1
Note 2.
Energy
2.1 Ideal gases: A simple system to play with
Ideal gases are a natural place to start learning about thermodynamics. We have some
intuition about how gases work and an “ideal” gas is the simplest model. In particular, an
ideal gas is a great model system to learn and test our understanding of thermodynamics
since all of the fundamental thermodynamic properties we will talk about can be found in
them and they allow us to study thermodynamics without getting lost in too much math.
Later on, we will study more realistic gases and see the nature of the differences.
2.1.1 Basic properties
Like many types of matter, we characterize an ideal gas by certain properties: volume
(
V
), pressure (
P
), temperature (
T
), and how many moles of atoms are in the gas (
n
).
There is a simple expression relating these quantities:
PV = nRT
This equation is called the equation
of state
for this system, since it relates the
state
variables
(
P
,
n
,
V
, and
T
) in this case. Actually, many gases behave like ideal gases in
certain conditions (this is called “ideal” conditions).
R is called the gas constant R and
R = 8.314 J/(mol K)
≈
0.082 L atm/(mol K)
You will learn later that R is related to the Boltzmann constant,
k
(=1.38
×
10
23
J/K) and
the Avogadro number
N
A
(=6.02
×
10
23
/mol) by
R =
k
N
A
This equation also tells that
you need only two variables
(two out of P, V and T)
to
define the state of a gas
(with a fixed n). This is true not just for ideal gas but also for
any gas system.
There are two types of properties here:
1.
Extensive properties
are properties which are related to “how much stuff” there is.
For example,
n
and
V
are extensive properties. If the system is duplicated, these variables
get doubled.
2.
Intensive properties
are independent of the size of the system. For example,
P
,
T
, are
intensive properties. So is the density
ρ
= N/V
, where
N
is the total number of atoms.
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2.1.2 A brief glimpse of phase transitions: Van der Waals equation
When gases cool, they condense to liquids. When liquids cool, they freeze into solids.
These are two examples of phase transitions. However, ideal gases cannot have phase
transitions. This is what is meant by “ideal.” Thus, far from the phase transition, they are
good models, but do not work well near the phase transition.
What makes a phase transition? Interaction between molecules. Gas particles start to
stick together at lower temperatures and form a liquid. How can we model this
interaction? We can modify the ideal gas equation to include interactions. We’ll do so in
two steps:
1. What’s the probability that a gas particle will bump into another one? The density
ρ
=
N/V
is a lot like a probability that we’ll find a given particle at a given spot. If the density
is high, then there is a high probability that the particle is there. The probability of finding
two particles at the same spot goes like the
2
. It’s like what’s the probability of flipping
two coins and having them both come up heads: it’s the probability of one coming up
heads squared.
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 Spring '10
 LIM
 Thermodynamics, Energy, Entropy, gas particles

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