Nicholas J. Giordano

www.cengage.com/physics/giordano

Chapter 15 Gases and Kinetic Theory

Introduction • Newton’s Laws can be used to analyze the behavior

of gases • Simplest phase of matter

• Newton’s Laws can be applied to a dilute gas • Simplifies the system enough to apply the laws • Atoms and molecules are far enough apart to collide

only rarely • Analysis of a dilute gas can lead to connections

between temperature and motion on an atomic scale • These ideas led to the notion of an ideal gas and the

area of physics called kinetic theory

Introduction

Molecular Picture of Gases

• The balloon contains identical molecules • Ideas also apply to

single atoms • The molecules are in

constant motion • They collide with other

gas molecules and the walls of the container

Section 15.1

Questions to be answered • How often does a gas molecule collide with other

molecules? • Are these collisions elastic or inelastic? • How are the microscopic properties of the gas

related to the macroscopic properties • Microscopic properties include molecular velocities • Macroscopic properties include temperature and

pressure • How many molecules are there in a typical sample of

gas?

Section 15.1

Avogadro’s Number • A typical sample of gas contains a very large number

of molecules • This large number allows statistical arguments to be

applied to the gas • Avogadro’s number, NA is 6.02 x 1023

• It is the number of particles in a mole • More exactly, it is the number of atoms in 12 g of C-12

• It is a pure number • Avogadro’s number plays an important part in

connecting the microscopic world to the macroscopic world

Section 15.1

Atomic Mass • The atomic mass of an element is the mass of a

natural sample of the element containing one mole of atoms • Atomic mass is usually given in grams • A capital M is used to denote atomic masses

• The mass of one atom is the atomic mass divided by Avogadro’s number:

• A typical sample of gas particles contains on the order of one mole of particles

Section 15.1

Ideal Gases – Experimental Perspective • Experimental studies led to the discovery of a

number of “gas laws” describing the macroscopic properties of gases

• These laws eventually gave rise to kinetic theory • Most of these laws apply only to a dilute gas

• Spacing between molecules is much larger than the size of the individual molecules

• In many cases, gases can be closely approximated as a dilute gas

Section 15.2

Avogadro’s Law • For a sample of gas at constant pressure and

temperature, the volume is proportional to the number of molecules in the sample • This tells us the average spacing between gas

particles is constant • So the density is constant

• Both results hold provided the pressure and temperature are held fixed

Section 15.2

Boyle’s Law • For a sample of gas at constant temperature, the

product of the pressure and volume is constant • P V = constant (at constant T) • If we increase the pressure, the volume must

decrease • Changing the pressure will change the average

spacing between the particles • The constant must be proportional to the number of

particles in order to be consistent with Avogadro’s Law

Section 15.2

Charles’ Law • For a sample of gas at constant pressure, if the

temperature changes by a small amount ΔT, the volume changes by an amount Δ V, with ΔV ∝ ΔT • Temperature is part of the constant in Boyle’s Law

Section 15.2

Gay-Lussac’s Law • For a sample of gas held in a container with constant

volume, changes in pressure are proportional to changes in temperature

• ΔP ∝ ΔT • Gay-Lussac’s Law was involved with gas

thermometers

Section 15.2

Measuring Temperature

• Charles and Gay-Lussac used reference temperatures determined by the properties of various substances

• Two convenient temperatures are those at which ice melts and water boils

• The points are used along with some thermal property of a material

• The mercury thermometer uses the property of thermal expansion

Section 15.2

Definition of Temperature • A scientific definition of temperature was still needed • According to Guy-Lussac’s Law, as temperature is

reduced, the pressure becomes smaller in a linear fashion • Until the gas condenses

• You can extrapolate to find the temperature at which the pressure would be zero

• If you do this with many different samples of gas, they all extrapolate to the same temperature

• This temperature is -273.15° C

Section 15.2

Definition of Temperature, cont.

• The Kelvin scale is defined so that T = 0 K when the pressure of a dilute gas becomes 0

• This is called the absolute zero of temperature

• This gives a universal way of measuring temperature using a gas thermometer

Section 15.2

Ideal Gas Law • All of the laws for a dilute gas are contained in a

single relation called the ideal gas law • For a dilute gas composed of any substance, P V = n R T

• R = 8.31 J/ mole.K and is called the universal gas constant

• n is the number of moles of gas present

Section 15.2

Ideal Gas Law, cont. • The ideal gas relation can be expressed in terms of

the number of gas particles, N, instead of the number of moles, n

• The total number of particles is nNA

• R/NA is called Boltzmann’s constant and given the symbol kB • kB = 1.38 x 10-23 J / K

• The ideal gas law becomes P V = N kB T

Section 15.2

Ideal Gases and Newton’s Laws – Pressure

• The molecules in a gas move in all directions with different speeds

• Look at one molecule colliding with a wall

• Such collisions are elastic

• From the Impulse Theorem,

Section 15.3

Pressure, cont. • For an elastic collision, the speed before and after

the collision will be the same and F = (2 m v) / Δt • Looking at the molecules that strike the wall in a

given period of time, Ftotal = (N m v2) / (3 L) • The pressure is F/A and the volume is L x A, so

Section 15.3

Temperature: Microscopic Basis • The result of the pressure can be compared to the

ideal gas law • Since KE = ½ m v2 we can write the KE in terms of

Boltzmann’s constant: KE = 3/2 kB T • This is the average kinetic energy of atoms or

molecules in a gas, liquid or solid • The KE is proportional to the temperature • This is how the macroscopic quantities in the ideal

gas law are related to the microscopic quantities that appear in Newton’s Laws

Section 15.3

Molecular Speeds • The speed of a molecule can be found from the KE

equations

• For air at room temperature, nitrogen molecules have a speed of approximately 510 m/s

Section 15.3

Kinetic Theory • Assumptions in kinetic theory

• Gas atoms and molecules spend more of their time moving freely, they move with a constant speed in a straight-line path • Collisions with other atoms and molecules in the gas are

very infrequent • Newton’s Law can be used to describe the motion of

individual gas particles • The collisions between gas atoms and molecules, and

with the walls of the container, are elastic

Section 15.4

Kinetic Theory, cont. • More assumptions

• Collisions • The average distance between collisions is called the mean

free path, ℓ • The mean free path depends on the density of the gas

particles, their size, and temperature • It is not the same as the average spacing between the

particles • Even a dilute gas contains a very large number of

particles, so statistical analysis can be used to calculate its properties

Section 15.4

Mean Free Path

• The mean free path depends on the particle size

• As the size of the molecules increases, the mean free path decreases

• The size of the molecule is smaller than the average spacing between them, so the mean free path is larger than the average spacing

Section 15.4

Internal Energy of an Ideal Gas • The mechanical energy of a particle in an ideal gas

is equal to its kinetic energy • The kinetic energy has translational and rotational

components • For a monatomic gas, the rotational kinetic energy

does not contribute to the gas properties • The total kinetic energy of a system of N particles is

KEtotal = N(KEtrans ) = 3/2 N kB T • For a monatomic gas, this is its internal energy, U

Section 15.4

Internal Energy, cont. • The internal energy depends only on the number of

particles present and the temperature of the gas • In terms of the number of moles, U = 3/2 n R T • These results apply only to monatomic gases • In other materials, the contribution from potential

energy becomes important

Section 15.4

Specific Heat of an Ideal Gas • Heat energy added to an ideal gas will cause the

molecular kinetic energy to increase and therefore increase the internal energy

• If energy in the amount of ΔU is added to an ideal gas, the internal energy will increase by that amount • ΔU = ΔQ = 3/2 N kB ΔT

• The heat capacity for one mole of gas is ΔQ / ΔT = 3/2 NA kB = 3/2 R • This is called the specific heat per mole at

constant volume and denoted as CV

Section 15.4

Polyatomic Gases • For polyatomic molecules, the energy associated

with rotational and vibrational motion of the molecules must also be included

• The internal energy of these types of gases can be found using kinetic theory • The analysis is much more complicated than for a

monatomic gas

Section 15.4

Distribution of Speeds

• The speeds of individual molecules in a gas are not all the same

• The speed we found earlier was the average speed of the molecules

• The distribution of speeds is called the Maxwell-Boltzmann distribution

• It gives the probability a molecule will have a given speed

Section 15.4

Distribution of Speeds, cont. • The typical speed, v, is found near the average

speed • A significant number of molecules have speeds that

are much larger or much smaller • The distribution varies with temperature

• If the temperature is increased, the entire distribution curve shifts to higher speeds

Section 15.4

Diffusion

• As a molecule moves through a gas, it follows a zigzag path as it collides with other molecules

• This type of motion is called diffusion

• Each particle follows a different path, but these paths can be described in an average way

Section 15.5

Diffusion, cont. • Assume a typical molecular speed v and an average

distance ℓ between collisions • The speed depends on the mass and temperature of

the molecule • The value of the mean free path (ℓ) depends on the

density of the gas • For N particles occupying a volume V,

• Due to all the collisions, the total distance traveled by

a molecule is much longer than the direct path between two points

Section 15.5

Diffusion, final • Statistical arguments give an accurate description of

an average random path • The magnitude of the average displacement, Δr of a

molecule after many steps taken over a time t is

• D is the diffusion constant • The value of the diffusion constant depends on both

v and ℓ, so it depends on temperature and mass of the diffusing particle as well as the properties of the medium

• Table 15.1 has values of some typical diffusion constants

Section 15.5

Gravity and Kinetic Theory • In the discussion of kinetic theory, gravity was

ignored • This was a good approximation since molecular

speeds are very high • For example, as mentioned, the speed of nitrogen

molecules in the atmosphere is ~500 m/s • The speed of a molecule obtained by falling 1 m in

the atmosphere is ~ 4 m/s • Gravitation effects on molecular motion is very small

Section 15.5

Diffusion in Medicine

• Transdermal drug delivery uses the process of diffusion

• A patch containing the drug is placed in contact with the skin

• The drug diffuses through the membrane, through the skin, and into the body

• The drug is delivered in a slow and steady manner

Section 15.5

Isotope Separation

• Diffusion is used in the separation of different isotopes • Nuclei that contain

different numbers of neutrons are called isotopes

• The average speed of the molecule decreases as the molecular mass increases

Section 15.5

Isotope Separation, cont. • A light atom will diffuse faster than a heavy one • The difference in speed can be used to separate the

isotopes • A higher concentration of the faster molecules will

develop on the far side of the membrane • Additional diffusion steps can be used to make the

separation more complete

Section 15.5

Bownian Motion • Although the diffusion of a molecule in a gas or liquid

cannot be traced, the diffusion of larger objects can be followed in detail

• Robert Brown carried out experiments with pollen grains in water

• He found that the grains followed the random zigzag paths shown for molecules • This motion is now called Brownian motion in his

honor • The motion was caused by the collisions of the

pollen grains with the molecules in the water

Section 15.5

Direction of Time

• An elastic collision between two molecules is time reversible

• Both forward and backward collision processes satisfy all the laws of physics

Section 15.5

Direction of Time, cont.

• Two gases, A and B are mixed

• At t = 0, the gases are separated

• After a period of time, the gases mix and there is an equal chance of finding either type of molecule anywhere in the container

Section 15.5

Direction of Time, final • If the process reversed, the mixed gases would

spontaneous separate and end up in separate regions of the container • Such unmixing does not occur in real life

• The direction of time does matter in this process • Newton’s Laws are time reversible, but many

processes in nature are not • Processes that are not time reversible are possible

in systems that contain a very large number of particles

• The time-reversible behavior of systems with many particles plays a role in the area of physics called thermodynamics Section 15.5

Puzzles • Certain properties of gases are not explained by

classical kinetic theory • These “failures” eventually led to the development of

quantum mechanics • A kinetic theory based on quantum mechanics

successfully explains the properties of dilute gases • Classical kinetic theory also does not address what

happens to a dilute gas or other substance when it is cooled to absolute zero

Section 15.6