Page 59 - Physical Chemistry

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Chapter 2 where K is the change in kinetic energy. The work–energy theorem (2.16) states that
The First Law of Thermodynamics
the work done on the particle by the force acting on it equals the change in kinetic
energy of the particle. It is valid because we defined kinetic energy in such a manner
as to make it valid.
Besides kinetic energy, there is another kind of energy in classical mechanics.
Suppose we throw a body up into the air. As it rises, its kinetic energy decreases,
reaching zero at the high point. What happens to the kinetic energy the body loses as
it rises? It proves convenient to introduce the notion of a field (in this case, a gravita-
tional field) and to say that the decrease in kinetic energy of the body is accompanied
by a corresponding increase in the potential energy of the field. Likewise, as the body
falls back to earth, it gains kinetic energy and the gravitational field loses a cor-
responding amount of potential energy. Usually, we don’t refer explicitly to the field
but simply ascribe a certain amount of potential energy to the body itself, the amount
depending on the location of the body in the field.
To put the concept of potential energy on a quantitative basis, we proceed as fol-
lows. Let the forces acting on the particle depend only on the particle’s position and
not on its velocity, or the time, or any other variable. Such a force F with F
x
F (x, y, z), F F (x, y, z), F F (x, y, z)is called a conservative force, for a reason
x y y z z
to be seen shortly. Examples of conservative forces are gravitational forces, electrical
forces, and the Hooke’s law force of a spring. Some nonconservative forces are air
resistance, friction, and the force you exert when you kick a football. For a conserva-
tive force, we define the potential energy V(x, y, z)asa function of x, y, and z whose
partial derivatives satisfy
0V 0V 0V
F , F , F z (2.17)
x
y
0x 0y 0z
Since only the partial derivatives of V are defined, V itself has an arbitrary additive
constant. We can set the zero level of potential energy anywhere we please.
From (2.13) and (2.17), it follows that
2 0V 2 0V 2 0V
w dx dy dz (2.18)
1 0x 1 0y 1 0z
Since dV ( V/ x) dx ( V/ y) dy ( V/ z) dz [Eq. (1.30)], we have
2
w dV 1V V 2 V V 2 (2.19)
2
1
1
1
But Eq. (2.16) gives w K K . Hence K K V V , or
2 1 2 1 1 2
K V K V 2 (2.20)
2
1
1
When only conservative forces act, the sum of the particle’s kinetic energy and poten-
tial energy remains constant during the motion. This is the law of conservation of
mechanical energy. Using E for the total mechanical energy, we have
mech
E mech K V (2.21)
If only conservative forces act, E mech remains constant.
What is the potential energy of an object in the earth’s gravitational field? Let the
x axis point outward from the earth with the origin at the earth’s surface. We have
F mg, F F 0. Equation (2.17) gives V/ x mg, V/ y 0 V/ z.
x
y
z
Integration gives V mgx C, where C is a constant. (In doing the integration, we
assumed the object’s distance above the earth’s surface was small enough for g to be
considered constant.) Choosing the arbitrary constant as zero, we get
V mgh (2.22)