5 Steps to a 5 AP Chemistry

Answer:

Using the equations

ΔE=hn and c=nl

we get

n=ΔE/h and l=c/n

Inserting the appropriate values:

n=ΔE/h=7.83 × 10 −^19 J/6.63 × 10 −^34 Js =1.18 × 1015 s−^1

Then:

l=c/n=(3.0 × 108 m/s)/(1.18 × 1015 s−^1 ) =2.5 × 10 −^7 m

This answer could have been calculated more quickly by combining the original two

equations to give:

l=hc/ΔE

Wave Properties of Matter

The concept that matter possesses both particle and wave properties was first postulated by de

Broglie in 1925. He introduced the equation l=h/mv, which indicates a mass (m) moving

with a certain velocity (v) would have a specific wavelength (l) associated with it. (Note that

this vis the velocity not νthe frequency.) If the mass is very large (a locomotive), the associated

wavelength is insignificant. However, if the mass is very small (an electron), the wavelength is

measurable. The denominator may be replaced with the momentum of the particle (p=mv).

Atomic Spectra

Late in the 19th century scientists discovered that when the vapor of an element was heated

it gave off a line spectrum,a series of fine lines of colors instead of a continuous spectrum

like a rainbow. This was used in the developing quantum mechanical model as evidence that

the energy of the electrons in an atom was quantized,that is, there could only be certain dis-

tinct energies (lines) associated with the atom. Niels Bohr developed the first modern atomic

model for hydrogen using the concepts of quantized energies. The Bohr model postulated a

ground statefor the electrons in the atom, an energy state of lowest energy, and an excited

state,an energy state of higher energy. In order for an electron to go from its ground state

to an excited state, it must absorb a certain amount of energy. If the electron dropped back

from that excited state to its ground state, that same amount of energy would be emitted.

Bohr’s model also allowed scientists to develop a method of calculating the energy associated

with a particular energy level for the electron in the hydrogen atom:

where nis the energy state. This equation can then be modified to calculate the energy dif-

ference between any two energy levels:

ΔE

nn

=− ×

⎛

⎝

⎜⎜

⎞

⎠

218 10.J−^181122 ⎟⎟

final initial

−

E

n n

=

−×218 10. −^18

2 joule

Spectroscopy, Light, and Electrons  139