gases and kinetic theory - faculty server...
TRANSCRIPT
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Nicholas J. Giordano
www.cengage.com/physics/giordano
Chapter 15 Gases and Kinetic Theory
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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