The Ideal Gas Law and the Kinetic Theory of Gases
We continue our discussion of temperature and heat
but we are going to approach it in a different way. From
last week, we looked at the global or bulk properties of
materials when I raise or lower their temperature. Now we
are going to discuss the affects of raising the temperature
on a molecular level.
Most of the quantities that we will look at are not
derived quantities but empirical quantities. That is, they
are laws that come about from experiments and not theory.
So, for the most part, the laws that we use are valid, except
in some wild extremes.
First, to discuss what happens to the molecules in a
gas, we need a way to quantify the number of molecules in
a certain amount of gas. Therefore, we now define the
mole.
To begin, the relative masses of the atoms of different
elements can be expressed in terms of their atomic masses,
which indicates the mass of one atom as compared to
another. The unit on this scale is called the atomic mass
unit. The molecular mass of a molecule is the sum of the
atomic masses of its atoms.
A mole, or one mole of a substance, contains as many
particles as there are atoms in 12 grams of carbon-12.
Experiment show that there are 6.022x1023 atoms in 12 g of
carbon-12. Therefore, one mole of hydrogen contains
6.022x1023 atoms of hydrogen. One mole of water contains
6.022x1023 molecules of water…and so on. Furthermore,
one mole of a substance has a mass in grams that is equal to
the atomic of molecular mass of the substance.
Example, how many grams are there in one mole of
water?
The Ideal Gas Law
The ideal gas law expresses the relationship between
the absolute pressure, the Kelvin temperature, the volume,
and the number of moles in a gas.
First, when we were discussing the Kelvin scale, we
saw that is was an extrapolation of what would happen to
the pressure in a gas, if the temperature kept decreasing.
We saw that the absolute pressure, P, is directly
proportional to the Kelvin temperature, T, for a fixed
volume of gas. (P T)
Also, we can infer what would happen to the pressure
inside a gas if we increased the number of molecules, or
moles. The absolute pressure of an ideal gas is
proportional to the number of molecules, or the number of
moles, n, of the gas. (P n)
Finally, it is possible to increase the pressure in a gas
if I reduce the volume, if the temperature and number of
molecules stay constant. So the absolute pressure in an
ideal gas is inversely proportional to its volume. (P 1/V).
We can express these as a single proportionality;
writing it down:
Also, the constant of proportionality is called R, the
universal gas constant.
If we replace the number of moles with the number of
particles, N, we can rewrite the ideal gas law as:
Where the constant, R/NA, is called Boltzmann’s
constant and has a value of 1.38x10-23 J/K, and is
represented by the symbol, k. So the ideal gas law
becomes:
Origin of the Ideal Gas Law
The work of several people led to the formulation of
the ideal gas law. The scientist Robert Boyle discovered
that at a constant temperature, the absolute pressure of a
gas is inversely proportional to the volume. Boyle’s Law
states:
P1V1 = P2V2
The curve that passes through the initial and final
points is called an isotherm (the gas expands slowly enough
to allow the system to remain in thermal equibrium).
Because it will take a certain amount of work to expand or
contract the system, the work done by the gas as its volume
changes is given by the integral:
Furthermore, there are several different paths you can
take to get from the initial to the final states. The work
done by a system depends on the initial and final states and
on the path followed by the system to these states.
Similarly, we can also account for the amount of heat
added or subtracted to the system as well. The energy
transferred by heat also depends on the initial, final and
intermediate states of the system. So we now have two
ways energy is transferred from the system to the
surroundings:
1. Work is done by the system on its surroundings
2. The system transfers heat to the surroundings.
This will lead us the first law of thermodynamics a little
later.
Another scientist, Jacques Charles, discovered the
relationship between the pressure and temperature in an
ideal gas. This Charles’ Law states:
V1/T1 = V2/T2
The Kinetic Theory of Gases
At any time in a container, the molecules are moving
at some speed for a given temperature. The physicist
James Clerk Maxwell was the first to find the distribution
of speeds within a large collection of molecules at a
constant temperature. In order to develop a model for the
motion of the particles in a gas, we have to define a few
assumptions:
1. The volume of the molecules is negligible compared
to the volume of the container.
2. The molecules obey Newton’s Laws…something that
only works for the bulk volume of a gas.
3. The molecules will undergo elastic collisions between
them and the walls.
4. The molecules do not interact except by
collisions…this is also not really correct.
5. All the molecules will be identical.
The following graphs show the average, or most probable,
distribution of speeds in a gas. The total pressure exerted
on the walls by the gas can be found to be:
P = 2/3 (N/V) (1/2 mv2)
Therefore, pressure is proportional to the number of
molecules per unit volume and to the average kinetic
energy of the molecules.
The average kinetic energy of the gas is then equal to:
½ mv2rms = 3/2 kT
This equation indicates that the Kelvin temperature is
directly proportional to the average kinetic energy per
particle in an ideal gas no matter the pressure or volume.
This leads to the question, does a single particle have a
temperature?
What we did not derive is that every degree of
freedom, contribute ½ kT to the energy. For our particles
in a box, they can move in the x, y and z directions, so they
have 3 degrees of freedom and thus the total energy is
3/2 kT.
What we have done previously is to describe ways to
measure the specific heat of a substance. However, it is
now necessary that we predict the specific heat from
theoretical grounds.
Let’s see how we can find how much heat is needed to
raise the temperature of a gas. From earlier, you should
remember that the amount of heat needed was given as:
Q = mcT
But now we know from the first law that the amount
of heat released by the system depends on the path taken to
get from the initial and final states. Therefore, now the
heat associated with a given change in temperature does not
have a unique value.
We can address this by defining specific heats that
frequently occur: the specific heat at constant volume and
the specific heat at constant pressure. Furthermore, we will
also change our mass in the above equation into the number
of moles present:
Q = nCVT
or
Q = nCpT
When we increase the temperature of a gas at constant
pressure, not only does the internal energy of the gas
increase but it also does work because the volume changes.
Therefore, the specific heat at constant pressure will always
be greater than at constant volume. If we notice the heat as
a change of energy, we can rewrite the equation for the
change in energy at a constant volume as:
CV 1/ndE/dT
We also know that the kinetic energy is given as:
dK = 3/2 k dT
Rewriting the kinetic energy with k = nR, we can
equate the above two equations:
3/2 nR dT = n CV dT
Therefore, for our ideal gas, the value of the specific
heat at constant volume should be:
CV = 3/2 R
And that is exactly what we find. However, this does
not hold for diatomic or polyatomic gasses…see table.
The problem is that for diatomic or polyatomic
molecules, there exists more degrees of freedom.
For a diatomic molecule like O2 or N2, there are 5
degrees of freedom. Not only can the molecule move in all
three dimensions but it can also rotate about its center of
mass. These extra two degrees of freedom will then
account for more energy and:
CV = 5/2 R
Doing the arithmetic and comparing to the table, we
see that we are very close for diatomic molecules.
Finally, vibrational motion can also contribute to the
heat capacity of gasses. Molecular bonds are not rigid but
can stretch and bend. However, for most diatomic gasses,
this does not contribute appreciably to the heat capacity.
However, the extra degrees of freedom can account for the
increase in the heat capacity of polyatomic molecules as
can be seen in the table. Vibrational energy can cause in
increase in the heat capacity to:
CV = 7/2 R
But since the heat capacity is very temperature
dependent at this point for a polyatomic species, the
specific heat is not well defined.
The question we can ask then is can we theoretically
determine the heat capacity for solids? That is, can we find
one heat capacity for all solids? At first, this seems like a
daunting task but we can approximate, in fact, find a limit
to the heat capacity of all solids.
If we assume that the atoms in the solid are attached
like “springs”, we expect each atom to have an average
kinetic energy of 3/2 kT AND an average potential energy
of 3/2 kT, or an average total energy of 3kT per atom. If
the crystal contains N atoms or n moles, the total energy is
3 nRT. From this we conclude that the molar heat capacity
of a crystal should be:
CV = 3 R
This is the rule of Dulong and Petit, which we
encountered as an empirical finding, where solids on
average have a specific heat capacity of about 25 J/mol-K.
We can look at a graph of several solids and see how solids
approach this limit at high T.
Thermodynamics
Thermodynamics is the branch of physics that is built
upon the fundamental laws of heat and work. In thermo,
the collection of objects upon which attention is being
focused is called the system, everything else in the
environment is called the surroundings. We start with a
fundamental law known as the Zeroth Law of
Thermodynamics:
Two systems individually in thermal equilibrium with
a third system are in thermal equilibrium with each other.
The First Law of Thermodynamics: The internal
energy of a system changes from an initial state Ui to a
final state Uf, due to heat and work:
U = Uf – Ui = Q-W
Q is positive when the system gains heat and negative
when it loses heat. W is positive when work is done by the
system and negative if work is done on the system.
Two special cases of the first law of thermodynamics
are worth mentioning. A process that eventually returns a
system to its initial state is called a cyclic process.
Therefore the total internal energy change is zero and
Q = W.
Another special case occurs in an isolated system, one
that does no work on its surroundings, and has no heat flow
to or from its surroundings. In this case, W = Q = 0 or the
change in internal energy is = 0. In other words, the
internal energy of an isolated system is constant. We will
now talk about different types of thermodynamic processes.
Adiabatic Process: An adiabatic process is defined as one
with no heat transfer into or out of the system: Q = 0. From
the first law then, U = -W. The compression stroke in an
internal-combustion engine is nearly adiabatic as there is a
change in the volume but very little loss of heat.
Isochoric Process: Is a constant volume process. When
the volume of a thermodynamic system is constant, it does
no work on its surroundings, then W = 0 so: U = Q. In an
isochoric process, all the energy added as heat remains in
the system as an increase in internal energy. Heating a gas
in a constant volume container is an example.
Isobaric Process: An isobaric process is a constant
pressure process. In general none of the three quantities are
zero but we can calculate the work nonetheless: W = pV.
Boiling water at a constant pressure is an isobaric process.
Isothermal Process: An isothermal process is a constant
temperature process. For a process to be isothermal, any
heat flow into or out of the system must occur slowly
enough that the thermal equilibrium is maintained. In
general, none of the three quantities are zero.
We can see all these processes on a pV diagram:
We can use the first law to determine the specific heat
capacities like we found before. We can derive a simple
relation between CP and CV for an ideal gas. If we consider
a constant volume process, an infinitesimal quantity of heat
dQ flows into the gas, and its temperature increases by an
amount dT. By definition of CV:
dQ = nCVdT
And because the pressure increases but not the volume, the
gas does no work so from the first law:
dU = nCVdT
If we now consider a constant pressure process, the amount
of heat is given by:
dQ = nCPdT
And the work done on the system is given by:
dW = pdV = nRdT
So now:
nCP dT = dU + nRdT
Solving for dU, we find:
nCPdT = nCVdT + nRdT
Dividing each term by ndT, we find:
CP = CV + R
And as expected we see that the specific heat capacity at
constant pressure is indeed greater than at constant volume.
We can finally form a quantity which is the ratio of specific
heats:
= CP/CV
For an ideal monotomic gas we find that  = 1.67 and for
diatomic gases, 1.40 both of which are in good agreement
with experiment.
For adiabatic processes, we can also find several equalities
that describe the temperature, volume, pressure and work
done on or by the system. The equality for temperature and
volume is given by:
T1V1-1 = T2V2-1
Therefore:
p1V1 = p2V2
The work done is given as:
W = nCV(T1-T2)
Or:
W = 1/ (p1V1 – p2V2)
The Second Law of Thermodynamics: Heat flows
spontaneously from a substance at a higher temperature to a
substance at a lower temperature and does not flow
spontaneously in the reverse direction.
This leads to the concept of the heat engine. A heat
engine is any device that uses heat to perform work. A
typical type of engine is a steam engine:
1. Heat is supplied to the engine at a relatively high
temperature.
2. Part of the input heat is used to perform work.
3. The remainder of the input heat is rejected at a
temperature lower than the input temperature.
This leads to the concept of efficiency. That is how
efficient can we make an engine?
= Work Done/Input Heat = W/QH
An engine must obey the principle of the conservation
of energy. Some of the input heat is used, converted to
work, and the remainder is rejected to a cold reservoir.
Conservation of energy requires that:
QH = W + QC
If we solve for W and use the equation of efficiency,
we find:
= 1 - QC /QH
So how efficient can you make an engine? A scientist
named Sadi Carnot found that the maximum efficiency
engine has:
QC /QH = TC /TH
Where T is given in Kelvins and is the temperatures
between the hot and cold reservoirs in an engine.
Therefore, since the cold reservoir can NEVER reach
absolute zero, you can never make and absolutely 100%
efficient engine. This leads to the third law of
thermodynamics:
It is not possible to lower the temperature of any
system to absolute zero in a finite number of steps.
Finally, we can restate the second law in terms of a
quantity called entropy. Entropy is a quantitative measure
of the disorder of a system. The entropy change in a
reversible process is given as:
Therefore, we can restate the second law as the
entropy of an isolated system may increase but can never
decrease. When a system interacts with its surroundings,
the total entropy change of system and surroundings can
never decrease. When the interaction involves only
reversible processes, the total entropy is constant and
S = 0, when there is any irreversible process, the total
entropy increases and S > 0.