Weíve established that in three dimensions, pressure ëPí, times volume ëVí, equals
number of molecules , ëNí, times one-third molecular mass ëmí, times mean-square velocity.
And, this equals ëNí times Boltzmannís constant ëkí times temperature ëTí.
Also, ëkTí equals molecular mass times mean-square velocity in a single dimension, so that Ω
ëkTí is the kinetic energy per degree of freedom.
Itís instructive to consider if this is true for a mixture of molecules with different
masses.
If so, all molecules would tend to have the same kinetic energy, but not the same mean-square
velocity.
There is a very elegant demonstration that collisions among balls of different masses
tends to equalize kinetic energy.
Details are given in the Feynman lectures, link at the bottom of the screen.
Hereís a quick outline.
Two balls have masses ëmí-1 and ëmí-2, and velocities, ëví-1 and ëví-2.
The center-of-mass velocity ëví-0 is the total momentum, ëmí-1 ëví-1 plus ëmí-2
ëví-2, divided by the total mass, ëmí-1 plus ëmí-2.
The relative velocity ëví-r is ëví-1 minus ëví-2.
This is the velocity of particle 1 as seen by particle 2.
This is also ëuí-1 minus ëuí-2, where the ëuís are the velocities in the center
of mass frame.
In the center of mass frame, the velocities are anti-parallel.
We can take them to be in the right and left directions, as shown.
After an elastic collision, the velocity magnitudes are unchanged.
But, the directions are rotated by an angle theta, which depends on the vertical offset
of the particles before the collision.
Over many collisions we expect that the angle alpha between the relative velocity, ëví-írí,
and the center-of-mass velocity, ëví-0, will take on all possible values.
So, averaged over all collisions the so-called inner product between those two velocities
will be zero.
Writing this out in terms of ëví-1 and ëví-2 we get two terms.
The first is the average of twice the difference in kinetic energy.
The second is the difference in masses times the average inner product of ëví-1 and ëví-2.
Over many collisions, we also expect the angle between these two velocities to take on all
possible values.
So this second term will vanish.
But then the first term must vanish, which requires, on average, the two particles to
have equal kinetic energy.
Letís verify this with a simulation using 16 red balls of mass ëmí-1, and 16 blue
balls with 4 times this mass.
The lighter balls are initially at rest and the heavier balls are given random initial
velocities.
As time goes on it does appear that, on average, the lighter red balls end up moving faster
than the heavier blue balls.
Hereís a plot of the total kinetic energy versus time, in green, and the fractions due
to the heavy and light balls, in blue and red, respectively.
Initially the blue balls contain all the kinetic energy.
As time goes, on energy spreads to the red balls, until each type accounts for roughly
half the total energy.
There are only sixteen of each type of ball, so the fractions fluctuate quite a bit.
But, on average over a long time period, the kinetic of each ball is the same.
The plots have many, very rapid, dips.
During a collision, when balls are in contact, they compress against each other.
This briefly converts kinetic energy into elastic potential energy, and results in a
dip in the kinetic energy curves.
For perfectly hard balls, these dips would have zero duration, and the total kinetic
energy curve would be a flat line.
We can now summarize our results in the Ideal Gas Law.
Pressure times volume equals number of molecules times Boltzmannís constant times absolute
temperature.
For ëNí and ëTí constant, the right-hand side is constant, so the left-hand side is
also.
This is Boyleís law.
Dividing by ëPí and ëTí we get ëVí over ëTí equals ëNí ëkí over ëPí.
For constant ëNí and ëPí, ëVí over ëTí is constant.
This is Charlesís law.
Dividing by ëVí and ëTí we get ëPí over ëTí equals ëNí ëkí over ëVí.
For constant ëNí and ëVí, ëPí over ëTí is constant.
This is the law of Gay-Lussac.
And, dividing by ëkí ëTí we get ëNí equals ëPí ëVí over ëkí ëTí.
For constant ëPí, ëVí and ëTí, ëNí is constant.
This is Avogadroís law.
In a macroscopic sample of gas, the number of molecules is huge.
Itís convenient to quantify this using Avogadroís constant.
This constant, also called Avogadroís number, was, until recently, defined as the number
of atoms in 12 grams of carbon 12.
In the revised international system of units, Avogadroís constant is defined as exactly
6.02214076 times 10 to the 23.
This number of molecules is called one mole of a substance.
The molar mass of an element is the mass in grams of one mole of atoms of that element.
In this periodic table, atomic number, the number of protons in the nucleus, is shown
above the elementís chemical symbol.
And, molar mass is shown below.
Molar mass essentially equals the number of protons plus the number of neutrons in the
nucleus.
Note that the molar mass of carbon is not 12.
This is because naturally occurring carbon is about 98.9% carbon 12, which has 6 protons
and 6 neutrons, and about 1.1% carbon 13, which has 6 protons and 7 neutrons.
The resulting molar mass is 12.011 grams.
Analogous isotope blends occur for other elements with the result that their molar-mass values
are also not integers.
In the revised international system of units, Boltzmannís constant is defined as exactly
1.380649 times 10 to the minus 23 joules per kelvin.
Itís convenient to combine the Avogadro and Boltzmann constants into a new constant, R,
called the gas constant.
To four digits this is 8.314 joules per mole kelvin.
This gives us a new form of the ideal gas law.
If we express the number of molecules as the number of moles, lower-case ëní, times Avogadroís
constant, then pressure times volume equals number of moles, times the gas constant times
temperature in kelvin.
If we use this to determine the volume of one mole of gas molecules at 20 degrees Celsius
and standard atmospheric pressure.
We find a volume of about 24 liters.
In the previous video we introduced the concept of heat capacity, and looked at the heat capacity
per gram, the so-called specific heat, of several substances.
We saw no discernible relation between the specific heats of different materials.
Letís consider the heat capacity of a monatomic idea gas.
Heat capacity is defined as the added heat energy ëQí required to raise the temperature
by one kelvin.
It has units of joules per kelvin, or alternately, calories per kelvin.
For a monatomic gas, the ideal gas law reads ëPí times ëVí equals ëNí times one-third
ëmí times the mean square velocity, which equals ëNí ëkí ëTí.
The internal, kinetic energy of the gas is ëNí times one-half ëmí times mean-square
velocity.
Multiplying the expression above by three halves we can express the internal energy
as three-halves ëNí ëkí ëTí.
The heat capacity is the change in energy when temperature increases by one kelvin,
which is three-halves ëNí ëkí.
More precisely, this is the heat capacity at constant volume, denoted with a subscript
ëVí.
We could also define a heat capacity at constant pressure and obtain a different value.
Weíll come back to this idea in a later video.
If we set ëNí to Avogadroís constant, we find the molar heat capacity at constant volume
is three-halves the gas constant, or 12.47 joules per mole kelvin.
This predicts that the heat capacity per mole of all monatomic gases will be the same.
Letís see if this is true.
Helium, which is monatomic, has a molar mass of 4.003 grams, and a specific heat of 3.1156
joules per gram kelvin.
Multiplying these we obtain a molar heat capacity of 12.47 joules per mole kelvin, agreeing
with our theory to four digits.
Doing the same for neon and argon we obtain the same molar heat capacity, even though
they have very different molar masses and specific heats.
This is an important insight.
Heat capacity, at least for a monatomic gas, depends only on the number of molecules.
Itís independent of the particular substance.
Looking at the diatomic gases nitrogen and oxygen, each composed of two-atom molecules,
we find different molar heat capacities, near 21 joules per mole kelvin.
So, in general, gases do not all have the same molar heat capacity.
The value is dependent in some way on molecular structure.
The monatomic value is three-halves ëRí, while the diatomic value is close to five-halves
ëRí.
We might have guessed that molecules with two atoms would have either the same or twice
the heat capacity of single-atom molecules, but that is not the case.
And, in contrast to the highly regular monatomic case, the molar heat capacities of these two
diatomic gases donít even agree to three digits.
Weíll treat the heat capacity of non-monatomic gases, along with liquids and solids, in future
videos.
For now weíll limit consideration to monatomic gases, which we seem to have a good model
for.