Final answer:
The de Broglie wavelength describes the wave nature of particles in quantum mechanics. The wavelength can be calculated using the equation λ = h / p. To find the de Broglie wavelength associated with the material photoelectric work function, use the equation λ = hc / (E - φ) and substitute the given values.
Step-by-step explanation:
The de Broglie wavelength is a concept in quantum mechanics that describes the wave nature of particles. According to Louis de Broglie's hypothesis, any object with mass and momentum, including particles like electrons, can exhibit wave-like properties. The de Broglie wavelength (λ) of a particle is given by the equation:
λ = h / p
Where:
λ = de Broglie wavelength
h = Planck's constant (6.626 x 10^-34 Js)
p = momentum of the particle
To obtain the expression for the de Broglie wavelength of the wave associated with the material photoelectric work function, we can use the equation:
λ = hc / (E - φ)
Where:
λ = de Broglie wavelength
h = Planck's constant
c = speed of light
E = energy of the photon
φ = work function of the material
In this case, the work function of the metal is given as 4.2 eV. The threshold wavelength, or the maximum wavelength that a photon can have to eject a photoelectron from the metal surface, can be determined by substituting the values into the equation and solving for λ:
λ = (6.626 x 10^-34 Js * 2.998 x 10^8 m/s) / (4.2 eV - 0)
Calculating the above expression gives the value of λ, which is the threshold wavelength.