Kinetic
Theory of Gases
Many of the properties of gases can be
explained by the fact that the molecules are bouncing around like mad. Pressure
on a surface, for example, is the force of the collisions against the surface
divided by the area. If a container had only a few gas molecules, one might
observe the pulsing nature of the force, but since we always deal with
gazillions of molecules, it seems like a steady push.
The mushy nature of gases (compressibility)
is due to the fact that there is a lot of space between molecules. Liquids and
solids don't have that space, so they require a lot of pressure to compress
them a little. (If you could muster up unlimited amounts of pressure, you could
put any amount of mass in as small a volume as you wanted to. Nature does this
sort of thing in black holes, but we don't have the means. Solids and liquids
are incompressible…almost. Matter must occupy space…usually.)
Another property is diffusion. Let loose
some gas in the corner of a room, and after awhile it will be everywhere in the
room.
In this article we will deduce that the
absolute temperature of a gas is a measure of the average kinetic energy of a
molecule, and we will apply this to specific heats of gases, liquids and
solids.
Before the absolute temperature scale was
found, the Celsius scale was invented, based on the freezing and boiling points
of water (0o C and 100o C). At constant pressure it was
found experimentally that the volume of a gas is related linearly to temperature
(straight line V vs T graph) in such way that the extension of the graph goes
to V = zero at -273o C. So the Kelvin scale was invented: add 273 to
the Celsius temp to get Kelvins, and this makes the volume proportional to T in
Kelvins at constant pressure. The volume does not really become zero at T = 0,
because the proportionality is not valid when the gas condenses to a liquid
(which happens to all gases).
The standard textbook derivation of an
equation for the pressure of a gas is based on the elastic collisions with the
wall of a container. The starting point is a single molecule bouncing around in
a box, and we find the average pressure due to one molecule. Then N molecules
are put in the box and we have N times as much pressure, which is clearly true
if they all hit the side of the box with the same regularity as the one. But
how can the equation apply to a volume in which it is clear that only a tiny
fraction of them are hitting
the side? The answer is blowin' in the wind or swept
under the rug.
So here is my derivation that avoids the
above pitfall and has the added advantage of showing how to handle a case where
the molecule cannot be regarded as a point mass.
1. Consider a flat surface in a gas. A wall, for example.
2. Consider a cube of space so small that it has, on average, one molecule of
the gas in it. For air under normal conditions, this would be 3.34 nanometers
on a side. Call the length of a side L and the volume of this cube V1
= L3, and imagine it nestled against the flat surface. If it were a
real box with one molecule in it, the molecule would bounce off the wall on the
left then bounce around the box and hit the left wall again, and the average
time between hits would be 2L/average vx.
In the case shown here, the box is not real (this defined space is called a
"control volume"), so the molecule is unlikely to hit the wall
repeatedly, but since there is an average of one molecule in that space, the
average time between collisions is the same as that noted above. Half the time vx is to the right and the other half of the
time vx is to the left….
3. This part is like the usual book derivation. The key steps are
a) Fx D t = change of x component of
the momentum of the molecule = 2mvx,
where vx is the average component of
velocity ^ to the surface. Since we
want the average Fx over time, take D t to be the
time between collisions,
2L/vx .
b) The 3-d Pythagorean theorem is
vx2 + vy2 + vz2
= v2 . There is no preferred direction in
space so on average, all are equal, so
vx2 = 1/3 v2
P is Fx divided by area L2
.
c) Play with these things and get
PV1 = 1/3
mv2 (Reme
where m is the mass of a molecule and v2
is the average of the squares of the speeds.
Now we are essentially finished. P is
independent of the area. Want to apply this to a large volume V that is N times
V1 where N is a huge nu
PV = 1/3Nmv2.
What Temperature Really Is
Experimentally it was found that PV
is proportional to nT, the nu
The equation relating PV to nT is
PV = nRT,
where R is a constant which has been found experimentally:
R = 8.31 Joule/(mole kelvin). This is often called the
gas law constant, but as we shall see, it applies to liquids and solids, too.
If you just want to deal with PV=nRT, you might want to go to the gas law for an easier
discussion. This equation is called the
ideal gas law, because old theories are commonly known as laws. A better name would be the gas approximation.
The two PV equations are good approximations
for gases at normal atmospheric pressure, but they deteriorate to poor for a
gas under high pressure. Go back to the derivation and you will realize that it
is valid only if the radius r of the molecule is negligible compared to L,
otherwise you need to use
2(L-r)/vx for D t, and we
end up with
PV(1-r/L) = 1/3Nmv2 =
nRT. Another thing not taken into account is the possible force of attraction
between molecules. Look up a gas equation concocted by van der
Waals:
(P+a/v2)(v-b)=RT, where a and b are constants which depend on the
particular gas and v is the volume per mole. None of these equations works well
for a wide range of T and P.
Heating Gases at Constant Volume
For a monatomic gas such as neon, just equate the right hand sides of the two
PV equations, then multiply both sides by 3/2 and get 1/2
Nmv2 = 3/2 nRT. When you add heat energy to
the gas, you are adding kinetic energy to it. The above equation clearly has
total kinetic energy of the gas on the left, so it is kinetic energy on the
right. Hence when you add heat energy D Q to it, the temperature rises according to
D
Q = 3/2 nRD T.
At constant volume the gas does no work, and therefore all of the energy is
there in the molecules. We call it internal energy change D U:
D
U = 3/2 nRD T (monatomic
gas)
With monatomic gases the energy of rotation
is insignificant. Air is mostly nitrogen and oxygen, both diatomic gases, du
D
Q = 3/2 nRD T + 2/3
(3/2 nRD T) or
D
Q = 5/2 nRD T. This is
also internal energy:
D
U = 5/2 nRD T. (diatomic
gas)
If you want to find out more about this, or how to handle other kinds of molecules,
look up equipartition of energy somewhere.
Heating Gases Having Constant Pressure
Expansion occurs so the gas does
work PD V = nRD T, so add nRD T to the
two cases above. For monatomic, the fraction becomes 5/2
and for diatomic, 7/2 . Do not
call this D U. When the gas does work, this transfers the energy
elsewhere, although sometimes it is hard to see where the energy goes.
To summarize all the above stuff on heat, it
is convenient to let C stand for the fraction times R. For the two cases, constant
volume and constant pressure, we have Cv and Cp, called
molar heat capacities. Thus
D
U = nCv D T and at constant pressure, D Q = nCp D T
monatomic gas: Cv = 3/2 R and Cp = 5/2R
diatomic gas: Cv = 5/2R and Cp = 7/2R
Heating Solids
If the molecules can be regarded as
little spheres, and if the temperature is not too low (where quantum effects
are significant), then the heat energy added becomes kinetic and potential,
equal amounts of both. An oscillating mass on a spring transfers energy back
and forth between KE and PE, so the averages are equal. Those atom oscillators
act like masses on springs.
When a gas is in contact with a solid or
liquid, the temperatures equalize. With repeated collisions, the kinetic
energies equalize. Aha! Same KE. But we know the KE of
a gas is 3/2nRT, so the D (KE) of a solid or liquid is
D
(KE) = 3/2 nRD T. Double
it to get the total change of energy, :
D
Q = 3nRD T.
(This is known as the law of Dulong and Petit.)
For these solids and liquids it is customary
to express the heat capacity on a per mass basis rather than per mole, and
write D Q = CmD T, where C is called the specific heat capacity.
Replace n by m/M , where M is atomic
mass, in the equation D Q = 3nRD T, and you will see that the specific heat capacity
is
C = 3R/M.
This works fine for the common metals at
room temperature- iron, aluminum, copper, zinc, tin, mercury…. Water? no. Carbon? no. Silicon? no.
Be sure the units check out. If you are
using calories, use
R = 1.99 cal /(mole kelvin).
Back to gases: sometimes we have a process with no heat transfer. The fancy name for
this is an adiabatic process. We define gamma (g) as the ratio of Cp/Cv. Now with some calculus involved,
For a Reversible Adiabatic
Process (DQ = 0), Show PVg = constant
g = Cp/Cv, notice that g >1 because Cp = Cv +R (monatomic: g = 5/3, diatomic: g = 7/5)
1) Play with the two equations above and show that R = Cv(g - 1)
2) PV = nRT → PdV + VdP = nRdT
3) dU = nCvdT
and dW = PdV but if dQ=0, dU = - dW,
or nCvdT = -PdV
From 2) and 3), eliminate dT, then from 1), eliminate R and show that
PdV + VdP = (1-g)PdV
Divide by PV, do a little algebra and show dP/P + gdV/V = 0.
Integrate: lnP + glnV = constant
Or PVg = another constant, valid for a reversible, adiabatic process in an ideal gas.
I have another discussion of related goodies
on specific heat
and one that tells about melting and vaporization: change of phase. How
molecular velocities vary: the Boltzmann factor and the
Maxwell-Boltzmann distribution.
Other hot topics found on my main page on fluids,
heat, etc.
Other main physics pages:
mechanics
vibrations and waves
quantum
or index for alphabetized stuff
comments, questions: fredrick.gram @ tri-c.edu (remove spaces)