Unit 15: Modern Physics
Weightage : 15–18%
A hypothetical one-electron atom in its highest excited state can only emit photons of energy \(2E\), \(3E\), and \(5E\). Which of the following is a possible energy-level diagram for the atom?
Correct Answer: \( \boxed{\mathrm{A}} \)
The energy of an emitted photon equals the difference between two energy levels:
\( E_{\gamma}=\Delta E \)
Since the atom can emit only photons of energies \(2E\), \(3E\), and \(5E\), the energy-level diagram must produce exactly these three energy differences.
For diagram (A), the levels are \(0\), \(3E\), and \(5E\):
\(5E-3E=2E\)
\(3E-0=3E\)
\(5E-0=5E\)
These are exactly the three photon energies given in the problem. The other diagrams produce additional energy differences or fail to produce all three required values.
Therefore, the correct answer is (A).
A hypothetical hydrogen-like atom has energy states as represented in the energy-level diagram above. The atom is in the ground state when it absorbs a photon with frequency \(2.18\times10^{15}\,\mathrm{Hz}\). What energy state will the atom be in after it absorbs the photon?
Correct Answer: \( \boxed{\mathrm{C}} \)
The energy of the absorbed photon is
\(E=hf\)
Substituting the given values,
\(E=(6.63\times10^{-34}\,\mathrm{J\,s})(2.18\times10^{15}\,\mathrm{Hz})=1.45\times10^{-18}\,\mathrm{J}\)
Converting to electron volts,
\(E=\dfrac{1.45\times10^{-18}}{1.60\times10^{-19}}\approx9.0\,\mathrm{eV}\)
The atom initially has energy \( -10\,\mathrm{eV} \) (ground state). After absorbing the photon, its energy becomes
\(-10\,\mathrm{eV}+9\,\mathrm{eV}=-1\,\mathrm{eV}\)
From the energy-level diagram, the energy level of \( -1.0\,\mathrm{eV} \) corresponds to
\(n=3\)
Since the absorbed photon exactly matches the energy difference between the ground state and the \(n=3\) level, the electron is excited to that state.
Therefore, the correct answer is (C).
An atom has its lowest four energy levels at \(-10\,\mathrm{eV}\), \(-5\,\mathrm{eV}\), \(-3.5\,\mathrm{eV}\), and \(-2\,\mathrm{eV}\). Which of the following photons could not be absorbed by the atom?
Correct Answer: \( \boxed{\mathrm{C}} \)
An atom absorbs a photon only if the photon’s energy exactly matches the energy difference between two allowed energy levels or the ionization energy.
Possible energy differences are:
\(-10\rightarrow-5:\quad 5\,\mathrm{eV}\)
\(-5\rightarrow0:\quad 5\,\mathrm{eV}\) (ionization)
\(-10\rightarrow0:\quad 10\,\mathrm{eV}\) (ionization)
\(-3.5\rightarrow-2:\quad 1.5\,\mathrm{eV}\)
However, there is no pair of energy levels whose difference is \(2.5\,\mathrm{eV}\).
Therefore, a \(2.5\,\mathrm{eV}\) photon cannot be absorbed by the atom.
Therefore, the correct answer is (C).
