In two separate experiments, photons of energy 116 eV and 20 eV are incident on hydrogen atoms in their ground state. Let ₁ and ₂ be the de Broglie wavelengths of the electrons emitted in the first and second experiments, respectively. If ₂

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In two separate experiments, photons of energy 116 eV and 20 eV are incident on hydrogen atoms in their ground state. Let ₁ and ₂ be the de Broglie wavelengths of the electrons emitted in the first and second experiments, respectively. If ₂ = x ₁ , the value of x is _____.

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The ionization energy required to eject an electron from the ground state of a hydrogen atom is 13.6 eV. Using the photoelectric equation, the kinetic energy of the emitted electron is K = E_ photon - 13.6 eV. For the first experiment ( E₁ = 116 eV): K₁ = 116 - 13.6 = 102.4 eV. For the second experiment ( E₂ = 20 eV): K₂ = 20 - 13.6 = 6.4 eV. The de Broglie wavelength of an electron is inversely proportional to the square root of its kinetic energy: = h 2mK 1 K Therefore, the ratio of the wavelengths is: ₂ ₁ = K₁ K

In two separate experiments, photons of energy 116 eV and 20 eV are incident on hydrogen atoms in their ground state. Let ₁ and ₂ be the de Broglie wavelengths of the electrons emitted in the first and second experiments, respectively. If ₂ = x ₁ , the value of x is _____.

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