Page 27 - Physical chemistry eng

4 CHAPTER 1 Fundamental Concepts of Thermodynamics

We have used the equality N = n N A where N is Avogadro’s number and n is the
A
number of moles of gas in the second part of Equation (1.7). Because particles travel in
either the +x or –x direction with equal probability, only those molecules traveling in
the +x direction will strike the area of interest. Therefore, the total number of collisions
is divided by two to take the direction of particle motion into account. Employing
Equation (1.7), the total change in linear momentum of the container wall imparted by
particle collisions is given by
¢p = (2mv )(N coll )
x
Total
nN Av ¢t
= (2mv )¢ ≤
x
A
x
V 2
nN A
2
= A¢t m8v 9 (1.8)
x
V
In Equation (1.8), angle brackets appear around v x 2 to indicate that this quantity
represents an average value since the particles will demonstrate a distribution of veloc-
ities. This distribution is considered in detail later in Chapter 30. With the total change
in linear momentum provided in Equation (1.8), the force and corresponding pressure
exerted by the gas on the container wall [Equation (1.3)] are as follows:
¢p nN
F = Total = A Am8v 9
2
x
¢t V
F nN A
P = = m8v 9 (1.9)
2
x
A V
Equation (1.9) can be converted into a more familiar expression once 1>2 m8v 9 is rec-
2
x
ognized as the translational energy in the x direction. In Chapter 31, it will be shown
that the average translational energy for an individual particle in one dimension is
m8v 9 k T
2
x
= B (1.10)
2 2
where T is the gas temperature.
Substituting this result into Equation (1.9) results in the following expression
for pressure:
nN A nN A nRT
2
P = m8v 9 = kT = (1.11)
x
V V V
We have used the equality N k = R where k B is the Boltzmann constant and R is
A
B
the ideal gas constant in the last part of Equation (1.11). Equation (1.11) is the ideal
gas law. Although this relationship is familiar, we have derived it by employing a clas-
sical description of a single molecular collision with the container wall and then scaling
this result up to macroscopic proportions. We see that the origin of the pressure exerted
by a gas on its container is the momentum exchange of the randomly moving gas mole-
cules with the container walls.
What physical association can we make with the temperature T? At the microscopic
level, temperature is related to the mean kinetic energy of molecules as shown by
Equation (1.10). We defer the discussion of temperature at the microscopic level until
Chapter 30 and focus on a macroscopic level discussion here. Although each of us has
a sense of a “temperature scale” based on the qualitative descriptors hot and cold, a
more quantitative and transferable measure of temperature that is not grounded in indi-
vidual experience is needed. The quantitative measurement of temperature is accom-
plished using a thermometer. For any useful thermometer, the measured temperature,
T, must be a single-valued, continuous, and monotonic function of some thermometric
system property such as the volume of mercury confined to a narrow capillary, the elec-
tromotive force generated at the junction of two dissimilar metals, or the electrical
resistance of a platinum wire.