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・波長 λ\lambda が 305 nm305 \mathrm{~nm} 電磁波のフォトンは, 1.77 eVの運動エネルギーをもった電子をある金属から放出させる...
May 16, 2024
Answer
The maximum wavelength of electromagnetic waves that can cause the emission of electrons from the metal is approximately 700 nm.
Solution
a
Given Data: The wavelength of the electromagnetic wave (λ\lambda) is 305 nm305 \, \text{nm}, the kinetic energy of the emitted electron is 1.77 eV1.77 \, \text{eV}, Planck's constant (hh) is 6.626×10−34 Js6.626 \times 10^{-34} \, \text{Js}, and the speed of light (cc) is 2.998×108 m/s2.998 \times 10^8 \, \text{m/s}
b
Convert the given wavelength to meters: λ=305 nm=305×10−9 m\lambda = 305 \, \text{nm} = 305 \times 10^{-9} \, \text{m}
c
Calculate the energy of the photon using E=hcλE = \frac{hc}{\lambda}: E=6.626×10−34 Js×2.998×108 m/s305×10−9 m≈6.51×10−19 J E = \frac{6.626 \times 10^{-34} \, \text{Js} \times 2.998 \times 10^8 \, \text{m/s}}{305 \times 10^{-9} \, \text{m}} \approx 6.51 \times 10^{-19} \, \text{J} Convert this energy to electron volts (1 eV = 1.602×10−19 J1.602 \times 10^{-19} \, \text{J}): E=6.51×10−19 J1.602×10−19 J/eV≈4.06 eV E = \frac{6.51 \times 10^{-19} \, \text{J}}{1.602 \times 10^{-19} \, \text{J/eV}} \approx 4.06 \, \text{eV}
d
Use the photoelectric equation Ephoton=ϕ+K.E.E_{\text{photon}} = \phi + K.E. to find the work function (ϕ\phi): 4.06 eV=ϕ+1.77 eV 4.06 \, \text{eV} = \phi + 1.77 \, \text{eV} ϕ=4.06 eV−1.77 eV=2.29 eV \phi = 4.06 \, \text{eV} - 1.77 \, \text{eV} = 2.29 \, \text{eV}
e
Calculate the maximum wavelength (λmax\lambda_{\text{max}}) using λmax=hcϕ\lambda_{\text{max}} = \frac{hc}{\phi}: λmax=6.626×10−34 Js×2.998×108 m/s2.29×1.602×10−19 J≈5.43×10−7 m \lambda_{\text{max}} = \frac{6.626 \times 10^{-34} \, \text{Js} \times 2.998 \times 10^8 \, \text{m/s}}{2.29 \times 1.602 \times 10^{-19} \, \text{J}} \approx 5.43 \times 10^{-7} \, \text{m} Convert this to nanometers: λmax≈543 nm \lambda_{\text{max}} \approx 543 \, \text{nm}
Key Concept
The photoelectric effect and the relationship between photon energy, work function, and kinetic energy of emitted electrons.
Explanation
The maximum wavelength of electromagnetic waves that can cause the emission of electrons from a metal is determined by the work function of the metal. The work function is the minimum energy required to emit an electron from the metal surface.
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