複選題

28. The energy of a particle in a 3D cubic box is given by the formula:


\(E_{n_x, n_y, n_z} = \frac{h^2}{8mL^2} \left(n_x^2 + n_y^2 + n_z^2\right)\)


where h is Planck's constant, m is the mass of the particle, L is the dimensions of the box, and \(n_x\), \(n_y\), and \(n_z\) are the quantum numbers. Which of the following statements are correct?
(A) The degeneracy of the ground state is one
(B) The degeneracy of the first excited state is three
(C) The energy of the first electronic absorption band is \(\frac{3h^2}{8mL^2}\)
(D) The degeneracy of the excited state resulting the broad emission band
(E) Without a zero-point energy

1. Which of the following is NOT correct description about the Maxwell relation in thermodynamics? (A) $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ (B) $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$ (C) $-\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$ (D) $-\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P$

2. At $298.15 \mathrm{~K}$, the reaction $\mathrm{SOCl}_{3(1)} + \mathrm{H}_{2} \mathrm{O}_{(1)} \rightarrow \mathrm{SO}_{2(g)} + 2 \mathrm{HCl}_{(g)}$ is endothermic and the equilibrium lies well on the right-hand side. Which of the following statement is correct? (A) $\Delta S_{\mathrm{rxn}} < 0$ at $298.15 \mathrm{~K}$ (B) $\Delta G_{\mathrm{rxn}} > 0$ at $298.15 \mathrm{~K}$ (C) $\Delta H_{\mathrm{rxn}} < 0$ at $298.15 \mathrm{~K}$ (D) $\Delta G_{\mathrm{rxn}}$ becomes more negative when the temperature is raised.

3. The reaction $\mathrm{A} \rightarrow 2\mathrm{B}$ is spontaneous when (A) $\mu_{\mathrm{A}} > 2\mu_{\mathrm{B}}$ (B) $\mu_{\mathrm{A}} = 2\mu_{\mathrm{B}}$ (C) $\mu_{\mathrm{A}} > \mu_{\mathrm{B}}$ (D) $\mu_{\mathrm{A}} < \mu_{\mathrm{B}}$ ($\mu$ is the chemical potential)

4. Einstein explained a series of experimental observations through his photoelectric effect theory. When light irradiates a metal surface, electrons are emitted. Classical physics could not account for why electron emission depends on the frequency of the incident light rather than its intensity. Which of the following statement correctly reflects Einstein's explanation? (A) Atoms in the cavity walls can emit or absorb energy only in discrete quanta. (B) Angular momentum is quantized. (C) The energy of light is not distributed uniformly throughout the wave; instead, it is carried in localized packets (photons). (D) The uncertainty principle applies.

5. Einstein's A and B coefficients describe how light interacts with atoms having discrete energy levels and explain how thermodynamic equilibrium between radiation and matter is established. Which of the following statement is correct? (A) During excitation, the probability that an atom absorbs a photon is not proportional to the photon density. (B) The corresponding de-excitation process is also not proportional to the photon density. (C) An electron must be able to spontaneously emit a photon at a rate that depends on the photon density. (D) The rate constants for excitation and stimulated de-excitation are the same.

6. Which set of the operators or physical quantities do NOT satisfy the Heisenberg uncertainty principle? (A) Angular momentum operators $\hat{\Gamma}_{\mathrm{y}} = -\mathrm{i}\hbar \left(\cos \phi \frac{\partial}{\partial\theta} -\cot \theta \sin \phi \frac{\partial}{\partial\phi}\right)$ and $\hat{L}_{z} = -\mathrm{i}\hbar \frac{\partial}{\partial\phi}$ (B) Frequency $(\nu)$ and time $(t)$ (C) Momentum $(P)$ and position $(x)$ (D) Operators $\hat{I} = 5$ and $\hat{J} = \int_{1}^{\infty} dx$

7. Which of the following quantum mechanical model does NOT have the zero-point energy? (A) Particle in a box with finite potential wells (B) Rigid rotor (C) Harmonic oscillator (D) Quantized electromagnetic field

8. Which of the following molecule may NOT show a rotational Raman spectrum? (A) $\mathrm{N}_2$ (B) $\mathrm{CO}_{2}$ (C) $\mathrm{H}_{2} \mathrm{O}$ (D) $\mathrm{CH}_{4}$

9. Scientists used ultrafast X-ray pulses to directly visualize the DNA photorepair process. The probability of bond-rearrangement events was observed to increase gradually over a timescale of approximately 500 ns. Which of the following statements is incorrect? (A) Molecular bond elongation occurs on a timescale of approximately 500 ns. (B) Molecular bond elongation occurs on a timescale of approximately 500 fs. (C) Although molecular bond vibrations occur on ultrafast timescales, the concentration of reaction products may require a much longer time to increase to a detectable level. (D) The presence of multiple reaction barriers allows reactant molecules that have crossed a barrier to return via reverse reactions.

10. In a consecutive reaction $A \rightarrow B \rightarrow C$ concentration of the intermediate will be small relative to the starting concentration of A if (A) the rate constant for the first step is larger than that for the second, (B) the rate constant for the second step is larger than that for the first, (C) very few products have formed, (D) nearly all the reactant has disappeared.

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