,
In gases, it is generally not a good assumption to hold the density constant, as we did when calculating the hydrostatic pressure of a liquid. Generally, the density of a sample of fluid increases as the pressure of the sample increases. We will consider the simplest relation between pressure and density, associated with the ideal gas.
It is found experimentally that dilute gases (such as air at atmospheric pressure) obey (very nearly) the ideal gas law:
,
where for a sample of a particular gas,
is pressure,
is the volume of the sample,
is a constant,
is the (absolute) temperature,
and 
is a mysterious quantity
called the "number of moles" in the sample. This equation is
discussed in detail later, but we must discuss it briefly now in order to
proceed with this application.
measures the quantity of
gas in the sample. It is easy experimentally (by measuring
,
, and
, and holding
constant) to add material to
the sample until 
; that is until
the sample contains one mole of gas. The mysterious part is that 1 mole of each
different chemical species of gas has a different mass.
The number of grams in one mole of a gas has been given the name of "molecular
weight" for that particular gas. This name is an unfortunate choice for
us, as we endeavor not to assume the existence of atoms. We will tell ourselves
that it is only a name, not an assumption, and soldier on. By the way, the
molecular weight is also the number of kilograms in a kilomole of the gas. We
will represent the molecular weight in kilograms with
.
The mass of a sample of
moles of any gas is given by 
.
For an ideal gas this lets us display the relation between pressure and density:
or
Returning to the zero-velocity Euler's equation with the y direction vertical,
.
As before, the pressure varies only in the vertical direction; horizontal planes are surfaces of constant pressure. The y component of this equation is
, which leads to
.
(We have made the simplifying assumption that temperature is constant.) As before we integrate both sides along a path of constant x and z,
so that
can be replaced with
in the integral.
Integrating from zero to 
,
we have
|
| Density of a gas decreases exponentially with height |
Since 
, this leads to
where has been replaced
with 
for simplicity. This says
that the density of a gas (such as oxygen) decreases exponentially with
increasing height above the earth's surface, falling to
of its surface value when
.
For larger the gas is
concentrated closer to the surface. Thus a gas with larger
seems to be pulled in the
direction of more effectively than a gas with low
. This effect can be used to
partially separate mixtures of gas. Using a centrifuge for "artificial
gravity," it is the basis for one method of separation of uranium isotopes
using uranium hexaflouride gas.
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Return to Euler's equation applications survey.
Go forward to streamline derivation of Bernoulli's equation.