path.

•Identify that the internal energy of an ideal monatomic gas is the

sum of the translational kinetic energies of its atoms.

of a

Learning outcome

•Distinguish between monatomic, diatomic, and polyatomic ideal

gases.

•For monatomic, diatomic, and polyatomic ideal gases, valuate the molar specific heats for a constant-volume process and a constant- pressure process.

•Calculate a molar specific heat at constant pressure C p by adding

R to the molar specific heat at constant volume C V ,and explain why

(physically) Cp is greater.

•Identify that for a given change in temperature, the change in the

internal energy of an ideal gas is the same for any process and is

most easily calculated by assuming a constant-volume process.

•For an ideal gas, apply the relationship between heat Q, number

of moles n, and temperature change %T, using the appropriate

molar specific heat.

Learning outcome

•Between two isotherms on a p-V diagram, sketch a

constant-volume process and a constant-pressure process, and for each

identify the work done in terms of area on the graph.

•Explain how Maxwell’s speed distribution law is used to find the

fraction of molecules with speeds in a certain speed range.

•Sketch a graph of Maxwell’s speed distribution, showing the

probability distribution versus speed and indicating the relative

positions of the average speed v avg , the most probable speed v P ,

and the rms speed v rms .

•Explain how Maxwell’s speed distribution is used to find the

average speed, the rms speed, and the most probable speed.

•For a given temperature T and molar mass M, calculate the

Learning outcome

7.1 Molecular Model of an Ideal Gas

Macroscopic vs Microscopic Descriptions

So far we have dealt with macroscopic variables:

Pressure Volume Temperature These can be related to a description on a microscopic level.

Matter is treated as a collection of molecules Applying

Newton’s laws of motion in a statistical manner to a collection of particles provides a reasonable description of thermodynamic

processes.

Pressure and temperature relate directly to molecular motion in

a sample of gas.

7.1 Molecular Model of an Ideal Gas

Avogadro's Number!

We ask the following question: How many carbon-12

atoms are needed to have a mass of exactly 12 g?

That number is NA - Avogadro's number

Careful measurements yield a value for

N A = 6.0221367x10^+23

1 mole = 6.022 x 1023 particles of any substance

7.1 Molecular Model of an Ideal Gas

Ideal Gas Assumptions

•The average separation between the molecules is large

compared with their dimensions

The molecules occupy a negligible volume within the container

This is consistent with the macroscopic model where we modeled the molecules as point-like particles

•The molecules obey Newton’s laws of motion, but as a

whole they move randomly

Any molecule can move in any direction with any speed

7.1 Molecular Model of an Ideal Gas

•The molecules interact only by short-range forces during

elastic collisions.

This is consistent with the macroscopic model, in which the molecules exert no long-range forces on each other.

•The molecules make elastic collisions with the walls.

These collisions lead to the macroscopic pressure on the walls of the container.

•The gas under consideration is a pure substance.

All molecules are identical.

Ideal Gas Assumptions Cont.

7.1 Molecular Model of an Ideal Gas

Ideal Gas Equation

7.1 Molecular Model of an Ideal Gas

The conditions 0 0C and 1 atm are called standard

temperature and pressure (STP).

Experiments show that at STP, 1 mole of an ideal gas occupies 22.414 L

R = 8.31 J/mol.K

Ideal Gas Equation, Cont.

7.1 Molecular Model of an Ideal Gas

Pressure and Kinetic Energy

Consider a collection of N molecules of an

ideal gas in a container of volume V.

Assume the container is a cube.

Only molecules moving toward the surface hit

the surface Assuming the surface is normal

to the x axis, half the molecules of speed v x

move toward the surface.

Only those close enough to the surface hit it

in time dt, those within the distance v x dt

The number of collisions hitting an area A in

2

N V







A v x dt

7.1 Molecular Model of an Ideal Gas

Assume perfectly elastic

collisions with the walls of the

container.

The molecule’s velocity

component perpendicular to the

wall is reversed.

The mass of the wall is much greater than the mass of the molecule.

The wall causes a change in the

molecule’s momentum:

2mv x

7.1 Molecular Model of an Ideal Gas

Momentum in time dt dp  2mv x1

2 

N V

7.1 Molecular Model of an Ideal Gas

N V

7.1 Molecular Model of an Ideal Gas

This equation relates the macroscopic quantity of pressure with a microscopic quantity of the average value of the square of the molecular speed

One way to increase the pressure is to increase the number of molecules per unit volume

The pressure can also be raised by increasing the speed (kinetic energy) of the molecules

This can be accomplished by raising the temperature of the gas

7.1 Molecular Model of an Ideal Gas

We can take the pressure as it relates to the kinetic

energy and compare it to the pressure from the

equation of state for an ideal gas

Therefore, the temperature is a direct measure of the

average molecular kinetic energy

Simplifying the equation relating temperature and

Molecular Interpretation of Temperature

K J

k B 1.38.1023 / 8.62.105 /





7.1 Molecular Model of an Ideal Gas

M is the mass of 1 mole

7.1 Molecular Model of an Ideal Gas

Internal Energy

For monatomic gas: the internal energy = sum

of the kinetic energy of all molecules:

7.1 Molecular Model of an Ideal Gas

Mean Free Path

Molecules collide elastically with other molecules

Mean Free Path : average distance between

two consecutive collisions

7.2 Molar Specific Heat

Several processes can change

the temperature of an ideal gas

Since ΔT is the same for each

process, ΔEint is also the same

The work done on the gas is

different for each path

The heat associated with a

particular change in

temperature is not unique

7.2 Molar Specific Heat

We define specific heats for two processes that frequently

occur:

Changes with constant pressure, isobaric

Changes with constant volume, isochoric

Using the number of moles, n, we can define molar specific

heats for these processes.

Molar specific heats:

CV for constant-volume processes → Q = n CV ΔT

CP for constant-pressure processes → Q = n CP ΔT

Q (constant pressure) must account for both the increase in

internal energy and the transfer of energy out of the system

by work.

Qconstant P > Qconstant V for given values of n and Δ T, CP > CV

7.2 Molar Specific Heat

Ideal Monatomic Gas

A monatomic gas contains one atom per molecule

When energy is added to a monatomic gas in a

container with a fixed volume, all of the energy goes

into increasing the translational kinetic energy of the

gas

There is no other way to store energy in such a gas

7.2 Molar Specific Heat

Solving for C V gives C V = 3/2 R = 12.5 J/mol . K

For all monatomic gasesThis is in good agreement with experimental results for monatomic gases

In a constant-pressure process, ΔEint = Q -W

(W is positive when work is done by the system and

negative when work is done on the system)

So: Q = ΔEint +W and PdV =nRΔT Then: Cp = C V + R

Cp = 5/2 R = 20.8 J/mol . K

7.2 Molar Specific Heat

Ratio of Molar Specific Heats

We can also define the ratio of molar specific heats

Theoretical values of C V , C P , and g are in excellent

agreement for monatomic gases

But they are in serious disagreement with the values

5 / 2

1.67

3 / 2

P V

   

7.3 Adiabatic Processes for an Ideal Gas

An adiabatic process is one in which no energy is

transferred by heat between a system and its surroundings

All three variables in the ideal gas law (P, V, T ) can change

during an adiabatic process

Assume an ideal gas is in an equilibrium state and so PV = nRT is valid.

The pressure and volume of an ideal gas at any time

during an adiabatic process are related by PV γ =

constant.

7.3 Adiabatic Processes for an Ideal Gas

The PV diagram shows

pV  = const.

7.4 Degrees of Freedom and Molar Specific Heats

With complex molecules,

other contributions to

internal energy must be

taken into account

One possible energy is the

translational motion of the

center of mass

The center of mass can

translate in the x, y, and z

directions

This gives three degrees of

freedom for translational

motion

7.4 Degrees of Freedom and Molar Specific Heats

Rotational motion about the

various axes also contributes

We can neglect the

rotation around the y axis

7.4 Degrees of Freedom and Molar Specific Heats

The molecule can also

vibrate

There is kinetic energy

and potential energy

associated with the

vibrations

The vibrational mode

adds two more degrees

of freedom

7.4 Degrees of Freedom and Molar Specific Heats

Taking into account the degrees of freedom from just

the translation and rotation contributions

Eint = 5/2 n R T and CV = 5/2 RThis gives CP = 7/2 R

This is in good agreement with data for diatomic molecules

However, the vibrational motion adds two more degrees

of freedom

Therefore, E = 7/2 nRT and C = 7/2 R

7.4 Degrees of Freedom and Molar Specific Heats

Molar Specific Heat: Agreement with Experiment

Molar specific heat is a function of temperature

At low temperatures, a diatomic gas acts like a

At high temperatures, the value increases to CV = 7/2 R

This includes vibrational energy as well as rotational and translational

7.4 Degrees of Freedom and Molar Specific Heats

7.4 Degrees of Freedom and Molar Specific Heats

For molecules with more than two atoms, the

vibrations are more complex

The number of degrees of freedom is larger

The more degrees of freedom available to a

molecule, the more “ways” there are to store

energy

This results in a higher molar specific heat

Complex Molecules

7.5 Boltzmann Distribution Law

The motion of molecules is extremely chaotic

Any individual molecule is colliding with others at an

enormous rate

Typically at a rate of a billion times per second

We add the number density n V (E )

This is called a distribution function

It is defined so that n V (E ) dE is the number of

7.5 Boltzmann Distribution Law

From statistical mechanics, the number

7.5 Boltzmann Distribution Law

Thus far we have discussed

the random nature of

molecular motion in terms of

the average (root mean

square) speed But how is

this speed distributed?

The observed distribution of

gas molecules in thermal

equilibrium is shown at right

7.5 Boltzmann Distribution Law

The fundamental expression that describes the

distribution of speeds in N gas molecules is

2

3/ 2

/ 2 2

Boltzmann’s constant and T is the absolute

temperature

7.5 Boltzmann Distribution Law

The peak shifts to the right as T increases.

This shows that the average speed increases with increasing

temperature

7.5 Boltzmann Distribution Law

Relationship between molar mass and molecular speed

The most probable speed increases as the molecular

mass decreases

The distribution broadens as the molecular mass

decreases

7.5 Boltzmann Distribution Law

The speed distribution for liquids is similar to that of gases

Some molecules in the liquid are more energetic than others

Some of the fast moving molecules penetrate the surface

and leave the liquid

This occurs even before the boiling point is reached

The molecules that escape are those that have enough

energy to overcome the attractive forces of the molecules in

the liquid phase

Evaporation

Key words of the chapter

Kinetic Theory of Gases; Molecular Model; Ideal Gas; Ideal

Gas Equation; Molar Specific Heat; Mean Free Path;

Monatomic Gas; Complex molecules; Degrees of Freedom; Boltzmann Distribution Law

Kinetic Theory of Gases The kinetic theory of gases relates

the macroscopic properties of gases to the microscopic properties

of gas molecules.

Avogadro’s Number One mole of a substance contains

NA is found experimentally to be NA = 6.02 10 23 mol -1

Ideal Gas An ideal gas is one for which the pressure p, volume

V, and temperature T are related by pV = nRT =NkT.

Pressure, Temperature, Kinetic energy, Internal energy and

N V

•The value of the specific heat depends on whether it is at

constant pressure or at constant volume

• Molar specific heat is defined by: Q=nCΔT

• For a monatomic gas at constant volume: CV =3/2 R

• For a monatomic gas at constant pressure: CP =5/2 R

•Degrees of Freedom and CV : CV =f/2 R (f is the number of

degrees of freedom).

•Adiabatic Process: When an ideal gas undergoes an adiabatic

(a change for which Q = 0), pV γ = constant (adiabatic process)

in which γ (= Cp/C V ) is the ratio of molar specific heats for the

Check your understanding 1

An ideal gas is made up of N diatomic molecules, each of mass

M All of the following statements about this gas are true

EXCEPT:

(A) The temperature of the gas is proportional to the average

translational kinetic energy of the molecules.

(B) All of the molecules have the same speed.

(C) The molecules make elastic collisions with the walls of the

container.

(D) The molecules make elastic collisions with each other.

(E) The average number of collisions per unit time that the

molecules make with the walls of the container

depends on the temperature of the gas.

Check your understanding 2

Which of the following statements is NOT a correct assumption of the classical model of an ideal gas?

(A) The molecules are in random motion.

(B) The volume of the molecules is negligible compared with the

volume occupied by the gas.

(C) The molecules obey Newton's laws of motion.

(D) The collisions between molecules are inelastic.

(E) The only appreciable forces on the molecules are those that

occur during collisions.

Ans D If the collisions were inelastic, the gas would change

its temperature by virtue of the collisions with no change in

pressure or volume.