Physical Chemistry Third Edition

9.2 A Model System to Represent a Dilute Gas 389

Exercise 9.3

Assume that a particle of massmmoves in thezdirection and is subject to a constant gravitational

force given byFz−mg. If the initial position of the particle isz(0)0 and its initial velocity

isvz(0)0, solve its equation of motion, findingvzandzas functions of time.

Newton’s third lawis called thelaw of action and reaction. It asserts that if one object

exerts a force on a second object, the second object exerts a force on the first that is

equal in magnitude and opposite in direction to the first force. IfF 21 is the force exerted

on the second object and ifF 12 is the force exerted on the first object, then

F 12 −F 21 (Newton’s third law) (9.2-5)

In order to jump into the air from a floor, you push on the floor with your feet, and the

floor throws you into the air because of Newton’s third law.

Potential Energy

If the force on a particle depends only on its position, the force can be derived from a

potential energy. For a force in thezdirection,

Fz−

∂V

∂z

(9.2-6)

where the potential energy is denoted byV. This derivative is a partial derivative, which

means thatxandyare treated as constants in the differentiation. Similar equations apply

for thexandydirections. If the force on a particle depends only on its position, the total

energy of the particle is the sum of its kinetic energyKand its potential energyV.If

the particle moves only in thezdirection,V depends only onz:

EK +V 

1

2

mv^2 z+V(z) (9.2-7)

Thelaw of conservation of energyasserts that the energy of a particle subject only

to forces from a potential energy is constant. We say that the energy isconservedand

that the energy is aconstant of the motion.

Exercise 9.4

a.Show that the solution to Exercise 9.3 conforms to the conservation of energy. The gravita-

tional potential energyVcorresponding toFz−mgis

Vmgz+constant

where the constant can take on any value without any physical effect.

b.Explain why the conservation of kinetic plus potential energy does not apply to the solution

of Example 9.1. Because the conservation of total energy is an accepted principle of science,

the decrease in kinetic energy in this example must correspond to an increase of some other

form of energy. What form is this?