課程名稱︰普通化學一
課程性質︰必修
課程教師︰鄭原忠
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰100年10月19日
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
General Chemistry (I)
Mid-term Exam #1 Date: 10/19/2011
106 points
10 bonus points
total 116 points
┌────────────────────┐
│Physical constants: │
│c = 3.0*10^8 m/s │
│mass of an electron = 9.1*10^(-31) kg │
│h = 6.626*10^(-34) Js │
│1 eV = 1.60*10^(-19) J │
│R = -13.6 eV │
└────────────────────┘
1. (10%) A 60.00 g sample of a dry-cleaning fluid was analyzed and found to
contain 10.80 g carbon, 1.36 g hydrogen, and 47.84 g chlorine. Answer
the following questions:
(a) Determine the empirical formula of the compound using the
following atomic masses: C: 12.0, H: 1.0, Cl: 35.5.
(b) Suggest a molecular formular for this compound that is stable in
air.
2. (10%) Sulfuric acid (H SO ) form in the chemical reaction (unbalanced):
SO + O + H O → H SO
2 2 2 2 4
(a) Balance the chemical equation and determine which compound is the
limiting reactant.
(b) What mass of H SO should be produced, and what masses og the
2 4
other reactants remain?
3. The time-independent Schrödinger equation (TISE) for a particle in two-
dimension is:
2 ┌ 2 2 ┐
-h │δ Ψ δ Ψ│
────│─── + ───│ + V(x,y) Ψ = EΨ
2 │ 2 2 │
8π m └ δx δx ┘
Consider a particle in a 2-D square box of length L.
(a) (3%)Clearly and concisely define the parameters and functions appear in
the TISE.
(b) (2%)What is the TISE of the 2-D particle-in-a-box system? Clearly
identify the boundary conditions for the stationary states.
(c) (5%)The stationary wave function are
n πx n πy
x y
Ψ (x,y) = N sin(───) sin(───)
n ,n L L
x y
Insert the expression into the TISE to derive an expression for
energy levels in terms of n , n , and L.
x y
(d) (5%)Contour plots of four stationary wave function are shown on the
right. The x and y directions of the box lie along the horizontal
and vertical directions, respectively. Identify the quantum
numbers n , n for states a-d.
x y
(e) (5%)Give the degeneracies of the states a-d. Sort the four states from
low to high energies.
4. (10%)Consider TISE for the harmonic oscillator model:
2 2
-h δΨ 1 2
──── ── + ── kx Ψ = EΨ
2 2 2
8π m δx 2
-αx
Show that a Gaussian wave function, Ψ(x) = Ne where α is a
positive parameter, is a stationary solution. Determine α and argue
that the Gaussian is the ground state. What is the zero-point energy?
5. (10%)The 2s radial function for a hydrogen-like atom is
3/2
1 ┌ Z ┐ ┌ Zr ┐ -Zr/2a
R (r) = ─────│──│ │1 - ─ │ e 0
2s sqrt(2) └ a ┘ └ 2a ┘
0 0
Consider an electron in the 2s orbital of a Helium atom.
(a) Sketch the radial part of the wave function as a function of r,
you should explicitly label the positions of node(s) and maxima.
(b) Sketch the radial probability density as a function of r. Again,
explicitly label the positions of node(s) and maxima.
6. On the right are figures of four hydrogen atomic orbitals including a
contour plot in the x-y plane, the radial function, and the radial
distribution function.
(a) (12%)Give the principle quantum number and angular quantum number for
each of the orbitals. Use the s, p, d, f, ... symbols in your
answers.
(b) (3%)Sort the orbitals from low energy to high energy.
7. Photoelectron spectroscopy studies of neon atoms excited by x-rays with
wavelength 9.890*10^(-10) m show three peaks in which the electron have
kinetic energy values 383.4 eV, 1205.2 eV, and 1232.0 eV.
(a) (5%)Calculate the ionization energy of the electrons in each peak.
(b) (5%)Assign each peak to an orbital of the neon atom, and use the
results to draw a energy-level diagram.
(c) (Bonus,5%)Clearly state the assumptions that are required in order to
use photoelectron spectrum to assign orbital energies.
(d) (Bonus,5%)Calculate the effective charge for each orbital and give
explanations of the trend.
8. Answer ture or false for the following statements (3 points each):
(a) The zero point energy is higher for a He atom in a box than for an
electron in the same box.
(b) Molecules with a longer π-conjugated system tend to absorb photons
with longer wavelengths.
(c) An example of vibrational isotope effect: ν / ν is roughly
H-Cl D-Cl
the square root of 2.
(d) Any bound system must have a non-zero zero-point energy.
(e) The wave function of a system must always satisfy the time-independent
Schrödinger equation.
(f) The magnitude of the angular momentum of a hydrogen-like atom is
proportional to sqrt[l(l+1)].
(g) An orbital can be used to represent an electronic state of a many-
electron atom.
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