9.2 A Model System to Represent a Dilute Gas
387
3. The particles do not exert any forces on each other.
4. The motions of the particles are governed by classical (Newtonian) mechanics.
5. The system is confined in a rectangular box with smooth hard walls.
The study of this model system is called gas kinetic theory. It was originated by
Bernoulli, who was the first to test the consequences of assuming that a gas was a
mechanical system made up of many tiny moving particles. The theory was brought
to an advanced state by Joule, Maxwell, and Boltzmann. The fundamentals were
worked out independently by Waterston, about 15 years prior to the work of Joule and
Maxwell.
Daniel Bernoulli, 1700–1782, was a
Swiss mathematician best known for
Bernoulli’s principle, which states that
the pressure decreases as the flow
velocity of a fluid increases. The
application of this principle allows
airplanes to remain in the air.
James Prescott Joule, 1818–1889, was
a great English physicist who pioneered
in the thermodynamic study of work,
heat, and energy while working in a
laboratory in his family’s brewery.
James Clerk Maxwell, 1831–1879, in
addition to his work on gas kinetic
theory, also derived the Maxwell
relations of thermodynamics and the
Maxwell equations of electrodynamics.
Ludwig Boltzmann, 1844–1906, was an
Austrian physicist who was one of the
inventors of gas kinetic theory. He
suffered from depression and
apparently killed himself, possibly
because his ideas were not widely
accepted during his lifetime.
John James Waterston, 1811–1883,
was a schoolteacher in India at the time
of his independent invention of kinetic
theory and was unable to get his work
published.
Classical Mechanics
According to classical mechanics, the state of a point-mass particle is specified by its
position and its velocity. If the particle moves in three dimensions we can specify its
position by the three Cartesian coordinates x, y, and z. These three coordinates are
equivalent to a three-dimensional vector, which we denote by r and call the position
vector. This vector is a directed line segment that reaches from the origin of coordinates to the location of the particle. We call x, y, and z the Cartesian components of
the position vector. We will denote a vector by a letter in boldface type, but it can
also be denoted by a letter with an arrow above it, as in − → r , by a letter with a wavy
underscore, as in r
126
, or by its three Cartesian components listed inside parentheses,
as in r (x, y, z). Appendix B contains an elementary survey of vectors.
The velocity of a particle is a vector v with the Cartesian components v x , v y , and v z .
These components are the time derivatives of the Cartesian coordinates:
v x
dx
dt
, v y
dy
dt
, v z
dz
dt
(9.2-1)
These three equations are equivalent to a single vector equation
v
dr
dt
(9.2-2)
Newton’s Laws of Motion
In classical mechanics, the motion of a particle is governed by Newton’s three laws,
which we accept as generalizations of experimental fact. Newton’s first law is called
the law of inertia: A stationary particle tends to remain stationary unless acted on by a
force, and a moving particle tends to continue moving with unchanged velocity unless
acted on by a force. The first law is a special case of Newton’s second law, which is
called the law of acceleration. If a particle moves only in the x direction, Newton’s
second law is
F x ma x m
dv x
dt
m
d 2 x
dt 2
(9.2-3)
In this equation F x is the force in the x direction acting on an object having mass m, x is
its coordinate, v x is its velocity component in the x direction, and a x is its acceleration
in the x direction. As indicated, the acceleration is the time derivative of the velocity
387
3. The particles do not exert any forces on each other.
4. The motions of the particles are governed by classical (Newtonian) mechanics.
5. The system is confined in a rectangular box with smooth hard walls.
The study of this model system is called gas kinetic theory. It was originated by
Bernoulli, who was the first to test the consequences of assuming that a gas was a
mechanical system made up of many tiny moving particles. The theory was brought
to an advanced state by Joule, Maxwell, and Boltzmann. The fundamentals were
worked out independently by Waterston, about 15 years prior to the work of Joule and
Maxwell.
Daniel Bernoulli, 1700–1782, was a
Swiss mathematician best known for
Bernoulli’s principle, which states that
the pressure decreases as the flow
velocity of a fluid increases. The
application of this principle allows
airplanes to remain in the air.
James Prescott Joule, 1818–1889, was
a great English physicist who pioneered
in the thermodynamic study of work,
heat, and energy while working in a
laboratory in his family’s brewery.
James Clerk Maxwell, 1831–1879, in
addition to his work on gas kinetic
theory, also derived the Maxwell
relations of thermodynamics and the
Maxwell equations of electrodynamics.
Ludwig Boltzmann, 1844–1906, was an
Austrian physicist who was one of the
inventors of gas kinetic theory. He
suffered from depression and
apparently killed himself, possibly
because his ideas were not widely
accepted during his lifetime.
John James Waterston, 1811–1883,
was a schoolteacher in India at the time
of his independent invention of kinetic
theory and was unable to get his work
published.
Classical Mechanics
According to classical mechanics, the state of a point-mass particle is specified by its
position and its velocity. If the particle moves in three dimensions we can specify its
position by the three Cartesian coordinates x, y, and z. These three coordinates are
equivalent to a three-dimensional vector, which we denote by r and call the position
vector. This vector is a directed line segment that reaches from the origin of coordinates to the location of the particle. We call x, y, and z the Cartesian components of
the position vector. We will denote a vector by a letter in boldface type, but it can
also be denoted by a letter with an arrow above it, as in − → r , by a letter with a wavy
underscore, as in r
126
, or by its three Cartesian components listed inside parentheses,
as in r (x, y, z). Appendix B contains an elementary survey of vectors.
The velocity of a particle is a vector v with the Cartesian components v x , v y , and v z .
These components are the time derivatives of the Cartesian coordinates:
v x
dx
dt
, v y
dy
dt
, v z
dz
dt
(9.2-1)
These three equations are equivalent to a single vector equation
v
dr
dt
(9.2-2)
Newton’s Laws of Motion
In classical mechanics, the motion of a particle is governed by Newton’s three laws,
which we accept as generalizations of experimental fact. Newton’s first law is called
the law of inertia: A stationary particle tends to remain stationary unless acted on by a
force, and a moving particle tends to continue moving with unchanged velocity unless
acted on by a force. The first law is a special case of Newton’s second law, which is
called the law of acceleration. If a particle moves only in the x direction, Newton’s
second law is
F x ma x m
dv x
dt
m
d 2 x
dt 2
(9.2-3)
In this equation F x is the force in the x direction acting on an object having mass m, x is
its coordinate, v x is its velocity component in the x direction, and a x is its acceleration
in the x direction. As indicated, the acceleration is the time derivative of the velocity