Questions & explanations
1. Given the Latimer diagram for manganese in acid: MnO4- (+0.56 V) -> MnO42- (+2.26 V) -> MnO2 (+0.95 V) -> Mn3+ (+1.51 V) -> Mn2+ (-1.18 V) -> Mn, predict if MnO2 will disproportionate into MnO4- and Mn2+.
To check if MnO2 disproportionates, look at the potentials on both sides of MnO2. The potential to go from MnO2 to MnO4- (via MnO42-) is the sum of steps: MnO2 to Mn3+ is +0.95 V, Mn3+ to Mn2+ is +1.51 V, but we need opposite direction? Actually, the potential for MnO2 to be reduced to Mn2+ is +0.95 V + (+1.51 V) = +2.46 V? That's not correct. We use the Frost diagram rule: the potential for MnO2 to go to Mn2+ is the difference in free energy. Alternatively, check if the potential for reduction to lower state is larger than for oxidation to higher state. For MnO2, reduction to Mn2+ has E° = 1.23 V, oxidation to MnO4- has E° = -0.56 V (reverse sign). Since reduction potential (1.23 V) > oxidation potential (0.56 V), disproportionation is thermodynamically favorable: MnO2 can become Mn2+ and MnO4-.
2. Give an example of a process that satisfies the Clausius inequality but is irreversible.
Consider heat conduction from a hot block to a cold block. The hot block loses heat Q at temperature T_hot, the cold block gains Q at T_cold. The total entropy change is Q/T_cold - Q/T_hot > 0. The Clausius inequality for the combined system gives ∫dQ/T = -Q/T_hot + Q/T_cold > 0, which is not zero. So it satisfies the inequality as a 'less than zero' on the cycle? Actually: For the process, dS > dQ/T locally? In the combined system, the process is irreversible because heat transfers across finite difference. The inequality holds: ΔS_total > 0. Another simple example: free expansion of a gas into vacuum. No heat exchange, so ∫dQ/T=0, but entropy increases, so ΔS > ∫dQ/T, satisfying inequality.
3. Why can a molecule with a center of symmetry not show an infrared absorption for a vibration that is symmetric with respect to that center?
In spectroscopy, a selection rule tells us which transitions are allowed. For a molecule with a center of symmetry, the Laporte rule says that transitions that conserve parity (symmetry with respect to inversion) are forbidden. A vibration symmetric about the center keeps the parity the same. Therefore, such a vibration cannot absorb infrared light because the transition dipole moment (the change in charge distribution during the vibration) is zero. This is why symmetric stretches in centrosymmetric molecules like CO₂ do not appear in the infrared spectrum. They may still be active in Raman spectroscopy, which has different selection rules.
4. Compare the spatial and temporal precision of photopharmacology versus optogenetics.
Both techniques offer high spatial and temporal precision. Optogenetics can control single neurons with sub-second timing using fiber optics or microscopes. Photopharmacology also uses focused light to activate drugs in small tissue regions. However, optogenetics requires genetic modification to express opsins, limiting its use to cells that can be engineered. Photopharmacology does not need genetic changes, so it works on native cells. Temporal precision is similar for both, but photopharmacology often relies on slower thermal cis-to-trans relaxation for off-switching, while optogenetics can use a second light for fast off switching.
5. Why might the initial rates method be less accurate for reactions with multiple steps?
For multi-step reactions, the initial rate may not reflect the true rate-determining step if there is a pre-equilibrium. The initial rate can be affected by rapid formation of intermediates. Also, if the reaction has a complex mechanism, the initial rate might be dominated by a fast step that is not rate-determining later. Additionally, measuring very early rates can be difficult experimentally due to mixing time or detection limits. The integrated method, using data over time, can sometimes handle these complexities better by fitting the full curve. However, initial rates remain a simple and powerful tool for many reactions.
6. Explain how the materials gap makes it difficult to apply model catalyst studies to real catalysts.
The materials gap refers to the difference between simple, well-defined model catalysts (like a single crystal metal) and complex real catalysts (like supported nanoparticles on an oxide). Model catalysts are flat and clean, while real catalysts have many active sites, defects, and interactions with the support. The support can change the metal's electronic properties and provide extra sites. Real catalysts may also have promoters or poisons. Thus, results from a platinum single crystal may not directly apply to platinum nanoparticles on alumina. Scientists use more complex model systems, like thin films, to reduce the gap.
7. Give an example of a study that used a model catalyst to mimic a real catalyst and reduced the materials gap.
Instead of a single crystal, researchers often use nanoparticles deposited on a flat oxide support, like gold nanoparticles on titania. This model system still has well-defined size and shape but includes the support effect. They can then study reactions like CO oxidation under controlled conditions. By varying nanoparticle size, they see how activity changes. This bridges the gap between simple single crystals and complex industrial catalysts. Another example: using a thin oxide film grown on a metal substrate to model the support. These models are simpler than real catalysts but more realistic than single crystals.
8. How are Marcus theory and the Landau-Zener formula used together?
Marcus theory gives the activation barrier for electron transfer, while the Landau-Zener formula provides the probability of crossing from the reactant to product surface when the system reaches the crossing region. The overall rate is proportional to both the probability of reaching the crossing (from Marcus) and the probability of staying on the product surface (from Landau-Zener). For strongly coupled systems, the transfer is adiabatic and Landau-Zener probability is near 1. For weak coupling, the probability is less than 1, and the rate is reduced. This combined approach is called semiclassical Marcus theory.
9. List the main forces that act between colloidal particles.
The main forces are van der Waals attraction, electrostatic repulsion, steric repulsion, depletion attraction, and excluded volume repulsion. Van der Waals forces arise from temporary dipoles and pull particles together. Electrostatic repulsion comes from like charges on particle surfaces pushing them apart. Steric repulsion happens when large molecules coating particles block close approach. Depletion attraction occurs when non-adsorbed molecules push particles together. Excluded volume repulsion keeps particles from overlapping in space. These forces together determine whether particles stay separate or clump.
10. Explain the mechanism: how does a homogeneous catalyst lower the activation energy?
A homogeneous catalyst provides an alternative reaction pathway with a lower activation energy. It does this by forming an intermediate with one or more reactants. This intermediate is more reactive and then decomposes to give the final product and regenerate the catalyst. For example, in the iodide-catalyzed decomposition of hydrogen peroxide, the catalyst first reacts with H2O2 to form an intermediate. This intermediate then reacts with another H2O2 molecule to produce oxygen and water, releasing the catalyst. The overall activation energy is lowered because the intermediate steps have lower energy barriers.
11. What is the grand canonical ensemble?
The grand canonical ensemble is a way to describe a system that can exchange both energy and particles with a large reservoir. It is used when the temperature and chemical potential are fixed, but the particle number can vary. The key quantity is the grand partition function, which sums over all possible numbers of particles and energy states. From it, we can calculate average properties like the average number of particles and the grand potential. The grand potential is related to pressure and volume. This ensemble is useful for systems where particles can be created or destroyed, like in chemical reactions.
12. If a mechanism has an elementary step that is termolecular, what is its rate law?
A termolecular elementary step involves three molecules colliding simultaneously. The rate law is rate = k[A][B][C] for three different molecules, or rate = k[A]^2[B] for two of one and one of another. For example, a step like 2NO + O2 → 2NO2 is sometimes proposed as termolecular, giving rate = k[NO]^2[O2]. Termolecular steps are rare because three molecules colliding at once is unlikely. They are usually only plausible if one of the molecules is a third body that carries away energy, like in recombination reactions. The rate law always reflects the product of the concentrations of the molecules in the step.