Chapter 7 Self-test questions

. A system is made up of a large number of particles that can each exist in three different quantum states 0, 1 and 2, all singly degenerate (g = 1), with energies ε₀ = 0 kJ mol^-1, ε₁ = 6 kJ mol^-1, and ε₂ = 12 kJ mol^-1. At a temperature of 320 K, what proportion of the particles will occupy the state 2?

1%

10%

33%

100%

. The vibrational partition function for a molecule of Br₂ (vibrational wavenumber: 323 cm^-1) can be calculated in the harmonic oscillator approximation. At 500 K, what is q_vib?

1

108

1.65

4.3

. For a reaction having a standard free energy change at 298 K of -30 kJ mol^-1, what is K at that temperature?

1.01

1.8 x 10⁵

5.5 x 10^-6

30

. A molecular dynamics simulation is carried out for a protein solvated in water, using NVT conditions and T = 310 K. Two different conformations, A and B, are visited during the simulation. At 1 ps intervals, the structure is assigned to either A or B, and after a simulation length of 24.6 ns, the number of times that conformation A was visited is n_A = 420. Assuming that equilibrium has been reached, what is the free energy of B relative to A?

-10.5 kJ mol^-1

20.6 kJ mol^-1

7.3 kJ mol^-1

-10,500 kJ mol^-1

. Enhanced sampling methods are often needed in conjunction with molecular dynamics or Monte Carlo simulations, because…

…they are newer methods so they are expected to yield less approximate results.

…for many systems, the transitions needed to reach equilibrium are 'rare events' that barely or never occur with normal sampling.

…computer simulation will never lead to the system crossing a transition state, as that is a quantum-mechanical event.

…normal sampling methods yield the wrong distribution of structures even if the simulation is long enough that key transitions occur often enough.

. Two candidate drug molecules A and B are being assessed. The free energy perturbation method is used to calculate their relative free energy of solvation in water, and their relative free energy of interaction with the relevant drug-binding site of a protein, in each case at 310 K. The calculations show that converting A to B in water leads to a free energy change of -1.4 kJ mol^-1, while converting A to B in the protein leads to a free energy change of -8.2 kJ mol^-1. Compared to A, B can be predicted to:

Bind roughly 10 times more strongly to the protein.

Bind roughly 100 times less strongly to the protein.

Bind roughly equally strongly to the protein.

Bind roughly 10,000 times more strongly to the protein.

. The dimer of nitrogen dioxide, N₂O₄, is in equilibrium with NO₂. The free energy of activation ΔG^‡ for the dissociation reaction at 298 K is calculated to be 35 kJ mol^-1. What is k_dissoc?

4.4 x 10⁶ M^-1 s^-1.

4.4 x 10⁶ s^-1.

6.1 x 10¹² s^-1.

0

. Concerning the molecular dynamics simulation for liquid butane described on pp. 113-115 and illustrated in Figs. 7.1 and 7.2, which of the following statements is correct?

The forcefield used suggests that butane should be chiral, since one of the mirror image gauche forms is observed more than the other one.

Anti-butane is lower in energy and so it is the only form present at most temperatures.

Isomerization of anti-butane to each of the two gauche forms occurs faster than the direct conversion of the gauche form with θ = 60° to the other gauche form with θ = 300°.

At 120 ps, the simulation time is too short to reach any conclusions about free energies in butane.

. M06/6-311G(d) calculations are performed for H₂, C₂H₄ and C₂H₆. Geometry optimization returns total energies of respectively -1.1686, -78.5301 and -79.7677 hartree (1 hartree = 2625.5 kJ mol^-1). Vibrational frequencies are computed, and statistical mechanics calculations are performed at 298 K using the rigid rotor, harmonic oscillator approximation, yielding G – G(0) values of respectively -0.0016, 0.0295 and 0.0528 hartree. What is the predicted ΔG° for the hydrogenation of C₂H₄?

-24 kJ mol^-1

-148 kJ mol^-1

-116 kJ mol^-1

-3200 kJ mol^-1

. A molecule of nitrogen oxide has a bond length of 1.151 Å. Using m_Nm_O/(m_N+m_O) = μ = 1.240 x 10^-26 kg and σ = 1, calculate the rotational partition function for this molecule at T = 350 K.

12

143

1

56

Chapter 7 Self-test questions

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