Q1 · 2005-2021 · 2 marks
The function e^{ax^2} (\alpha > 0) is not an acceptable wave function for bound system, because
- (a) it is not continuous
- (b) it is multivalued
- (c) it is not normalisable
- (d) all of these
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Answer: (c)
GATE Chemistry (CY) — Atomic Structure previous year questions
25 questions · 2005 · all with verified answer · last checked 2026-11-06
Q2 · 2005-2021 · 2 marks
The 2s orbital of H – atom has radial node at 2a_0 because \psi_{2s} is proportional to
- (a) \left(\frac{1}{2} + \frac{r}{a_0}\right)
- (b) \left(2 + \frac{r}{a_0}\right)
- (c) \left(2 - \frac{r}{a_0}\right)
- (d) \left(2 - \frac{r}{2a_0}\right)
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Answer: (c)
Q3 · 2005-2021 · 2 marks
As per Huckel theory, \pi-electron energy levels of cyclobutadiene are
- (a) \alpha+2 \beta, \alpha+\beta, \alpha-\beta, \alpha-2 \beta
- (b) \alpha+2 \beta, \alpha-\beta, \alpha-\beta, \alpha-2 \beta
- (c) \alpha+2 \beta, \alpha, \alpha, \alpha-2 \beta \alpha+\beta, \alpha-\beta, \alpha-\beta, \alpha-2 \beta
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Answer: (a)
Q4 · 2005-2021 · 2 marks
To demonstrate the variational principle, a trial function \psi = C_1\frac{\psi_{2s}+\psi_{3s}}{\sqrt{2}} + C_2\frac{\psi_{2s}-\psi_{3s}}{\sqrt{2}} where C_1 and C_2 are the variational parameters and \psi_{2s} and \psi_{3s} are the 2s, and 3s orbitals of the hydrogen atom, is constructed. The corresponding secular determinant for the hydrogen atom (in eV) is
- (a) \begin{vmatrix} 3.4(1+4/9)2-E & 3.4(1-4/9)/2 \\ 3.4(1-4/9)/2 & 3.4(1+4/9)2-E \end{vmatrix}
- (b) \begin{vmatrix} 3.4(1+4/9)2-E & 3.4(1-4/9)/2 \\ 3.4(1-4/9)/2 & 3.4(1-4/9)2-E \end{vmatrix}
- (c) \begin{vmatrix} 3.4(1+4/9)2-E & 3.4(1+4/9)/2 \\ 3.4(1-4/9)/2 & 3.4(1-4/9)2-E \end{vmatrix}
- (d) \begin{vmatrix} 3.4(1+4/9)2-E & 0 \\ 0 & 3.4(1+4/9)2-E \end{vmatrix}
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Answer: (a)
Q5 · 2005-2021 · 2 marks
The first excited state wavefunction for a particle in a box that spans from –a to +a is
- (a) \sqrt{\frac{1}{a}}\cos\left(\frac{\pi x}{a}\right)
- (b) \sqrt{\frac{1}{a}}\sin\left(\frac{\pi x}{a}\right)
- (c) \sqrt{\frac{2}{a}}\cos\left(\frac{2\pi x}{a}\right)
- (d) \sqrt{\frac{2}{a}}\sin\left(\frac{2\pi x}{a}\right)
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Answer: (b)
Q6 · 2005-2021 · 2 marks
A perturbation V = \delta(x - a/2) is introduced in the box. The first order energy correction to the first excited state is.
- (a) 0
- (b) 2/a
- (c) 1/a
- (d) 1/2a
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Answer: (c)
Q7 · 2005-2021 · 2 marks
For a homonuclear diatomic molecule, the bonding molecular orbital is
- (a) \sigma_{\mathrm{u}} of lowest energy
- (b) \sigma_{\mathrm{u}} of second lowest energy
- (c) \pi_{\mathrm{g}} of lowest energy
- (d) \pi_{\mathrm{u}} of lowest energy
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Answer: (d)
Q8 · 2005-2021 · 2 marks
The ionisation potential of hydrogen atom is 13.6 eV. The first ionisation potential of a sodium atom, assuming that the energy of its outer electron can be represented by a H – atom like model with an effective nuclear charge of 1.84, is
- (a) 46.0 eV
- (b) 11.5 eV
- (c) 5.1 eV
- (d) 2.9 eV
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Answer: (c)
Q9 · 2005-2021 · 2 marks
The lifetime of a molecule in an excited electronic state is 10^{-10} s. The uncertainty in the energy (eV) approximately is
- (a) 2 \times 10^{-5}
- (b) 3 \times 10^{-6}
- (c) 0
- (d) 10^{-14}
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Answer: (b)
Q10 · 2005-2021 · 2 marks
A particle is confined to a one-dimensional box of length 1 mm. If the length is changed by 10^{-9}\,\mathrm{m}, the % change in the ground state energy is
- (a) 2 \times 10^{-4}
- (b) 2 \times 10^{-7}
- (c) 2 \times 10^{-2}
- (d) 0
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Answer: (a)
Q11 · 2005-2021 · 2 marks
The operation of the commutator [x, d/dx] on a function f(x) is equal to
- (a) 0
- (b) f(x)
- (c) -f(x)
- (d) x\,df/dx
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Answer: (c)
Q12 · 2005-2021 · 2 marks
The de-Broglie wavelength of the electron is \frac{\lambda}{\mathrm{pm}} = \left(\frac{a}{\varphi}\right). The value of 'a' in volts is
- (a) 1.5 \times 10^{-18}
- (b) 1.5 \times 10^{6}
- (c) 6.63 \times 10^{-5}
- (d) 2.5 \times 10^{18}
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Answer: (b)
Q13 · 2005-2021 · 2 marks
The degeneracy of a quantum particle in a cubic box having energy four times that of the lowest energy is
- (a) 3
- (b) 6
- (c) 1
- (d) 4
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Answer: (c)
Q14 · 2005-2021 · 2 marks
The de-Broglie wavelength for a He atom travelling at 1000\ \mathrm{m\,s^{-1}} (typical speed at room temperature) is
- (a) 99.7 \times 10^{-12}\ \mathrm{m}
- (b) 199.4 \times 10^{-12}\ \mathrm{m}
- (c) 199.4 \times 10^{-18}\ \mathrm{m}
- (d) 99 \times 10^{-6}\ \mathrm{m}
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Answer: (a)
Q15 · 2005-2021 · 2 marks
For eigen functions \psi_1 = \sqrt{\frac{1}{b}}\sin\left(\frac{\pi x}{b}\right) and \psi_2 = \sqrt{\frac{2}{b}}\sin\left(\frac{2\pi x}{b}\right) os particle in a 1 – D box of length b\,(0 \leq x \leq b)
- (b) \psi_2 is normalized but not orthogonal to \psi_1
- (a) \psi_1 is normalized but not orthogonal to \psi_2
- (c) \psi_2 is normalized and orthogonal to \psi_1
- (d) \psi_2 is neither normalized nor orthogonal to \psi_1
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Answer: (c)
Q16 · 2005-2021 · 2 marks
An electron of mass ‘m’ is confined to a one dimensional box of length ‘b’. If it makes a radiative transition from second excited state to the ground state, the frequency of the photon emitted is
- (a) \frac{9h}{8mb^{2}}
- (b) \frac{3h}{8mb^{2}}
- (c) \frac{h}{mb^{2}}
- (d) \frac{2h}{mb^{2}}
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Answer: (c)
Q17 · 2005-2021 · 2 marks
The wave function for a quantum mechanical particle in a 1-dimensional box of length 'a' is given by \psi = A\sin\frac{\pi x}{a}. The value of 'A' for a box of length 200 nm is
- (a) 4 \times 10^{4}\ (\mathrm{nm})^{2}
- (b) 10\sqrt{2}\ (\mathrm{nm})^{1/2}
- (c) \sqrt{2}/10\ (\mathrm{nm})^{-1/2}
- (d) 0.1\ (\mathrm{nm})^{-1/2}
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Answer: (d)
Q18 · 2005-2021 · 2 marks
The wavefunction of a 1 – D harmonic oscillator between x=+\infty and x=-\infty is given by \psi(x)=N(2x^2-1)e^{-x^2/2}. The value of N that normalizes the function \psi(x) is: \left(\text{Given }\int_{-\infty}^{+\infty}x^{2n}e^{-x^2}\,dx=\frac{1\cdot3\cdot5\cdots(2n-1)}{2^n}\sqrt{\pi}\right)
- (a) \left(\frac{1}{8\sqrt{\pi}}\right)^{\frac{1}{2}}
- (b) \left(\frac{1}{3\sqrt{\pi}}\right)^{\frac{1}{2}}
- (c) \left(\frac{1}{2\sqrt{\pi}}\right)^{\frac{1}{2}}
- (d) \left(\frac{1}{4\sqrt{\pi}}\right)^{\frac{1}{2}}
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Answer: (c)
Q19 · 2005-2021 · 2 marks
The possible value of E are
- (a) \alpha+\sqrt{2} \beta, \alpha, \alpha-\sqrt{2} \beta
- (b) \alpha+2 \sqrt{2} \beta, \alpha, \alpha-2 \sqrt{2} \beta
- (c) \alpha+\beta, \alpha, \alpha-\beta
- (d) \alpha+2 \beta, \alpha, \alpha-2 \beta
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Answer: (a)
Q20 · 2005-2021 · 2 marks
e^{-2x^2} is an eigen function on the operator \left(\frac{d^2}{dx^2} - 16x^2\right). The corresponding eigen value is
- (a) +4
- (b) −4
- (c) +2
- (d) −2
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Answer: (b)
Q21 · 2005-2021 · 2 marks
The value of the magnetic quantum number of a p_x orbital is
- (a) -1
- (b) 0
- (c) +1
- (d) undefined
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Answer: (d)
Q22 · 2005-2021 · 2 marks
When the operator, -\hbar^2 d^2/dx^2, operates on the function e^{-ikx}, the result is
- (a) k^2\hbar^2 e^{-ikx}
- (b) ik^2\hbar^2 e^{-ikx}
- (c) i\hbar^2 e^{-ikx}
- (d) \hbar^2 e^{-ikx}
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Answer: (a)
Q23 · 2005-2021 · 2 marks
The experimental ionization energies of hydrogen and helium atoms in their ground states are, respectively, 13.6 eV and 24.6 eV. The ground state energy of helium atom, in eV, is
- (a) -\frac{1}{2}(13.6) - 24.6
- (b) -4(13.6) - 24.6
- (c) -\frac{1}{4}(13.6) - 24.6
- (d) -2(13.6) - 24.6
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Answer: (b)
Q24 · 2005-2021 · 2 marks
For ortho-hydrogen, the nuclear wavefunction and the rotational quantum number, respectively, are
- (a) antisymmetric and even
- (b) symmetric and odd
- (c) symmetric and even
- (d) antisymmetric and odd
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Answer: (b)
Q25 · 2005-2021 · 2 marks
Among the following linear combination of atomic orbitals, the CORRECT representation of the lowest unoccupied \pi-molecular orbital of butadiene is:
- (a) \Psi = 0.602\phi_1 - 0.372\phi_2 - 0.372\phi_3 + 0.602\phi_4
- (b) \Psi = -0.372\phi_1 + 0.602\phi_2 - 0.602\phi_3 + 0.372\phi_4
- (c) \Psi = 0.602\phi_1 + 0.372\phi_2 - 0.372\phi_3 - 0.602\phi_4
- (d) \Psi = 0.372\phi_1 + 0.602\phi_2 + 0.602\phi_3 + 0.372\phi_4
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Answer: (a)
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