Page 99 - Essentials of physical chemistry

The First Law of Thermodynamics 61

So far this is a general equation and we will use it more later, but at this point let us take a short cut
and specify that we are using an ideal gas with the extended definition given above.

qU
Then we can use the relationship that ¼ 0 and cancel the last two terms we are left with
qV
T

qV RT qV R
(C P C V ) Ideal ¼ [0 þ P] and V ¼ so ¼ .
qT P qT P
P n¼1 p

qV R
And thus for an ideal gas we have (C P C V ) Ideal ¼ P ¼ P ¼ R. This is only
qT P
p

3 qU 3
R so that
2 qT 2
for an ideal gas but it means that if U ¼ RT then C V ¼ ¼
V

3 5
R. Now we can add to the properties of an ideal gas.
C P ¼ R þ R ¼
2 2
1. PV ¼ nRT

qU qU
2. ¼ ¼ 0
qV T qP T
3. C P C V ¼ R
ADIABATIC PROCESSES
Next we come to another key word in thermodynamics, ‘‘adiabatic.’’ Under some circumstances
q ¼ 0. One way to achieve this condition is to surround the ‘‘system’’ by a nonconductive layer of
insulation such as a vacuum layer (Dewar flask, thermos bottle) or use some sort of glass wool or
fiberglass mat around the system as is done with modern refrigerators. Another method is to carry
out the process quickly because heat flow tends to be relatively slow. For instance you can put a
metal poker into the hottest part of a fireplace for a few moments because the heat will not be
conducted up the metal shaft to your hand in the short time it is in the heat. Another more dramatic
process is in the combustion chamber of an internal combustion engine. Although such engines
appear to be moving rapidly, the explosion of fuel in such an engine is much quicker than the
motion of the mechanical parts. For instance an internal combustion engine (ICE) of the four-cycle
type (intake, compression, power, exhaust) requires two full revolutions per explosion. Thus, an
ICE engine running at a typical 3000 revolutions=minute (rpm) experiences a power stroke duration
of about 1=2 of the second revolution of the full four steps, two revolution cycles, so the mechanical
(60 s= min )
time for the explosion is about ¼ 0:020s=rev and only about half of every other cycle is
(3000 rpm)
the power stroke so the mechanical time for the explosion is about 0.01 s. That is a very long time
compared to chemical reactions in the gas phase. We know from the previous discussion of the
28
3
Boltzmann KMTG that the binary collision number can be of the order of (10 =cm s) at 1 atm
pressure so in a compressed gas the collision rate will be even higher. Thus, chemical considerations
are needed to find a fuel whose burn time more nearly matches the mechanical timing of the engine.
A combustion reaction that is too fast for the mechanical parts of the internal combustion engine
causes a noticeable sound described as a ‘‘ping’’ or ‘‘knock’’ in the engine and can damage the
internal parts of the engine.
For the sake of thermodynamics let us consider a diesel engine. The diesel engine was developed
by Rudolf Diesel (1853–1913) who received an initial patent in Europe in 1893 and a U.S. patent in
1898 for an internal combustion engine, which used high compression of air to ignite almost any
combustible fuel including crude oil. While he was involved in patent disputes, it certainly seems he
was the prime inventor of the documented engines. However, there is evidence that a similar
principle has been in use for unrecorded ages by Fiji natives as a fire starter illustrated by a wooden