Thermodynamics
Now let us turn our attention to gases. If we compress a gas while
keeping its temperature constant, we find that the pressure increases as the
volume decreases. The rate of increase in pressure is inversely proportional to
the decrease in volume, so that
pV = constant
(21.1)
This is called Boyle's law. This is true for all gases at low densities. But it had
been found that the absolute temperature of a gas at low densities is proportional
to the pressure at constant volume. Similarly, the absolute temperature (i.e. the
temperature in Kelvin) is proportional to the volume of a gas if the pressure is
kept constant. Thus at low densities, the product pV is approximately
proportional to the temperature T
pV = CT
where C is a constant of proportionality. In order to determine the constant,
consider two containers, each holding the same amount of gas at the same
temperature. If each container has a volume V, then when we combine the two
containers, the resulting new container has a volume of 2V, yet the pressure and
temperature remain unchanged. Thus C must be proportional to the number of
molecules in the gas,
pV = NkT
(21.2)
where k is called Boltzmann's constant. It has a value of
k = 1.381x10-23 J/K
Ideal Gas Law
Measuring the number of molecules in a gas is extremely difficult. It is
much easier to measure the mass of the gas and convert over to the number of
moles in the gas. A mole (mol) of any substance is the amount of that substance
that contains Avogadro's number of atoms or molecules. Avogadro's number is
defined as the number of carbon atoms in 12 g of 12C. It has a value of
NA = 6.022x1023 molecules/mol. If we have n moles of a substance, the number
of molecules is
N = nNA
Thus (21.2) becomes
pV  nN A kT
 nRT
(21.3)
This is known as the ideal gas law. Here R is called the universal gas
constant, and it has a value of R = 8.314 J/mol-K. The mass of 1 mol is called
the molar mass M. The molar mass of 12C is 12 g/mol. Thus the mass of n
moles of gas is given by m = nM. So we see that the number of moles in the gas
can be determined by knowing the chemical composition and measuring the
mass of the gas. The ideal gas law is an example of an equation of state. If the
amount of gas is constant, then the state of the gas is determined by any two of
the three variables p, V or T. One of the side effects of the ideal gas law is to
note that if we have a fixed amount of gas, then the ratio pV/T is a constant and
can be used to compare the system when it is in two different states. One last
thing to note is that all of our discussion has been for an ideal gas. This is a gas
whose density is low enough so that there is essentially no interactions between
the molecules in the gas. This is important since the interactions introduce
corrections to the equation of state.
Example:
5 kg of CO2 occupy a volume of 500 L at a pressure of 2 atm. What is the
temperature? If the volume is increased to 750 L and the temperature is kept
constant, what is the new pressure?
We can find the temperature once we know the number of moles of CO 2
present. The molar mass of CO2 is 12 + 16 + 16 = 44 g/mol, the number of
moles is
m
M
5000 g 

g
44 mol
n


 113.64 mol
The absolute temperature is then
T

pV
nR
2 atm 500 L 
L - atm
113.64 mol 0.0821 mol

-K
 107.18 K
The new pressure can be found from the fact that
pV
T
pV
pV
 1 1  2 2
T1
T2
nR 
p2 


V1 T2
p1
V2 T1
V1
p1
V2
500 L  2 atm 
750 L 
 1.33 atm
Kinetic Theory of Gases
We now turn to understanding the underlying cause of temperature.
Remember that all materials are made up of collections of molecules and atoms.
In a gas, these molecules are free to bounce around inside the containing vessel.
In doing so they have collisions and exchange momentum and energy. To start
with, lets make the following assumptions:
1. The gas consists of a large number of molecules that make elastic
collisions with each other and with the walls of the container.
2. The molecules are separated, on the average, by distances that are large
compared with their diameters, and they exert no forces on each other
except when they collide.
3. In the absence of external forces (the molecules are moving fast enough
so that we can neglect gravity), there is no preferred position for a
molecule in the container, and there is no preferred direction for the
velocity vector.
Look at a rectangular container with volume V. If it has N molecules with
a mass m and speed v, then we can calculate the force that these molecules
exert on any one wall. Consider the wall perpendicular to the x-axis. The x
component of the momentum of a molecule is mvx before it hits the wall, and
-mvx after it hits. Thus the molecule changes momentum by 2mvx. The total
change in the momentum of the gas in a time t is given by the number of
molecules colliding with the wall times the change in momentum for one
molecule. In a time t, the number of molecules that hit the wall is 12  VN vx At  ,
where the ½ comes from the fact that an equal number of molecules in the
volume vxAt are moving away from the wall as towards it. The impulse received
by the wall from all the molecules is
Fx t  px
1 N


vx tA 2mvx 
2 V

N
 mvx2 At
V
This allows us to determine the pressure exerted on the wall
Fx
A
N
 mvx2
V
p
or
PV  2 N
1
2
mvx2
(21.4)
where I have replaced vx2 with<vx2> to account for the fact that not all the
molecules are moving at the same speed. Using the ideal gas law, (21.4) can be
rewritten as
NkBT = 2N<K>
or
<K> = ½kBT
(21.5)
But, there is nothing special about the x direction, so we can repeat the process
for the y and z directions. Since we assumed there was no preferred direction of
motion, we see that
v 2  vx2  vy2  vz2
 3 vx2
so in three dimensions the average kinetic energy of a molecule is given by
K 
3
kBT
2
(21.6)
Conduction, Convection and Radiation
Let's turn back to heat once more, and now ask how the thermal energy is
transferred from one place to another. There are three methods of heat
transportation: conduction, convection and radiation.
In conduction, thermal energy is transferred by interactions among atoms
and/or molecules. Thus the systems that are exchanging heat are in contact with
one another. The rate at which thermal energy is transferred is given by
Q
T
 kA
t
x
(21.7)
where k is called the thermal conductivity. The ratio T/x is known as a
temperature gradient. A temperature gradient must exist for heat exchange to
occur between two systems.
Convection is the transport of thermal energy via direct mass transport.
In other words, as the material which is at temperature T is moved, the thermal
energy in the matter moves as well, allowing it to be transferred to a system not
in direct contact with the heat source. Thermal energy is not actually transferred
to another system until that system comes in contact with the mass carrying the
thermal energy. Thus the rate at which the thermal energy is transferred is the
same as that for conduction.
Radiation is the last mechanism of thermal energy transport. Here the
energy is emitted and absorbed by the system in the form of electromagnetic
radiation. When a system is in thermal equilibrium it is emitting and absorbing
radiation at the same rate. The rate at which thermal energy is radiated is given
by
Q
 eAT 4
t
(21.8)
where e is the emissivity of the system and  is a constant known as Stefan's
constant. This is known as the Stefan-Boltzmann law. The emissivity of the
system is a number between 0 and 1 and depends on the composition of the
surface of the object. Stefan's constant has a value of
 = 5.6703x10-8 W/m2K4
When an object absorbs all of the radiation that strikes it, it has an emissivity of 1
and is known as a black body.
Three Laws of Thermodynamics
Finally, just as we found that there were three main laws that guide
mechanics, namely Newton's laws, there are three main laws that guide
thermodynamics. They can be stated as
1. The net heat added to a system equals the change in the internal energy
of the system plus the work done by the system,
Q = U + W
2. It is impossible for a heat engine working in a cycle to produce no other
effect than that of extracting from a reservoir and performing an equivalent
amount of work.
3. The entropy of a system approaches a constant value as the temperature
approaches zero.
In order to understand the second and third laws, we must introduce some
new concepts. A heat engine is a device that operates by extracting heat from a
high temperature reservoir as its energy source. The second law says that there
will always be some heat left over that must be ejected into the environment.
Finally, we have entropy. Entropy can be most easily described as a
measurement of the disorder in a system. This is not a very enlightening
description, so a better one is that entropy is a measurement of the number of
states that the system can possibly be in. The third law basically states that as
the temperature goes to zero, the number of states available to a system reduces
to one.