http://www.ck12.org Chapter 5. Electrons in Atoms
that described the wave nature of any particle. He determined that the wavelength (λ) of any moving object is given
by:
λ=
h
mvIn this equation,his Planck’s constant,mis the mass of the particle in kg, andvis the velocity of the particle in m/s.
The problem below shows how to calculate the wavelength of an electron.
Sample Problem 5.4: de Broglie Equation
An electron with a mass of 9.11× 10 −^31 kg is moving at nearly the speed of light. Using a velocity of 3.00× 108
m/s, calculate the wavelength of this electron.
Step 1: List the known quantities and plan the problem.
Known
- mass (m) = 9.11× 10 −^31 kg
- Planck’s constant (h) = 6.626× 10 −^34 J•s
- velocity (v) = 3.00× 108 m/s
Unknown
- wavelength (λ)
Apply the de Broglie wave equationλ=
h
mv
to solve for the wavelength of the moving electron.Step 2: Calculate.
λ=
h
mv=
6. 626 × 10 −^34 J·s
( 9. 11 × 10 −^31 kg)×( 3. 00 × 108 m/s)= 2. 42 × 10 −^12 mStep 3: Think about your result.
This very small wavelength is about 1/20thof the diameter of a hydrogen atom. Looking at the equation, we can see
that the wavelengths of everyday objects will be even smaller because their masses will be much larger.
Practice Problem- Calculate the wavelength of a 0.145 kg baseball thrown at a speed of 40. m/s.
The above practice problem results in an extremely short wavelength on the order of 10−^34 m. This wavelength is
impossible to detect even with advanced scientific equipment. Indeed, while all objects move with wavelike motion,
we never notice it because the wavelengths are far too short. On the other hand, particles that are extremely small,
such as the electron, can have measurable wavelengths. The wave nature of the electron proved to be a key insight
that led to a new way of understanding how the electron functions. An electron that is confined to a particular space
around the nucleus of an atom can only move around that atom in such a way that its electron wave “fits” the size
of the atom correctly (Figure5.12). This means that the frequencies of electron waves are quantized. Based on
theE=hνequation, the fact that only certain quantized frequencies are allowed for a given electron means that
electrons can only exist in an atom at specific energies, as Bohr had previously theorized.