96. The energy of an electron in the ground state of hydrogen is 13.6 eV. The energy required to excite it to the first excited state is ________.
Energy Required to Excite an Electron from the Ground State to the First Excited State of Hydrogen
Understanding Energy Levels in the Hydrogen Atom
According to Bohr’s atomic model, the energy of an electron in the nth orbit is given by
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En = −13.6/n² eV
where:
n = Principal quantum number
13.6 eV = Ionization energy of the hydrogen atom from the ground state
The negative sign indicates that the electron is bound to the nucleus. An electron must absorb energy to move to a higher energy level or to escape completely from the atom.
Step 1: Energy of the Ground State
For the ground state,
n = 1
Therefore,
E₁ = −13.6 eV
Step 2: Energy of the First Excited State
The first excited state corresponds to
n = 2
Substituting into the energy equation,
E₂ = −13.6/2²
E₂ = −13.6/4
E₂ = −3.4 eV
Step 3: Calculate the Excitation Energy
The energy required is the difference between the energies of the two levels.
ΔE = E₂ − E₁
ΔE = (−3.4) − (−13.6)
ΔE = −3.4 + 13.6
ΔE = 10.2 eV
Thus, the electron must absorb 10.2 eV of energy to move from the ground state to the first excited state.
Why Is the Energy Positive?
Although the energy levels themselves are negative, the excitation energy is always positive because the electron must absorb energy from an external source to move to a higher orbit. During excitation, the electron becomes less tightly bound to the nucleus, so its energy increases from −13.6 eV to −3.4 eV.
The positive value of 10.2 eV represents the amount of energy supplied to the atom during this transition.
Difference Between Excitation Energy and Ionization Energy
Students often confuse excitation energy with ionization energy. Excitation energy is the energy required to move an electron from one bound state to another bound state. In contrast, ionization energy is the minimum energy required to completely remove the electron from the atom, taking it from the ground state to infinity.
For a hydrogen atom, the ionization energy is 13.6 eV, whereas the excitation energy from the ground state to the first excited state is only 10.2 eV because the electron still remains bound to the nucleus after excitation.
Detailed Concept
The energy levels in a hydrogen atom become closer together as the principal quantum number increases. Therefore, larger amounts of energy are required for transitions involving lower energy levels, while transitions between higher energy levels require comparatively smaller amounts of energy.
The transition from the ground state (n = 1) to the first excited state (n = 2) is one of the most important electronic transitions in atomic physics and plays a significant role in explaining the spectral lines of hydrogen.
Final Answer
Energy required to excite the electron to the first excited state = 10.2 eV
