8. For an one-dimensional harmonic oscillator with force constant k and mass m, which statement is wrong?
(A) ω =.
(B) Energy levels are equally spaced.
(C) Zero-point energy isℏω.
(D) ⟨x⟩ = 0 for every stationary state.
(E) Each vibrational energy level is doubly degenerate.

1. If the reaction 2HI → H2 + I2 is second order in HI, which plot is linear? (A) [HI] vs t (B) ln[HI] vs t (C) 1/[HI] vs t (D) 1/ln[HI] vs t (E) [HI]² vs t

2. For a reaction at constant pressure, the standard Gibbs energies at two temperatures are: ΔrG°(298K) = −12.0 kJ mol⁻¹, ΔrG°(348K) = −5.0 kJ mol⁻¹. Using the Gibbs-Helmholtz equation:what can be concluded about the sign of ΔrH°? (assuming that ΔrH° does not depend on T) (A) ΔrH° = 0 (B) ΔrH° ＞ 0 (endothermic) and ΔrH° is around +54 kJ mol⁻¹ (C) ΔrH° ＜ 0 (exothermic) (D) Cannot be inferred from ΔrG°(T) (E) ΔrH° ＞ 0 (endothermic) and ΔrH° is around +5.4 kJ mol⁻¹

3. An enzyme follows Michaelis-Menten kinetics with Vmax = 120 μM min⁻¹ and Km = 30 μM. What is the initial rate v0 when the substrate concentration is [S] = 30 μM? (A) 15 μM min⁻¹ (B) 30 μM min⁻¹ (C) 60 μM min⁻¹ (D) 90 μM min⁻¹ (E) 120 μM min⁻¹

4. A reaction proceeds by the mechanism: Assuming the steady-state approximation for the intermediate I (i.e., d[I]/dt ≈ 0), what is the rate law for product formation r = in terms of [A] and [B]? (A) r = k2[A][B] (B) (C) (D) r = k1[A][B] (E) r = k2[I][B]

5. At the same temperature, the ratio of root-mean-square speeds (urms) of helium (M = 4.0 g mol⁻¹) to nitrogen (M = 28.0 g mol⁻¹), i.e., urms(He) / urms(N2), is closest to: (A) 0.38 (B) 0.53 (C) 1.0 (D) 2.6 (E) 7.0

6. For a reversible adiabatic expansion of an ideal gas: (A) q = 0 and ΔSsys ＞ 0. (B) q = 0 and ΔSsys = 0. (C) q ＜ 0 and ΔU = 0. (D) q = 0 and ΔH = 0. (E) w = 0 and ΔU ＜ 0.

7. The vapor pressure of a pure liquid is P1 = 20.0 kPa at T1 = 300 K. Its enthalpy of vaporization is approximately constant at ΔHvap = 40.0 kJ mol⁻¹. Using the Clausius-Clapeyron equation, what is the vapor pressure P2 at T2 = 330 K (closest value)? (A) 25 kPa (B) 33 kPa (C) 45 kPa (D) 60 kPa (E) 90 kPa

9. Which function is not acceptable as an one quantum-mechanical wavefunction on (−∞, ∞)? (A) ψ(x) = Ae⁻|x| (B) ψ(x) = Asin(πx) e⁻x² (C) ψ(x) = A (D) ψ(x) = Ae+x² (E) ψ(x) = A e⁻x²/2

10. A vibrational mode can be approximated as a harmonic oscillator with energy spacing ΔE = hν. At T = 300 K, for ΔE = 6.0 kJ mol⁻¹ and equal degeneracies, what is the ratio Nv=1/Nv=0? (A) 0.02 (B) 0.06 (C) 0.09 (D) 0.22 (E) 0.55

11. For a particle in an one-dimension box of width L with two infinite potential wells, which statement is incorrect? (A) The energy eigenvalues scale as n² (B) The ground state has nonzero energy (C) The wavefunction has n − 1 interior nodes for level n (D) If L → ∞, the energy spectrum becomes continuous (E) Shifting the box in space changes the energy eigenvalues

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