19. The rate equation of species $A_{T}$ is
(A) $[B] = \frac{k_3}{k_1 + k_2 + k_3} [A_T]_0 e^{-(k_1 + k_2 + k_3)t}$
(B) $[A_T] = [A_T]_0 e^{-(k_3)t}$
(C) $[A_T] = \frac{k_3}{k_1 + k_2 + k_3} [A_T]_0 [1 - e^{-(k_3)t}]$
(D) $[A_T] = \frac{k_3}{k_1 + k_2 + k_3} [A^*]_0 [1 - e^{-(k_1 + k_2 + k_3)t}]$

20. To increase the fluorescence quantum yield, the chemist may design a molecule with the following properties. Which of the following is incorrect?(A) A shorter fluorescence lifetime(B) Slower internal conversion rate(C) Slower intersystem crossing rate(D) Faster radiative decay

複選題21. In ultrafast photochemical processes, molecules are often excited by femtosecond laser pulses to electronically excited states, followed by rapid nuclear motion and electronic relaxation. In such situations, the validity of the Born–Oppenheimer approximation (BOA) becomes a central issue. The BOA exploits the large mass difference between nuclei and electrons to simplify the molecular Schrödinger equation by separating electronic and nuclear motions. Which of the following statements regarding the Born–Oppenheimer approximation (BOA) are correct?(A) The BOA assumes that electrons respond instantaneously to nuclear motion due to their much smaller mass.(B) Within the BOA, the total molecular wavefunction is expressed as a product of an electronic wavefunction and a nuclear wavefunction.(C) Zero-point vibrational energy is neglected in the BOA.(D) Immediately after vertical photoexcitation, the BOA is typically valid because the nuclear geometry has not yet changed.(E) Within the BOA, ultrafast population transfer between different electronic states can be fully described without considering the interaction between nuclear motion and electronic states.

複選題22. If we consider the three non-hybridized $p$ orbitals ($\psi_{1}, \psi_{2}$ and $\psi_{3}$) and their electrons using the molecular orbital (MO) theory, we can determine three $\pi$ MOs $\psi_{\pi}^{(1)}, \psi_{\pi}^{(2)}$ and $\psi_{\pi}^{(3)}$ with linear combination of $\psi_{1}, \psi_{2}$ and $\psi_{3}$ for that $\psi_{\pi}^{(1)} = C_A \psi_1 + C_B \psi_2 + C_A \psi_3$; $\psi_{\pi}^{(2)} = C_N (\psi_1 - \psi_3)$; $\psi_{\pi}^{(3)} = C_A \psi_1 - C_B \psi_2 + C_A \psi_3$, which obey orthonormal conditions ($\int \psi_{\pi}^{(i)*} \psi_{\pi}^{(i)} d\tau = 1$; $\int \psi_i' \psi_i d\tau = 1$; $\int \psi_i' \psi_j d\tau = 0$ for $i \neq j$). Which of the following statements regarding the linear combination coefficients $C_A$, $C_B$ and $C_N$ are correct?(A) $C_A > C_B$(B) $C_B = C_N$(C) $C_A < C_N$(D) $C_A = C_B < C_N$(E) $C_A > C_B = C_N$

複選題23. The molar Gibbs free energy of a certain gas is given by $G_{m} = A_{1} + A_{2}P + RT\ln P$, where $A_{1}, A_{2}$ are constants, $T$ is temperature, and $P$ is pressure. One mole of this gas at temperature $T$ is expanded isothermally and reversibly from pressure $P_{1}$ to $P_{2}$. Which of the following statements regarding this process are correct?(A) The equation of state for the gas is $PV_{m} - A_{2} = RT$(B) The fugacity $(f)$ of the gas is $f = P \times e^{\frac{A_2P}{RT}}$(C) The internal energy change $\Delta U = 0$(D) The entropy change $\Delta S = R \ln \frac{P_2}{P_1}$(E) The enthalpy change $\Delta H = A_{2}(P_{2} - P_{1})$注意:背面有試題台灣聯合大學系統115學年度碩士班招生考試試題類組：化學類 科目：物理化學(1004)共 6 頁 第 5 頁

複選題24. The following half-cells for which the reduction reaction and its corresponding standard electromotive force $(\mathrm{E}^{\circ})$ is given byHalf-cell (a): $\mathrm{Fe(OH)_{2(s)} + 2e^{-} \rightarrow Fe_{(s)} + 2OH^{-}(aq)}$ $\quad \mathrm{E}^{\circ} = -0.877\,\mathrm{V}$Half-cell (b): $\mathrm{Al}^{3+}(aq) + 3e^{-} \rightarrow \mathrm{Al}_{(s)}$ $\quad \mathrm{E}^{\circ} = -1.66\,\mathrm{V}$Half-cell (c): $\mathrm{AgBr}_{(s)} + \mathrm{e}^{-} \rightarrow \mathrm{Ag}_{(s)} + \mathrm{Br}^{-}_{(aq)}$ $\quad \mathrm{E}^{\circ} = +0.071\,\mathrm{V}$The electrochemical cell 1 $(\mathbf{E}_1)$ is constructed with the half-cell (a) and (b), whereas the electrochemical cell 2 $(\mathbf{E}_2)$ is constructed with the half-cell (a) and (c). The activity of all species is one for both cells. Which of the following statements are correct?(A) The electromotive force of the electrochemical cell is positive when the electrochemical reaction is spontaneous.(B) In $\mathrm{E}_1$, $\mathrm{Fe}_{(s)}$ is oxidized.(C) In $\mathrm{E}_2$, $\mathrm{Fe}_{(s)}$ is reduced.(D) If $\mathrm{E}_1$ is spontaneous, the overall reaction is $3\mathrm{Fe(OH)}_{2(s)} + 2\mathrm{Al}_{(s)} \rightleftharpoons 3\mathrm{Fe}_{(s)} + 6\mathrm{OH}^{-}_{(aq)} + \mathrm{Al}^{3+}_{(aq)}$.(E) If $\mathrm{E}_2$ is spontaneous, the overall reaction is $\mathrm{Fe(OH)}_{2(s)} + 2\mathrm{Ag}_{(s)} + 2\mathrm{Br}^{-}_{(aq)} \rightleftharpoons \mathrm{Fe}_{(s)} + 2\mathrm{OH}^{-}_{(aq)} + 2\mathrm{AgBr}_{(s)}$.

複選題25. Experimental observations show that the fluorescence decay curves of anthracene measured at two different wavelengths do not exhibit simple exponential behavior. Instead, the decay curves display oscillatory features that are $180^{\circ}$ out of phase. Which of the following statements are correct? (A) Multiple excited states in anthracene are excited non-coherently. (B) Interference between coherently excited states leads to periodic modulation of the excited-state population, producing oscillations in the fluorescence intensity, a phenomenon known as quantum beating. (C) In a simple two-level system with incoherent population decay, fluorescence follows a single exponential decay. When coherence is present, the fluorescence intensity contains both exponential decay and oscillatory components, resulting in non-exponential decay behavior. (D) Fluorescence detected at different wavelengths corresponds to emission from different vibronic transitions. Due to interference, population dynamics between these emitting states can be anticorrelated, leading to oscillations that are $180^{\circ}$ out of phase between the two detection wavelengths. (E) Fluorescence from the non-coherently excited states leads interference resulting the oscillations in the fluorescence intensity, a phenomenon known as quantum beating.

複選題26. If the fluorescence quantum yield $(\varnothing_{f})$ of anthracene is $10\%$, and the fluorescence lifetime is $50\,\mathrm{ns}$, $k_{r}$ is the radiative rate constant, and $k_{nr}$ is the non-radiative rate constant. Which of the following are TRUE? (A) $k_{r} = 2 \times 10^{6}\,\mathrm{sec}^{-1}$ (B) $k_{r} = 2 \times 10^{7}\,\mathrm{sec}^{-1}$ (C) $k_{nr} = 2 \times 10^{6}\,\mathrm{sec}^{-1}$ (D) $k_{nr} = 1.8 \times 10^{7} \sec^{-1}$ (E) $k_{nr} = 9k_{r}$

複選題27. There are three methods for hydrogen production(1) Methane pyrolysis:$\mathrm{CH_4(g)} \rightarrow \mathrm{C(s)} + 2 \mathrm{H_2(g)}$; $\Delta \mathrm{H} = +76 \mathrm{kJ}$, corresponding to $+38 \mathrm{kJ}$ per mole of $\mathrm{H_2}$ produced.(2) Water electrolysis:$2 \mathrm{H_2O(l)} \rightarrow \mathrm{O_2(g)} + 2 \mathrm{H_2(g)}$; $\Delta \mathrm{H} = +572 \mathrm{kJ}$, corresponding to $+286 \mathrm{kJ}$ per mole of $\mathrm{H_2}$ produced.(3) Steam methane reforming:$\mathrm{CH_4(g)} + 2 \mathrm{H_2O(g)} \rightarrow \mathrm{CO_2(g)} + 4 \mathrm{H_2(g)}$; $\Delta \mathrm{H} = +252 \mathrm{kJ}$, corresponding to $+63 \mathrm{kJ}$ per mole of $\mathrm{H_2}$ produced.Based solely on the reaction enthalpies $(\Delta \mathrm{H})$, methane pyrolysis [method (1)] appears to be the most energy-efficient pathway for hydrogen production. However, this conclusion is misleading in practice and does not adequately represent the true energetic and technological challenges of the process. Which of the following statements correctly explain the reasons?(A) Although the net enthalpy requirement is only $38\mathrm{kJ}$ per mole of $\mathrm{H}_{2}$, a much higher activation energy must be overcome to cleave the strong C-H bonds in methane.(B) Non-catalytic methane pyrolysis requires temperatures above approximately $1200^{\circ}\mathrm{C}$, with activation energies typically in the range of $300 - 420\mathrm{kJ / mol}$.(C) Catalytic methane pyrolysis employing metal (e.g., Ni, Fe, Cu) or carbon catalysts can lower the required operating temperature to approximately $600 - 900^{\circ}\mathrm{C}$ and reduce the activation energy to about $150 - 200\mathrm{kJ / mol}$.(D) In methane pyrolysis reactors, heat transfer to methane is generally less efficient than in steam methane reforming [method (3)], which benefits from the use of high-temperature water vapor as an effective heat-transfer medium.(E) In [method 3], the total bond enthalpy of the products is less than that of reactants.

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 statements about atomic electronic structure is correct? (A) The effective nuclear charge (Zeff) experienced by a valence electron increases down a group (B) Electrons in $d$ orbitals shield outer s electrons more effectively than inner s electrons (C) The energy of $4s$ orbitals is always lower than that of $3d$ orbitals for all transition-metal atoms (D) For a given shell (same n), $p$ orbitals generally have higher energy than $s$ orbitals due to poorer penetration (E) The Pauli exclusion principle allows two electrons in the same atom to have identical sets of four quantum numbers

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