Case: Read the passage and answer the question.
The energy of an atom is the least (largest negative value) when its electron is revolving in the orbit closest to the nucleus, i.e., the one for which n = 1. For n = 2, 3, ... the absolute value of the energy E is smaller; hence the energy is progressively larger in the outer orbits. The lowest state of the atom, called the ground state, is that of the lowest energy, with the electron revolving in the orbit of smallest radius, the Bohr radius \( a_0 \). The energy of this state (n = 1), \( E_1 \), is \( -13.6\,\mathrm{eV} \). Therefore, the minimum energy required to free the electron from the ground state of the hydrogen atom is 13.6 eV. It is called the ionisation energy of the hydrogen atom. This prediction of Bohr’s model is in excellent agreement with the experimental value of the ionisation energy.
At room temperature, most of the hydrogen atoms are in the ground state. When a hydrogen atom receives energy by processes such as electron collisions, the atom may acquire sufficient energy to raise the electron to higher energy states. The atom is then said to be in an excited state. For n = 2, the energy \( E_2 \) is \( -3.40\,\mathrm{eV} \). It means that the energy required to excite an electron in a hydrogen atom to its first excited state is \( E_2-E_1=-3.40\,\mathrm{eV}-(-13.6\,\mathrm{eV})=10.2\,\mathrm{eV} \). Similarly, \( E_3=-1.51\,\mathrm{eV} \) and \( E_3-E_1=12.09\,\mathrm{eV} \); that is, to excite the hydrogen atom from its ground state (n = 1) to the second excited state (n = 3), 12.09 eV of energy is required, and so on. From these excited states the electron can fall back to a state of lower energy, emitting a photon in the process. Thus, as the excitation of the hydrogen atom increases (that is, as n increases), the minimum energy required to free the electron from the excited atom decreases.
A hydrogen atom in its ground state is struck by an electron of kinetic energy 12.5 eV. The highest energy state to which the atom can be raised in this collision is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → the second excited state (n = 3)
In a collision, the incoming electron can give the atom any part of its kinetic energy. But the atom takes up only an amount that lifts it to one of its allowed levels. So the atom reaches the highest level whose excitation energy does not exceed 12.5 eV. The passage gives 12.09 eV for n = 3. For n = 4, with \( E_n=-13.6\,\mathrm{eV}/n^{2} \):
\( \begin{aligned} E_4 &= \frac{-13.6\,\mathrm{eV}}{16} \\ &= -0.85\,\mathrm{eV} \end{aligned} \)
\( \begin{aligned} E_4-E_1 &= -0.85\,\mathrm{eV}-(-13.6\,\mathrm{eV}) \\ &= 12.75\,\mathrm{eV} \end{aligned} \)
\( 12.09\,\mathrm{eV}\lt12.5\,\mathrm{eV}\lt12.75\,\mathrm{eV} \)
From the above, the electron has enough energy to lift the atom to n = 3, but not to n = 4. Hence, the atom reaches the second excited state.