Physical Chemistry Third Edition

652 14 Classical Mechanics and the Old Quantum Theory

Summary of the Chapter

According to classical mechanics, the state of a single particle is specified by its position

and velocity. Newton’s three laws are the basis of classical mechanics, and determine

how the position and velocity of a particle depend on time. If the position and velocity of

all particles of a system are specified for some initial time, the positions and velocities

are determined for all times.

Newton’s second law,Fma, provides an equation of motion for a system that

obeys classical mechanics. The solution of the classical equation of motion for the

harmonic oscillator provides formulas for the position and velocity that correspond to

uniform harmonic motion. The solution of the classical equation of motion for a flexible

string prescribes the position and velocity of each point of the string as a function of

time. These solutions are deterministic, which means that if the initial conditions are

precisely specified, the motion is determined for all times.

The “old quantum theory” consists of theories with arbitrary assumptions of quan-

tization that were devised to explain phenomena that classical physics could not explain.

The old quantum theory includes the black-body radiation theory of Planck, the

photoelectric effect theory of Einstein, and the hydrogen atom theory of Bohr.

ADDITIONAL PROBLEMS

14.34The classical equation of motion of a particle of massm
falling in a vacuum near the surface of the earth is

m

d^2 z

dt^2

−mg

wheremis the mass of the particle andgis the

acceleration due to gravity, equal to 9.80ms−^2.

a.Solve this equation to obtain a general solution.

How does your solution depend onm?

b. Obtain the solution for the case thatz0att 0

andvz0att0.

c.Is this a conservative system? That is, is the total

energy a constant of the motion?

14.35Identify each statement as either true or false. If a
statement is true only under special circumstances, label
it as false.
a.A classical equation of motion gives the positions and
velocities of all particles of a system for all times if the
initial positions and velocities are specified.
b.The energy of a harmonic oscillator is quantized
according to classical mechanics.

c.The solution to the equation of motion of a harmonic

oscillator can be a linear combination of

solutions.

d.The Bohr theory of the hydrogen atom is a hybrid

theory, maintaining elements of classical mechanics

along with quantization.

e.Planck’s constant appears in all three of the theories of

the old quantum theory.

14.36Consider the earth and the sun to be described by the Bohr

theory of the hydrogen atom, with the centripetal force

provided by gravity instead of electrostatic attraction. The

mass of the earth is 5. 983 × 1024 kg, and the mass of the

sun is larger by a factor of 332,958. The gravitational

constant,G, is equal to 6. 673 × 10 −^11 m^3 kg−^1 s−^2.

a.Find the value of the Bohr radius.

b.The mean radius of the earth’s orbit is

1. 495 × 108 km. Find the value of the quantum

numbernfor the earth’s orbit.

c.Find the value of the angular momentum for this

orbit.