Take-home Exam 1: Mini-projects

CHEM 291-A · Chemical Thinking

Project 1: The sample that looks too dark

Chemical question: Can you report both unknown concentrations without using a calibration beyond the range tested?

Recognition prompts

• What do the standard solutions hold fixed, and what changes from vial to vial?
• Why does adding water change concentration but not the amount of solute?
• Which observed quantity is an instrument reading, and which is the chemical quantity you want?

A colorimeter was calibrated with five known concentrations of one dissolved dye. Absorbance is dimensionless. The numbers below are simulated, chosen so you can examine the reasoning.

Concentration (mM)	0.0	1.0	2.0	3.0	4.0
Absorbance	0.020	0.160	0.300	0.440	0.580

Unknown U: absorbance 0.370. Unknown V: absorbance 0.720. Equal volumes of V and pure water are then mixed; the diluted V reads 0.370.

1. Make the observation visible. Plot the five standards as absorbance versus concentration; mark U and the undiluted V as horizontal reading levels.
2. Build the simplest useful model. Find the slope and baseline, with units where appropriate. Write absorbance as a function of concentration and verify one standard you did not use to find the slope.
3. Find the unknowns. Determine U from its reading. Determine V from its diluted reading, explicitly showing the dilution calculation.
4. Make a decision. Explain why calculating a number from V = 0.720 using the line does not, by itself, justify reporting it as calibrated.
5. Test your answer. Predict the absorbance of the diluted V from your reported original concentration and explain why U and diluted V can have identical readings.

Project 2: The reaction that halves

Chemical question: What mathematical description explains a constant fractional loss, and when is a short polynomial an acceptable shortcut?

Recognition prompts

• Do equal five-minute intervals remove equal amounts or equal fractions?
• What does a straight line hide or reveal when you change what is on the vertical axis?
• Why should a local approximation be checked before using it for a longer time?

A dissolved species A disappears during a simplified reaction. The following simulated concentration observations are exact for this teaching model.

Time (min)	0	5	10	15
[A] (M)	0.800	0.400	0.200	0.100

For this project, use the model [A](t) = [A]0 exp(−kt). Use a dimensionless ratio inside the logarithm. An exponent of −0.10 corresponds to a short time interval Δt = 0.10/k. Compare exp(−0.10) with 1 − 0.10 and 1 − 0.10 + (0.10)²/2.

1. Recognize the pattern. Compute successive differences and successive ratios. Decide whether a straight line in ordinary [A] versus time is a suitable model.
2. Change the representation. Plot or tabulate ln([A]/[A]₀) against time. Extract k, state its units, and explain the meaning of the negative slope.
3. Predict. Estimate [A] at 12 min from the model; check that your prediction lies between the 10- and 15-minute observations.
4. Approximate locally. Evaluate the linear and quadratic approximations to exp(−0.10), compare each to your calculator, and decide which achieves absolute error below 0.001.
5. State a boundary. Explain why an approximation tested at −0.10 should not be assumed accurate at −2.

Project 3: The dipole and the fixed meter

Chemical question: Why can turning an unchanged molecule change what a fixed directional sensor measures?

Recognition prompts

• Which parts of the molecular description are lengths, and which are directions?
• Does changing the direction of the whole molecule change its internal geometry?
• If the meter and molecule both turn together, what should remain the same?

Model water in a 2-D drawing: O = (0,0), H₁ = (0.80, 0.60) Å, H₂ = (0.80, −0.60) Å. Assign each O→H bond a dipole contribution of magnitude 1.50 D, directed along its bond. This is an idealized vector model, not a measured water dipole.

Object	x coordinate / component	y coordinate / component
O	0.00 Å	0.00 Å
H1	+0.80 Å	+0.60 Å
H2	+0.80 Å	−0.60 Å

The meter reads the dot product of the total dipole with a fixed unit sensing direction ex = (1,0). The molecule is rotated 90° counterclockwise using the matrix R = [[0, −1], [1, 0]].

1. Draw before computing. Sketch both O→H vectors, their individual dipole contributions, and the fixed +x meter direction.
2. Add the physical contributions. Find each bond unit vector, each bond dipole vector, and the total dipole μ. Give components and magnitude in D.
3. Let a matrix act. Multiply Rμ. Then separately compute Rμ₁ + Rμ₂ and compare with R(μ₁ + μ₂).
4. Ask what the meter reads. Compute μ·e_x before rotation and (Rμ)·e_x after rotation.
5. Test what stays the same. Check the dipole magnitude before and after; now rotate the sensing direction by R too and recompute the dot product. Explain the difference between turning the molecule and merely redescribing the same geometry.

Project 4: Two colors, two readings

Chemical question: When two substances both affect both signals, can two instrument readings distinguish their amounts?

Recognition prompts

• What is mixed physically, and what is mixed in the instrument response?
• Why can the reading at one wavelength be explained by several different mixtures?
• What does the row-by-column product actually calculate?

A sample contains dyes P and Q. A detector reads absorbance at wavelengths λ₁ and λ₂. For this simulated linear model, the calibration sensitivities (in mM⁻¹) are:

Wavelength	Effect of 1 mM P	Effect of 1 mM Q
λ1	0.40	0.10
λ2	0.15	0.30

Three prepared controls contain (P,Q) in mM: C₁ = (2,0), C₂ = (0,2), C₃ = (2,1). A blinded mixture produces the readings (A₁, A₂) = (0.90, 0.60). Absorbance is dimensionless.

1. Draw the coupling. Sketch arrows from both dyes to both wavelength readings; explain why a single calibration line for P would not suffice.
2. Represent the mapping. Form the 2×2 sensitivity matrix M and a 2×3 concentration matrix C whose columns are the three controls. Label rows, columns, and units.
3. Multiply with meaning. Compute MC by row-by-column multiplication. Explain one product entry in a complete chemical sentence.
4. Solve the blind mixture. Write the two simultaneous equations and solve by substitution or elimination. Verify your concentrations by multiplying M by your result.
5. Expose a tempting error. If you pretend Q does not absorb at λ₁, what P concentration would you report? Compare it with the two-wavelength result and explain the discrepancy.

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