A beam of light with a wavelength of 500 nm is incident on a metal surface. The work function of the metal is 2.5 eV.
a) Calculate the energy of one photon of light with this wavelength.
b) Determine the maximum kinetic energy of an ejected electron if the intensity of the incident light is doubled.
c) If the intensity of the incident light remains constant, how would the number of emitted electrons change when the frequency of the incident light is tripled?
Answer:
a) The energy of one photon can be determined using the equation:
E = hf,
where E is the energy, h is Planck's constant (6.63 × 10^-34 J·s), and f is the frequency of the light.
The wavelength of the light is given as 500 nm. To find the frequency, we can apply the equation:
c = λf,
where c is the speed of light (3.00 × 10^8 m/s) and λ is the wavelength. Rearranging the equation, we have:
f = c/λ.
Substituting the given values:
f = (3.00 × 10^8 m/s) / (500 × 10^-9 m) f = 6.00 × 10^14 s^-1.
Now, we can calculate the energy:
E = (6.63 × 10^-34 J·s) × (6.00 × 10^14 s^-1) E = 3.98 × 10^-19 J.
Therefore, the energy of one photon with a wavelength of 500 nm is approximately 3.98 × 10^-19 J.
b) The maximum kinetic energy of an ejected electron can be determined using the equation:
KE_max = hf - φ,
where KE_max is the maximum kinetic energy, hf is the energy of one photon, and φ is the work function of the metal.
From part a, we already found the energy of one photon (3.98 × 10^-19 J). The work function is given as 2.5 eV, which can be converted into joules by multiplying by the conversion factor 1 eV = 1.6 × 10^-19 J:
φ = (2.5 eV) × (1.6 × 10^-19 J/eV) φ = 4.0 × 10^-19 J.
Substituting the values into the equation:
KE_max = (3.98 × 10^-19 J) - (4.0 × 10^-19 J) KE_max = -2.0 × 10^-21 J.
The maximum kinetic energy of the ejected electron is approximately -2.0 × 10^-21 J.
Note: The negative sign indicates that the energy is insufficient to eject any electrons.
c) The number of emitted electrons is determined by the intensity (or number of photons) and not the frequency of the incident light. Therefore, if the intensity remains constant, tripling the frequency of the incident light will not change the number of emitted electrons.