1.2 The Ideal Gas
7
However, we are not yet done, since we still need some refinements. Let us now
examine them. If we wished to apply our expression obtained above directly to
calculate the pressure for the gas as a whole, we would have to assume that all
the atoms in the cylinder between the piston face and the end wall travel with the
same x-component of velocity and that they all move in the same direction. This
is, of course, unrealistic, since neither assumption would hold, as we well know.
How do we correct for such assumptions? In the first place, since all the atoms
have v 2
x -values that are different, we must ‘take an average’; we shall denote this
average by v 2
x . Further, we must remember that on average, as many atoms will
have x-components of velocity that could be designated by −v x , and that since
(−v x ) 2 = v 2
x , when we replace v 2
x by its average, we obtain a value that is twice as
large as we should have, so that the pressure should be given now by
P = nm
v
2
x
.
Thus far, we have only considered motion along the x-axis, but as far as atoms
are concerned, there is nothing special about the x-direction: it will thus be true that
v 2
x = =v 2
y = =v 2
z . Now, if this is true, we can employ the trick
v
2
x
=
1
3
v
2
x + v
2
y + v
2
z
=
1
3
v
2
and obtain for the pressure the relation
P = nm
1
3
v
2
=
2
3 n
1
2 mv
2
.
(1.2.4)
Because n = N/V , an equivalent form for this equation is
P V =
2
3 N
1
2 mv
2
,
in which N
1
2 mv 2 represents the total kinetic energy of the atoms in the gas (for
hard spheres, this is also the total energy). Let us call the total energy U , so that we
can also write 2
P V =
2
3 U .
(1.2.5)
We might now ask, ‘What about the factor
2
3 ?’. Is it always that, independent of the
gas in the cylinder? In fact, it is found to vary with the kind of gas that one is using
in the cylinder, so that to be general enough, we should allow for this factor to vary.
2 In thermodynamics, one refers to U as the internal energy, meaning the energy contained within
the thermodynamic system at rest, in contrast to the external energy associated, for example,
with the kinetic energy of translation and rotation of a macroscopic system as a whole. The
thermodynamic internal energy thus consists of the total kinetic and potential energies of the
constituent particles comprising the macroscopic system.
7
However, we are not yet done, since we still need some refinements. Let us now
examine them. If we wished to apply our expression obtained above directly to
calculate the pressure for the gas as a whole, we would have to assume that all
the atoms in the cylinder between the piston face and the end wall travel with the
same x-component of velocity and that they all move in the same direction. This
is, of course, unrealistic, since neither assumption would hold, as we well know.
How do we correct for such assumptions? In the first place, since all the atoms
have v 2
x -values that are different, we must ‘take an average’; we shall denote this
average by v 2
x . Further, we must remember that on average, as many atoms will
have x-components of velocity that could be designated by −v x , and that since
(−v x ) 2 = v 2
x , when we replace v 2
x by its average, we obtain a value that is twice as
large as we should have, so that the pressure should be given now by
P = nm
v
2
x
.
Thus far, we have only considered motion along the x-axis, but as far as atoms
are concerned, there is nothing special about the x-direction: it will thus be true that
v 2
x = =v 2
y = =v 2
z . Now, if this is true, we can employ the trick
v
2
x
=
1
3
v
2
x + v
2
y + v
2
z
=
1
3
v
2
and obtain for the pressure the relation
P = nm
1
3
v
2
=
2
3 n
1
2 mv
2
.
(1.2.4)
Because n = N/V , an equivalent form for this equation is
P V =
2
3 N
1
2 mv
2
,
in which N
1
2 mv 2 represents the total kinetic energy of the atoms in the gas (for
hard spheres, this is also the total energy). Let us call the total energy U , so that we
can also write 2
P V =
2
3 U .
(1.2.5)
We might now ask, ‘What about the factor
2
3 ?’. Is it always that, independent of the
gas in the cylinder? In fact, it is found to vary with the kind of gas that one is using
in the cylinder, so that to be general enough, we should allow for this factor to vary.
2 In thermodynamics, one refers to U as the internal energy, meaning the energy contained within
the thermodynamic system at rest, in contrast to the external energy associated, for example,
with the kinetic energy of translation and rotation of a macroscopic system as a whole. The
thermodynamic internal energy thus consists of the total kinetic and potential energies of the
constituent particles comprising the macroscopic system.