110 年 國立臺灣聯合大學系統（清華、政治、陽明交通、中央四校聯招）研究所化學類《物理化學》

According to Marcus theory, which of the following factors does NOT influence the electron transfer rate constant?

(A) Reorganization energy.
(B) Gibbs energy difference between the product and reactant.
(C) Angular momentum of electron.
(D) Distance between electron donor and acceptor.
(E) Type of spacer between electron donor and acceptor.

The energy levels of the linear harmonic oscillator are

(A) all nondegenerate
(B) $n$-fold degenerate
(C) $(n+\frac{1}{2})$-fold degenerate
(D) $(2n+1)$-fold degenerate
(E) $n^2$-fold degenerate

The illustrated wave function represents the state of the linear harmonic oscillator

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with $n=$

(A) $2$
(B) $3$
(C) $4$
(D) $5$
(E) $6$

There is one state at $\varepsilon=0$ and there are two states at $\varepsilon=300\ \mathrm{cm^{-1}}$. According to the Boltzmann distribution, the population of the second state is $0.15$. What is the temperature ($\mathrm{K}$)?

(A) $180\ \mathrm{K}$
(B) $300\ \mathrm{K}$
(C) $480\ \mathrm{K}$
(D) $90\ \mathrm{K}$
(E) $>1000\ \mathrm{K}$

The rate constant of a chemical reaction is NOT determined by

(A) steric requirement
(B) encounter rate
(C) minimum energy requirement
(D) temperature
(E) standard reaction free energy

Spherical polar coordinates are used in the solution of the hydrogen atom Schrödinger equation because

(A) the Laplacian operator has its simplest form in spherical polar coordinates.
(B) Cartesian coordinates would give particle-in-a-box wave functions.
(C) the Schrödinger equation is then separable into 3 ordinary differential equations.
(D) otherwise the atomic orbitals would violate the Pauli Exclusion Principle.
(E) Schrödinger first used this coordinate system; any other coordinate system would be equally convenient.

Consider the thermal decomposition of $1\ \mathrm{atm}$ of $\mathrm{(CH_3)_3COOC(CH_3)_3}$ to acetone $\mathrm{(CH_3)_2CO}$ and ethane ($\mathrm{C_2H_6}$), which occurs with a rate constant of $0.02\ \mathrm{sec^{-1}}$.

After initiation of the reaction, at what time would you expect the pressure to be $2.0\ \mathrm{atm}$?

(A) $693\ \mathrm{sec}$
(B) $34.65\ \mathrm{sec}$
(C) $263\ \mathrm{sec}$
(D) $699\ \mathrm{sec}$
(E) Cannot be determined.

Molecules are known to absorb radiation in which region of the electromagnetic spectrum:

(A) Ultraviolet
(B) visible
(C) infrared
(D) microwave
(E) all of the above

The van der Waals equation is expressed as

$\left(P+\frac{n^2a}{V^2}\right)(V-nb)=nRT.$

Which of the following statements is true?

(A) Compression factor $Z<1$ when $\frac{a}{RT}>b$.
(B) Compression factor $Z<1$ when $\frac{a}{RT}<b$.
(C) Compression factor $Z<1$ when $\frac{a}{RT^2}<b$.
(D) Compression factor $Z<1$ when $a<\frac{b}{RT}$.
(E) All of the above is incorrect.

Which of the following is “NOT” a correct aspect of the Born–Oppenheimer approximation?

(A) The electrons in a molecule move much faster than the nuclei.
(B) Excited electronic states have the same equilibrium internuclear distance as the ground electronic state.
(C) The electronic and vibrational motions of a molecule are approximately separable.
(D) Electronic energy curves serve as potential energy functions for nuclear vibrational motions.
(E) The typical amplitude of nuclear vibration is much smaller than that characterizing the motion of electrons.

The benzene molecule $\mathrm{C_6H_6}$ has how many vibrational modes?

(A) $6$
(B) $12$
(C) $24$
(D) $30$
(E) $36$

For the reaction $A+B\rightarrow\text{product}$, the reaction rate is first order to $A$ and first order to $B$. Integrate the rate law.

$[A]_0$ and $[B]_0$ are the initial concentrations for $A$ and $B$, respectively, and $[A]_0$ is not equal to $[B]_0$. $t=$ time.

(A) $kt=\frac{1}{[A]_0-[B]_0}\ln\frac{[A]_0[B]_0}{[A][B]}$
(B) $kt=\frac{-1}{[A]_0-[B]_0}\ln\frac{[A][B]_0}{[A]_0[B]}$
(C) $kt=\frac{1}{[A]_0-[B]_0}\ln\frac{[A][B]}{[A]_0[B]_0}$
(D) $kt=\frac{1}{[A]_0-[B]_0}\ln\frac{[A][B]_0}{[A]_0[B]}$
(E) All of the above is incorrect.

The normal boiling temperature of ethanol is $78.5\,^\circ\mathrm{C}$ and its molar enthalpy change of vaporization at this temperature is $40.3\ \mathrm{kJ/mol}$.

Find $\Delta U$ when $3.000\ \mathrm{mol}$ of ethanol are vaporized at $78.5\,^\circ\mathrm{C}$ at a constant pressure of $1.000\ \mathrm{atm}$. Neglect the volume of the liquid compared with that of the vapor and assume the ethanol vapor as an ideal gas.

(A) $\Delta U=120.9\ \mathrm{kJ}$
(B) $\Delta U=0.0\ \mathrm{kJ}$
(C) $\Delta U=-8.77\ \mathrm{kJ}$
(D) $\Delta U=130.1\ \mathrm{kJ}$
(E) $\Delta U=112.1\ \mathrm{kJ}$

The ionization energy for hydrogen atom is $13.6\ \mathrm{eV}$. The ionization energy for the ground state of $\mathrm{Li^{2+}}$ is approximately

(A) $13.6\ \mathrm{eV}$
(B) $27.2\ \mathrm{eV}$
(C) $40.8\ \mathrm{eV}$
(D) $54.4\ \mathrm{eV}$
(E) $122.4\ \mathrm{eV}$

Which of the following is NOT a correct consequence of the Heisenberg Uncertainty principle:

(A) The shorter the lifetime of an excited state of an atom, the less accurately can its energy be measured.
(B) An electron in an atom cannot be described by a well-defined orbit.
(C) The momentum of an electron cannot be measured exactly.
(D) Measurement of one variable in an atomic system can affect subsequent measurements of other variables.
(E) A harmonic oscillator possesses a zero-point energy.

The Joule coefficient, $\mu_J$, is defined as

$\mu_J=\left(\frac{\partial T}{\partial V}\right)_U.$

$\alpha$ is the thermal expansion coefficient. $\kappa_T$ is the isothermal compressibility. Which of the following equations is correct?

(A) $\mu_J C_V=p+\frac{\alpha T}{\kappa_T}$
(B) $\mu_J C_V=p-\frac{\alpha T}{\kappa_T}$
(C) $\mu_J C_V=p+\frac{\kappa_T T}{\alpha}$
(D) $\mu_J C_V=p-\frac{\kappa_T T}{\alpha}$
(E) $\mu_J C_V=\frac{\alpha T}{\kappa_T}$

The orbital degeneracy (excluding spin) of hydrogen atom energy levels equals

(A) $n-1$
(B) $n$
(C) $n+1$
(D) $2n+1$
(E) $n^2$

The change in enthalpy is given by $dH=C_p\,dT+V\,dp$. In combination with the Clapeyron equation, find out $d(\Delta H/T)$ and the two phases remain in equilibrium.

(A) $d(\Delta H/T)=\Delta C_p\,d(\ln T)$
(B) $d(\Delta H/T)=R\,d(\ln T)$
(C) $d(\Delta H/T)=\Delta S\,d(\ln T)$
(D) $d(\Delta H/T)=C_p\,d(\ln T)$
(E) All of the above is incorrect.

Which of the following statements about the hydrogen atom ground state is INCORRECT:

(A) It is described by the quantum numbers $n=1$; $l=0$; $m=0$.
(B) The electron's angular momentum equals Planck constant, $h$.
(C) The wave function is spherically symmetrical.
(D) The wave function decreases exponentially as a function of $r$.
(E) The radial distribution function has its maximum at the Bohr radius.

At the solid-liquid-vapor triple point of a pure substance, what is the slope, $dp/dT$, of phase transition?

(A) solid-vapor line $>$ liquid-vapor line
(B) solid-vapor line $=$ liquid-vapor line
(C) solid-vapor line $<$ liquid-vapor line
(D) solid-vapor line $=$ liquid-vapor line $=0$
(E) all of the above is incorrect

Based on statistical thermodynamics, which of the following statements about heat capacity is true?

(A) The $C_p$ for $\mathrm{Ar}$ is the same as that for $\mathrm{Ne}$, even though the atomic weight of $\mathrm{Ar}$ is twice that of $\mathrm{Ne}$.
(B) The heat capacity for monoatomic atom $\mathrm{F}$ is considerably larger than that for $\mathrm{Ne}$ even though they have about the same atomic weight.
(C) The heat capacity of liquid water is much larger than that of water vapor.
(D) $\mathrm{Br_2(g)}$ has a larger $C_p$ value than that of $\mathrm{Cl_2(g)}$.
(E) The $C_p$ values for $\mathrm{Ar}$, $\mathrm{O_2(g)}$, and $\mathrm{Cl_2(g)}$ are in the order of $\mathrm{Cl_2(g)}>\mathrm{O_2(g)}>\mathrm{Ar}$.

The velocity probability of Maxwell–Boltzmann distribution in one direction is expressed as:

$f(v_y)=k'e^{-mv_y^2/(2kT)}.$

Which of the following statements about Maxwell–Boltzmann distribution at a finite $T$ is true?

(A) The most-probable velocity for a one-dimensional gas velocity distribution is zero.
(B) The Maxwell–Boltzmann speed (three dimensions) distribution approaches zero at high speeds.
(C) The Maxwell–Boltzmann speed (three dimensions) distribution approaches zero at low speed.
(D) The total energy of one mole perfect gas according to the Maxwell–Boltzmann distribution is $1.5RT$.
(E) Most-probable speed $>$ Mean speed $>$ Root mean square speed.

Consider the collision-induced dissociation of $\mathrm{N_2O_5(g)}$ via the following mechanism:

$\mathrm{N_2O_5(g)}+\mathrm{N_2O_5(g)}\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}}\mathrm{N_2O_5^*(g)}+\mathrm{N_2O_5(g)}$

$\mathrm{N_2O_5^*(g)}\xrightarrow{k_2}\mathrm{NO_2(g)}+\mathrm{NO_3(g)}$

The $*$ symbol in the first reaction indicates the reactant is activated through collision.

Experimentally, it is observed that the reaction can be either first or second order under different conditions. The rate law is expressed as

$R=\frac{k'[\mathrm{N_2O_5}]^m}{1+k''[\mathrm{N_2O_5}]}.$

Which of the following statements is true?

(A) $m=2$.
(B) $k'=k_1$.
(C) $k''=k_{-1}/k_2$.
(D) When $k_{-1}[\mathrm{N_2O_5}]\gg k_2$, the reaction will be first order in $\mathrm{N_2O_5(g)}$.
(E) When $k_{-1}[\mathrm{N_2O_5}]\ll k_2$, the reaction will be half order in $\mathrm{N_2O_5(g)}$.

Assume the molar Gibbs free energy of a gas is given by the equation:

$G_m-RT\ln\left(\frac{P}{p_0}\right)+(a+bT)P=0,$

where $p_0$ is the standard pressure and $a$ and $b$ are gas-dependent constants.

Derive an equation of state for this gas. Show the final result in the form of $PV=?$.

The bonding in Hydrogen Iodide ($\mathrm{HI}$) can be described by a two-orbital interaction between the $1s$ orbital on Hydrogen and the $5p$ orbital on Iodine.

The Hamiltonian matrix elements (Coulomb integrals, exchange integral and overlap integral) are:

$H_{\mathrm{HH}}=-14\ \mathrm{eV},\qquad H_{\mathrm{II}}=-10\ \mathrm{eV},\qquad H_{\mathrm{HI}}=-3\ \mathrm{eV},\qquad S_{\mathrm{HI}}=0.$

Assume that the molecular wave function is:

$\psi=c_{\mathrm{H}}1s_{\mathrm{H}}+c_{\mathrm{I}}5p_{\mathrm{I}}.$

(a) (4 pts) Set up the Secular Determinant and solve for the energy of the Bonding and Anti-Bonding Orbitals.

(b) (4 pts) Use your answer in part (a) to calculate the coefficients, $C_{\mathrm{H}}$ and $C_{\mathrm{I}}$, in the Normalized anti-bonding orbital. If you are not sure of the energy, then use $E_{\mathrm{antibond}}=-8.0\ \mathrm{eV}$ to work out this part.

(c) (2 pts) Assume for this part that the bonding orbital is:

$\psi=N\left[(0.5)\cdot1s_{\mathrm{H}}+(0.9)\cdot5p_{\mathrm{I}}\right].$

What fraction of the charge of an electron in the bonding orbital is on the Iodine atom?

Consider a transition between the ground state ($\Psi_0$) and the $i$'th excited state ($\Psi_i$). Time dependent perturbation theory can be used to show that the intensity of the absorption is proportional to the square of the “transition moment”, $M_{0i}$.

The $x$, $y$ and $z$ components of the transition moment are:

$M_{0i}^x=e\langle\Psi_i|x|\Psi_0\rangle$

$M_{0i}^y=e\langle\Psi_i|y|\Psi_0\rangle$

$M_{0i}^z=e\langle\Psi_i|z|\Psi_0\rangle$

In the case of Carbene, it has $C_{2v}$ symmetry, has 8 electrons and the Triplet Ground state takes the electronic configuration $(a_1)^2(a_1)^2(b_2)^2(a_1)^1(b_1)^1$ as shown in the following diagram, where the shapes of the molecular orbitals are also included.

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Please determine if the electronic transition (only the highest two unpaired electrons are shown and the paired inner orbitals are neglected) is UV allowed or forbidden.

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For your information, the character table for $C_{2v}$ symmetry is: