Thermodynamics
Now let us turn our attention to gases.
If we compress a gas while
keeping its temperature constant, we find that the pressure increases as the
volume decreases.
The rate of increase in pressure is inversely proportional to
the decrease in volume, so that
pV
= constant
(21.1)
This is called
Boyle's
law
.
This is true for all gases at low densities.
But it had
been found that the absolute temperature of a gas at low densities is proportional
to the pressure at constant volume.
Similarly, the absolute temperature (i.e. the
temperature in Kelvin) is proportional to the volume of a gas if the pressure is
kept constant.
Thus at low densities, the product
pV
is approximately
proportional to the temperature
T
pV
=
CT
where
C
is a constant of proportionality.
In order to determine the constant,
consider two containers, each holding the same amount of gas at the same
temperature.
If each container has a volume
V
, then when we combine the two
containers, the resulting new container has a volume of 2
V
, yet the pressure and
temperature remain unchanged.
Thus
C
must be proportional to the number of
molecules in the gas,
pV
=
NkT
(21.2)
where
k
is called
Boltzmann's
constant
.
It has a value of
k
= 1.381x10
23
J/K
Ideal Gas Law
Measuring the number of molecules in a gas is extremely difficult.
It is
much easier to measure the mass of the gas and convert over to the number of
moles in the gas.
A
mole
(mol) of any substance is the amount of that substance
that contains Avogadro's number of atoms or molecules.
Avogadro's
number
is
defined as the number of carbon atoms in 12 g of
12
C.
It has a value of
N
A
= 6.022x10
23
molecules/mol.
If we have n moles of a substance, the number
of molecules is
N
=
nN
A
Thus (21.2) becomes
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pV
nN kT
nRT
A
=
=
(21.3)
This is known as the
ideal
gas
law
.
Here
R
is called the
universal
gas
constant
, and it has a value of
R
= 8.314 J/molK.
The mass of 1 mol is called
the
molar
mass
M
.
The molar mass of
12
C is 12 g/mol.
Thus the mass of
n
moles of gas is given by
m
=
nM
.
So we see that the number of moles in the gas
can be determined by knowing the chemical composition and measuring the
mass of the gas.
The ideal gas law is an example of an
equation
of
state
.
If the
amount of gas is constant, then the state of the gas is determined by any two of
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 Spring '09
 Knott
 Physics, Thermodynamics, Energy, Thermal Energy

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