Physical Chemistry Third Edition

640 14 Classical Mechanics and the Old Quantum Theory

What would you have to do to produce the correct

pitch at a different temperature?

b.In a musical instrument such as a clarinet with an

open-ended “pipe,” the fundamental wave has a

wavelength approximately equal to four times the

length of the pipe. Find the length of the pipe for A

above middle C at 25◦C.

c.The overtones for the instruments of parts a and b

have different patterns, since each overtone wave for

an open pipe has a node at the closed end of a pipe

and a maximum at the open end. Find the frequency

at 25◦C of the first overtone for an organ pipe and for

a clarinet for A above middle C.

14.12The speed of sound in sea water at 25◦Cis1531ms−^1.

If a SONAR system on a submarine uses a frequency of

500 s−^1 find the wavelength of the sound waves.

14.13Assume that the fundamental and the first overtone are

simultaneously excited in a flexible string such that

z(y,t)A 1 sin(πy/L) sin(πct/L)

+A 2 sin(2πy/L) sin(2πct/L)

where

A 1  4 A 2

Construct a graph representing the shape of the string at

timet0, at timetL/(4c), and at timet 3 L/(4c).

Comment on the differences in the shapes.

14.14Show that the relationshipνc/λfor a traveling wave

also holds for the standing wave in Eq. (14.3-39).

14.15a.A violin string has a mass per unit length of

3.0gm−^1. Find the tension force necessary to

produce a fundamental frequency of 440 s−^1 if the

length of the string is 50.0 cm.

b.If the string were infinitely long with the same tension

force, what would be the speed of a traveling wave of

wavelength 1.00 m?

14.16For liquid water at 25◦C, the permittivity depends on

frequency, and for fairly low frequency is equal to

6.954× 10 −^10 C^2 N−^1 m−^2. The refractive index is

defined as the ratio:

refractive indexncvacuum/cmedium

The refractive index of water is equal to 1.33 for visible

light. Find the speed of light in water and the permittivity

for frequencies corresponding to visible light. To four

significant digits, the permeability is the same as that of a

vacuum for these frequencies.

14.17The wavelength of yellow light given off in air by

sodium in flame tests is 589.2 nm.

a.Find the frequency of this light. The index of

refraction of air is 1.00027.

b.Find the wavelength of this light in carbon disulfide at

20 ◦C. The index of refraction of carbon disulfide at

20 ◦C is equal to 1.62546.

14.4 The Old Quantum Theory

The Atomic Nature of Matter

All of ordinary matter is made up of atoms. Although the idea of atoms was introduced

by the Greek philosopher Democritus, it became part of chemistry only after 1803,

when Dalton proposed his atomic theory. This theory asserts that each element has its

own characteristic type of atoms, and that atoms combine as units to produce com-

pounds. Dalton’s theory did not include any information about the structure of atoms

and how they could bind to other atoms to make molecules. Dalton even proposed that

atoms might have little hooks that could snag similar hooks on other molecules.

John Dalton, 1766–1844, was an
English schoolmaster and chemist.
After becoming famous, he continued to
teach young pupils elementary subjects.

Joseph John Thomson, 1856–1940,
was a British physicist who received the
1906 Nobel Prize in physics for his
discovery of the electron and his
measurement of its charge-to-mass
ratio.

In the 1870s Thomson showed that the “cathode rays” emitted by negatively charged

metals in a vacuum consisted of negatively charged particles, now known as electrons.

Thomson pictured an atom as containing stationary electrons imbedded in a positive

matrix, like raisins in an English plum pudding. However, Rutherford discovered in

1911 that an atom contains a very small positive nucleus, so that the electrons must

orbit the nucleus. The charge on the electron was measured by Millikan in a series of

experiments carried out between 1908 and 1917. Moseley discovered the concept of