Appendix C — Problem Set 1 | Seeing the Structure

Companion to the Prologue — Seeing

Purpose

Before calculating, decide what kind of problem reality has given you. The central objects in this course are COLLECTION, ARRANGEMENT, DIRECTION, PROXIMITY, SAMENESS, CHANGE, RATE, ACCUMULATION, and SPREAD.

A. Recognition

1. Name the primitive

For each question, choose the primary primitive and justify the choice in one sentence.

  1. How many distinguishable ways can four ligands occupy four sites around a metal center?
  2. What component of an electric field lies along an O–H bond?
  3. Does a concentration function approach a finite value as t\to\infty?
  4. Which molecular property is unchanged when the coordinate axes rotate?
  5. What is the instantaneous disappearance speed of a reactant?
  6. What total charge passed through an electrode during 30 minutes?
  7. How broadly are molecular speeds distributed at 500 K?

2. Object, representation, or process?

Classify each statement as primarily about a state, a process, a constraint, or a representation. Some cases admit two labels; if so, state which is primary.

  1. “The sample contains 0.150 mol of ethanol at 298 K.”
  2. “The reaction vessel is sealed and rigid.”
  3. “The same molecular geometry is reported in Cartesian and spherical coordinates.”
  4. “The temperature rises from 298 K to 315 K.”
  5. “Total carbon atoms are conserved.”

B. Quantitative structure

3. State vector for a gas sample

A rigid vessel contains a single gas phase described by \mathbf{s}=(n,T,V,p)=(0.500\ \mathrm{mol},300.0\ \mathrm K,12.5\ \mathrm L,0.985\ \mathrm{atm}).

  1. Which entries are extensive and which are intensive?
  2. The gas is divided equally between two identical vessels without changing temperature. Write the state vector for either half, assuming ideal behavior.
  3. Identify the quantities that provide SAMENESS between the original equilibrium state and either half, and those that do not.

4. Description versus physical change

The position of an atom is \mathbf r=(1.20,-0.50,0.80)\ \text{Å}. A second investigator moves the coordinate origin by \mathbf a=(0.20,0.10,-0.30)\ \text{Å}, so that \mathbf r'=\mathbf r-\mathbf a.

  1. Calculate \mathbf r'.
  2. Did the atom move? Defend your answer.
  3. Give one atom-to-atom quantity that must remain unchanged under this coordinate translation.

5. Dimensional constraint

Suppose a proposed diffusion timescale is \tau=L/D, where L has units m and D has units \mathrm{m^2\,s^{-1}}.

  1. Test the dimensions of the proposal.
  2. Construct a dimensionally valid monomial \tau=L^aD^b. Solve for a and b.
  3. Estimate \tau for L=10\ \mu\mathrm m and D=1.0\times10^{-9}\ \mathrm{m^2\,s^{-1}}.

6. Scale decides the model

A solute diffuses with D=8.0\times10^{-10}\ \mathrm{m^2\,s^{-1}}. Use \tau\sim L^2/D to estimate diffusion times over:

  1. L=1.0\ \mathrm{nm},
  2. L=10\ \mu\mathrm m,
  3. L=1.0\ \mathrm{cm}.

Explain why “diffusion is fast” is not a scientific statement until a length scale is supplied.

C. Model judgment

7. One graph, two claims

An instrument gives A=0.013+0.842c, where absorbance A is dimensionless and concentration c is in \mathrm{mmol\,L^{-1}}.

  1. State the units of the intercept and slope.
  2. Find c when A=0.350.
  3. A student says “doubling concentration doubles absorbance.” Is that exactly true for this calibration? Show the relevant test.

8. Synthesis: choose before calculating

For each task below, name the object, the governing constraint, and the operation you would use. Do not perform the calculation.

  1. Balance a redox equation.
  2. Determine a molecular dipole from bond dipoles.
  3. Recover a rate constant from concentration–time data.
  4. Determine the probability that a molecule exceeds an activation energy.
  5. Compute work from a pressure–volume path.