14. What is the expression of entropy change $(\Delta S)$ for the isothermal reversible expansion at temperature $T$ from $V_{1}$ to $V_{2}$ of $n$ moles of a van der Waals gas?
(A) $nRT\ln \frac{V_1 - nb}{V_2 - nb} + an^2\left(\frac{1}{V_1} -\frac{1}{V_2}\right)$
(B) $-nR\ln \frac{V_1 - nb}{V_2 - nb} -\frac{an^2}{T}\left(\frac{1}{V_1} -\frac{1}{V_2}\right)$
(C) $nR\ln \frac{V_1 - nb}{V_2 - nb} +\frac{an^2}{T}\left(\frac{1}{V_1} -\frac{1}{V_2}\right)$
(D) $-nRT\ln \frac{V_1 - nb}{V_2 - nb} -an^2\left(\frac{1}{V_1} -\frac{1}{V_2}\right)$

15. The equation of $C_{P,m} - C_{V,m}$ can be expressed in terms of $V_{m}, T,$ and the gas constants $R, a$ and $b$ as follows: $C_{P,m} - C_{V,m} = R / \left(\Gamma - \frac{\Theta a(V_m - b)^\Phi}{RTV_m^\Omega}\right)$ , where $\Gamma, \Theta, \Phi$ and $\Omega$ are real integers. Which of the description about $\Gamma, \Theta, \Phi$ and $\Omega$ is correct? (A) $\Gamma \neq \Omega$ (B) $\Theta \neq \Phi$ (C) $\Theta = \Phi = \Omega = 1$ (D) $\Omega = \Phi$

16. The rate equation of $A^{*}$ is (A) $[A^{*}] = [A]_{0}e^{-(k_{1} + k_{2})t}$ ; (B) $[A^{*}] = [A^{*}]_{0}e^{-(k_{3})t}$ ; (C) $[A^{*}] = [A^{*}]_{0}e^{-(k_{1} + k_{2} + k_{3})t}$ ; (D) $[A^{*}] = [A]_{0}e^{-(k_{1} + k_{2} + k_{3})t}$

17. The fluorescence lifetime of $A^{*}$ is (A) $\frac{1}{k_1}$ (B) $\frac{1}{k_1 + k_2}$ (C) $\frac{1}{k_1 + k_2 + k_3}$ (D) $\frac{1}{k_2}$

18. The branching percentage of $A_{hot}$ , is (A) $\frac{k_2}{k_1 + k_2}$ (B) $\frac{k_2}{k_1 + k_2 + k_3}$ (C) $\frac{k_1}{k_2}$ (D) $\frac{k_2}{k_1}$

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 $\mathrm{Fe(OH)}_{2(s)} + 2\mathrm{Ag}_{(s)} + 2\mathrm{Br}^{-}_{(aq)} \rightleftharpoons \mathrm{Fe}_{(s)} + 2\mathrm{OH}^{-}_{(aq)} + 2\mathrm{AgBr}_{(s)}$.

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