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Sunday, March 22, 2009

THERMAL PHYSICS III

Suppose there are N atoms in a box of volume, and the average speed of an atom in the
direction is v . By symmetry, half are moving in the and half in the directions.
So the number o
x x +x −x
( ) 2
2
f atoms that hit the wall in time is 1 v . Hence the change in
2
momentum is 2 v 1 v v . From this, we can calculate the
2
pressure 1 v . Hence v
x
x x x
x x
t N tA
V
p m N tA Nm At
V V
P F p Nm PV Nm
A A t V
Δ Δ
Δ = ×⎛⎜ Δ ⎞⎟= Δ
⎝ ⎠
Δ
= = = =
Δ
2 2
2
2 1 v .
2
5. Let us return to our intuitive understanding of heat as the random energy of small particles.
Now, 1 v is the average kinetic energy of a particle on account of it
2
x
av
x
av
N m
m
= ⎛⎜ ⎞⎟
⎝ ⎠
⎛ ⎞
⎜ ⎟
⎝ ⎠
2
2
s motion in the
direction. This will be greater for higher temperatures, so 1 v , where is the
2
absolute temperature. So we write 1 v , which you can think of as t
2
x
av
x B
av
x mT T
m kT
⎛⎜ ⎞⎟ ∝
⎝ ⎠
⎜⎛ ⎟⎞ =
⎝ ⎠
he
definition of the Boltzmann constant, . So we immediately get , the equation
obeyed by an ideal gas (dilute, no collisions). Now, there is nothing special about any
particular d
B B k PV = Nk T
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
2 2 2
2 2 2 2 2
2
irection, so v v v and therefore the average total velocity is
v v v v 3 v . So the total average kinetic energy of one atom is,
1 v 3 , and the ene
2 2
x av y av z av
av x av y av z av x av
av B
av
K m kT
= =
= + + =
=⎛⎜ ⎞⎟ =
⎝ ⎠
( )
2
2
rgy of the entire gas is 1 v 3 .
2 2
From this, the mean squared speed of atoms in a gas at temperature is v 3 .
B
av
B
av
K N m NkT
k T
m
= ⎛⎜ ⎞⎟ =
⎝ ⎠
=
6. Heat, which is random kinetic energy, causes changes of
phase in matter:
a) Water molecules attract each other, but if water is
heated then they can escape and water becomes steam.
b) If steam is heated, the water molecules break up and
water becomes separated hydrogen and oxygen atoms.
c) If a gas of atoms gets hot enough, then the atoms collide
so violently that they lose electrons (i.e. get ionized).
d) More heat (such as inside the core of a star) will break up
atomic nuclei into protons and neutrons.
7. Evaporation: if you leave water in a glass, it is no longer there after some time. Why?
Water molecules with high enough KE can break free from the water and escape. But
how does a water molecule get enough energy to escape? This comes from random intermolecular
collisions, such as illustrated below. Two molecules collide, and one slows
down while the other speeds up. The faster molecule might escape the water's surface.
8. Heat needs to be supplied to cause a change of phase:
a) To melt a solid (ice, wax, iron) we must supply the "latent heat of fusion" for unit
mass of the solid. For mass we must su
F L
m pply an amount of heat to overcome
the attraction between molecules. No temperature change occurs in the melting.
b) To vapourize a substance (convert to gas or vapour), we must suppl
Q=mLF
y the "latent heat
of vapourization" for unit mass of the substance. For mass we must supply an
amount of heat . No temperature change occurs in the vapourization .
c) T
V
V
L m
Q=mL
o raise the temperature of a substance we must supply the "specific heat" C for unit
mass of the substance by 1 degree Kelvin. To change the temperature of mass by ,
we must supply h
m TΔ
eat Q=mCΔT.
In the above diagram you see what happens as ice
is heated. The amount of heat needed is indicated
as well. Note that no change of temperature happens
until the phase change is completed. The same
physical situation is represented to the right as
well where the temperature is plotted against Q.
no increase in
temperature
added heat
melts ice
no increase in
temperature
added heat
vaporizes water
i Q=cmΔT F Q=mL w Q=c mΔT V Q=mL s Q=cmΔT
5-mph interval
Speed
9. A very important concept is that of . It was first introduced in the context of
thermodynamics. Then, when the statistical nature of heat became clear a century later,
it was underst
entropy
ood in very different terms. Let us begin with thermodynamics: suppose a
system is at temperature and a small amount of heat is added to it. Then, the small
increase in the entropy of the s
T dQ
ystem is . What if we keep adding little bits of
heat? Then the temperature of the system will change by a finite amount, and the change
in entropy is got by adding up all the small chang
dS dQ
T
=
2
1
es: .
Now let's work out how much the entropy changes when we add heat to an ideal gas.
From the First Law, . The entropy change
is
T
T
V B
V
S dQ
T
dQ
dQ dU dW dU PdV C dT Nk T dV
V
dS dQ C dT Nk
T T
Δ =
= + = + = +
= = +

2 2 2 2
1 1 1 1
2 1
, and for a finite change we simply integrate,
ln ln 3 ln + ln . Note that the following:
2
a)The entropy increases if we heat a gas ( ).
b)The entr
B
V B B B
dV
V
S dQ C T Nk V Nk T Nk V
T T V T V
T T
Δ = = + =
>

2 1 opy increases if the volume increases ( ).
10. In statistical mechanics, we interpret entropy as the degree of disorder. A gas with all
atoms at rest is considered ordered, while a hot gas havi
V >V
ng atoms buzzing around in all
directions is more disordered and has greater entropy. Similarly, as in the above example,
when a gas expands and occupies greater volume, it becomes even more disordered and
the entropy increases.
11. We always talk about averages, but how do you
define them mathematically? Take the example
of cars moving along a road at different speeds.
( )
( )
( )
( )
Call v the number of cars with speed v ,
then v is the total number of cars and
v v
the average speed is defined as v .
v
We define the probability of finding a car with
i i
i
i
i i
i
i
i
n
N n
n
n
=
=
Σ
Σ
Σ
( ) ( )
( )
( )
( )
v v
speed v as v . We can
v
also write the average speed as, v v v .
i i
i i
i
i
i i
i
n n
P
n N
P
≡ =
=
Σ
Σ

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