25 practice questions on quantum-numbers-and-orbitals from the Structure of Atom section of the UPSC Prelims syllabus. 25 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.
How many electrons can be accommodated in the 4p subshell?
Correct answer: A - 6
The 4p subshell has 3 orbitals (m_l = -1, 0, +1), each holding 2 electrons, totaling 6. Options 2, 3, and 4 assume incorrect number of orbitals or electron capacity.
Which statement correctly describes the shape of a d orbital?
Correct answer: A - D orbitals have a cloverleaf shape.
D orbitals have a cloverleaf shape. S orbitals are spherical, p orbitals are dumbbell-shaped, and 'double-sphere' is not a standard shape.
What is the shape of a p orbital and how many orientations does it have?
Correct answer: C - Dumbbell-shaped, 3 orientations
The correct option is (b). P orbitals are dumbbell-shaped and have three orientations corresponding to px, py, and pz. Option (a) describes s orbitals. Options (c) and (d) provide incorrect shapes and numbers of orientations.
A $4s$ orbital and one other orbital of the same atom have the same total number of nodes. Which is the other orbital?
Correct answer: D - $4d$
The total number of nodes in an orbital is $n - 1$, whatever the value of $l$. A $4s$ orbital therefore has $3$ nodes, and so does every other orbital of the fourth shell, including $4d$, which has two angular nodes and one radial node. Both $3d$ and $3p$ have $n - 1 = 2$ nodes and $5s$ has $4$, so only the orbital from the same shell matches.
According to the $(n + l)$ rule, which of the following subshells has the highest energy in a multi-electron atom?
Correct answer: C - $5s$
By the $(n + l)$ rule the subshell with the larger value of $n + l$ has the higher energy, and when two subshells share the same sum the one with the larger $n$ lies higher. Here $4s$ gives $4$, while $3d$, $4p$ and $5s$ all give $5$. Among those three $5s$ has the largest $n$, so it is highest, and the filling order runs $4s$, then $3d$, then $4p$, then $5s$.
A chemist observes that the ionization energy decreases as she moves down a group in the periodic table. Which quantum number mainly contributes to this trend?
Correct answer: D - Principal quantum number (n)
The correct option is (a). The principal quantum number (n) increases down a group, leading to larger atomic radii and thus lower ionization energy. Options (b), (c), and (d) do not directly affect the ionization energy trend across groups.
Which of the following orbitals is spherically symmetrical about the nucleus?
Correct answer: B - $2s$
Every $s$ orbital, including $2s$, is spherically symmetrical, so the probability of finding the electron depends only on the distance from the nucleus and not on the direction. The $p$ orbitals such as $2p_x$ and $3p_z$ are dumb-bell shaped with two lobes along one axis, and $3d_{xy}$ has four lobes lying between the $x$ and $y$ axes. None of those three is spherical.
Consider these statements about the quantum numbers of an electron in an atom. 1. For a given $n$, the azimuthal quantum number $l$ may take values from $0$ to $n - 1$. 2. For a given $l$, the magnetic quantum number $m_l$ may take $2l + 1$ values. 3. The spin quantum number $m_s$ depends on the value of $l$. Which are correct?
Correct answer: C - Statements 1 and 2 only
Within a shell of principal quantum number $n$, the azimuthal quantum number runs from $0$ to $n - 1$, so statement 1 is right, and each value of $l$ gives $2l + 1$ possible orientations, so statement 2 is right as well. Statement 3 is wrong: the spin quantum number takes only $+\frac{1}{2}$ or $-\frac{1}{2}$ whatever the value of $l$, because spin is an intrinsic property of the electron.
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