26 50 mL of 0.1 M diprotic acid (H₂A; pK_a1 = 10^-4, pK_a2 = 10^-8) reacted with 50 mL of 0.1 M NaOH. What is the pH of the resulting solution?
(A) 8.0
(B) 7.0
(C) 6.0
(D) 5.0
(E)4.0

1Given a free particle in a box, which of the following quantum mechanical conditions is not true?(A) Shifting the box in space will change the wavefunction.(B) The precise position of the particle is unpredictable.(C) If the size of the box is infinite, the energy of the particle is not quantized.(D) Shifting the box in space will change the energy of the particle.(E) The energy of the particle depends on the shape of the box.

2Which one of the following is the correct form for the lowest-energy eigenfunction for a particle in a1D box with infinite barriers at x= -L/2 and L/2?(A)sin(x/L) (B)cos(πx/L) (C) (D)(E)None of the above.

3 Given a wave function φ(x) = x(x-1) in the region 0≤ x ≤ 1. Where N is the normalization constant to be determined. What is the possible value of N?(A)(B)(C)(D) (E) None of the above. 

4The wavefunction on the right illustrates the quantum state of a harmonic oscillator with its vibrational quantum number v = (A) 1 (B) 2 (C) 3 (D) 4(E)5.

5 Which of the following statements is true for a given set of harmonic oscillator wavefunctions corresponding to the same potential V(x)? (A) The ground-state energy is called zero-point energy and its value is zero. (B) The mean displacement from the equilibrium distance <x>≠0 for the first excited vibrational state v= 1.(C) The wavefunctions must have zero amplitude beyond the classical turning points.(D)(E) The wavefunctions involve Gaussian-type functions and thus are all even functions.

6Which of the following about the spherical harmonics is not true? m (A) The number of angular nodes in is l. m (B) The complex give non-zero eigenvalues of which represents the operator of the z-component of angular momentum. (C) is a constant everywhere in space. (D) serves as eigenfunctions for.(E) cosθ correlates with all p₂ orbitals.

7Which statement about H-atom is not true? (A) Spherical polar coordinates are the natural choice for H-atom Schrödinger equation because the potential term is spherically symmetric. (B) H-atom Schrödinger equation can be solved by separation of variables where the radial part Rni(r) and the angular partare linked through I(I+1) while Θ(θ) andΦ (Φ) are linked through m². (C) The operator for the z-component of angular momentum has the simplest form in spherical polar coordinates as opposed to the x and y-components. (D) The rules for allowed angular momentum magnitudes and orientations are shared by the rigid rotor and the H-atom. (E) By changing the variable sinθ into x, the Θ differential equation can be recognized as being the associated Legendre equation.

8In the above plot, the letters a, b, and c label three different radial distribution functions (RDF's) r²R² of H-atom from the same n level. Which of the following statements is true? (A) The maximum of curve a corresponds to the radii of maximum probability rmp = n²a0, (a0 is Bohr 2 radius). (B) Curve b corresponds to the RDF derived from a radial function R that is proportional to r² exp(-r/3a0). (C) Curve c corresponds to angular momentum quantum number I = 2. (D) Curve c shows two angular nodes. (E) The principal quantum number n = 2.

9The term "intersystem crossing" is used for(A) a transition from an excited singlet state to another excited singlet state.(B) a transition from the ground singlet state to an excited singlet state.(C) a transition from an excited singlet state to an excited triplet state.(D) a transition from an excited triplet state to another excited triplet state.(E) a transition from an excited triplet state to the ground singlet state.

10 Which of the following statements is not true for the experimental HCl rotational-vibrational spectrum below? (A) The splitting in each line is due the isotope effect. (B) In the middle of the spectrum a forbidden line is missing by the selection rules. (C) The bond length for HCl can be determined from the intervals between neighboring lines. (D) The separation between lines decreases in the R-branch and increases in the P-branch. (E) The lines connected to transitions with △J = +1 form the P-branch.

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