Which equilibrium model fits the evidence?
Week 7 · After-midterm problem workshop
Bring one question from Week 6
Open your Lab 13 notes. Which step would you like to see worked again?
Take one minute to write the step and the value or sign you expected. Compare with a partner, then bring one question to the class discussion.
To get started, consider identical electrodes with unequal ion activities. How could you predict which way the reversible cell would run?
Plan for the workshop
| Minutes | Work with a partner |
|---|---|
| 0–15 | Resolve a question from Week 6. |
| 15–50 | Calculate two phase fractions and check which phases coexist. |
| 50–80 | Account for the gas inventory in two vessels. |
| 80–90 | Break. |
| 90–135 | Predict the direction of a cell reaction and check its voltage. |
| 135–172 | Change one condition, run a lab, and exchange results. |
| 172–180 | Complete the Week 7 exit ticket (5 min) and the after-midterm feedback survey (3 min). |
Which fraction of the feed becomes vapor?
At 350 K, an ideal binary mixture has P_1^{\rm sat}=150 kPa and P_2^{\rm sat}=60 kPa. The feed contains z_1=0.40 and the pressure is 90 kPa.
- Use P=x_1P_1^{\rm sat}+(1-x_1)P_2^{\rm sat} and y_1=x_1P_1^{\rm sat}/P to find the liquid and vapor compositions.
- Find the mole fraction of the feed that is vapor, \beta=(z_1-x_1)/(y_1-x_1).
- Open Teaching binary · synthetic in Lab 01. Its vapor pressures are supplied by the lab. Select the ideal liquid, set the feed and conditions above, and select Calculate equilibrium. Verify the overall composition on paper.
What if the compositions came from an experiment?
Suppose the measured values at 350 K are x_1=1/3, y_1=5/9 and P=90 kPa. Assume ideal vapor, negligible Poynting correction, and a pure-liquid standard state.
- Use \gamma_1=y_1P/(x_1P_1^{\rm sat}). Compare the values inferred using 150 and 120 kPa for P_1^{\rm sat}. Which measured quantity changed?
- Before choosing an Antoine correlation, identify its log base, temperature and pressure units, and valid temperature range.
- A model fits these data closely. What additional evidence would you seek across compositions to assess thermodynamic consistency?
Which fraction forms the second liquid?
Now consider a different binary mixture. The one-parameter Margules model has A=3 and the overall mole fraction is z_1=0.18.
- In Lab 04 choose Margules · 1 parameter, enter A=3 and z_1=0.18, then select Find equilibrium. Read both liquid compositions from the common tangent.
- Use the lever rule to calculate the mole fraction of the feed in the liquid richer in component 1. Reconstruct the feed from the two endpoint compositions.
- Compare the common tangent with the homogeneous curve at z_1=0.18. Which state has lower Gibbs energy?
Check the phases against the feed
| Result | Independent check |
|---|---|
| Flash at 90 kPa. x_1=0.3333, y_1=0.5556, \beta=0.3000. | 0.70(0.3333)+0.30(0.5556)\approx0.4000. |
| LLE with A=3. The liquids have x_1=0.07072 and 0.92928. The fraction in the richer liquid is 0.1273. | 0.8727(0.07072)+0.1273(0.92928)\approx0.1800. The common tangent lies below the homogeneous liquid at this feed. |
Week 2–3 data check. \gamma_1=1.00 or 1.25. The measured x_1,y_1,P,T are the same. A small fitting residual measures agreement with the chosen model. Apply a Gibbs–Duhem consistency test appropriate to the experimental path and inspect the local residuals.
Can the vessel reach 10 kPa?
A closed vessel contains 0.0086712 mol of A at 300 K. The free gas volume is 1.00 L. Treat the gas as ideal and use R=8.31446 L kPa mol^{-1} K^{-1}.
Vessel A. Pure solid A can sublime. Its sublimation pressure is 10.0 kPa at this temperature. Calculate how many moles are needed in the gas to reach this pressure. How many remain as bulk solid?
Vessel B. The inventory of A can occupy the gas or adsorb onto 10.0 g of a separate adsorbent. Use absolute Langmuir uptake q=q_s bP_A/(1+bP_A), with q_s=2.00 mol kg^{-1} and b=0.100 kPa^{-1}. Try P_A=5.00 kPa. Does the inventory close?
Test what happens when adsorbent mass doubles
In Lab 11, select Reset synthetic parameters. Use Langmuir at 300 K, 1 L free gas volume, 10 g adsorbent, 0.0086712 mol total adsorbate, and zero inert gas.
- Select Compare & solve vessel and record P_A, the gas moles, and the adsorbed moles.
- Predict the pressure at 20 g relative to its value at 10 g. Change only the adsorbent mass, recalculate, and check n_A^g+m_sq.
- Compare the roles of the two solids. What sets P_A while bulk solid A remains in vessel A? Which equation sets it in vessel B?
Follow the material in each vessel
| Vessel | P_A (kPa) | Gas (mol) | Bulk solid or adsorbed A (mol) |
|---|---|---|---|
| A · Pure solid present | 10.00 | 0.0040091 | 0.0046621 in the solid |
| B · 10 g adsorbent | 5.000 | 0.0020045 | 0.0066667 adsorbed |
| B · 20 g adsorbent | 2.389 | 0.0009578 | 0.0077134 adsorbed |
For vessel B at 10 g, q(5.00)=0.6667 mol kg^{-1}. Multiplying by 0.0100 kg gives the amount on the surface. Check that each row accounts for 0.0086712 mol of A.
Which electrode would gain metal?
At 298.15 K, both electrodes use \mathrm{M^{2+}/M}. Take oxidation on the left, reduction on the right, and zero junction potential. If the circuit is closed, deposition can occur at the electrode undergoing reduction. Use a_L=0.040 and a_R=0.40.
- Write the reaction, keeping each species in its own compartment. Which ion activity belongs in the numerator of Q?
- The standard cell potential is zero. What is K? Compare Q with K to predict the direction and which electrode gains metal.
- Calculate E=E^\circ-(0.05915935\ {\rm V}/n)\log_{10}Q and \Delta_rG=-nFE. Use n=2 and F=96485 C mol^{-1}.
Run the three cell conditions
In Lab 13, select Concentration-cell example, then Direct dimensionless activities. In each \mathrm{M^{2+}} row, edit the value column. Keep \gamma=1 and the pure-metal entries at 1.
| Case | Left activity | Right activity | Before selecting Calculate cell |
|---|---|---|---|
| Original | 0.040 | 0.40 | Record your predicted sign and voltage. |
| Exchanged | 0.40 | 0.040 | Predict which electrode now gains metal. |
| Equal | 0.40 | 0.40 | Predict Q and E. |
Compare the lab’s potential with your own RT\ln(Q/K) check on the reaction Gibbs energy.
Check the direction before rounding
For left oxidation and right reduction, the dissolved ion is produced on the left and consumed on the right, so Q=a_L/a_R. Identical electrodes give E^\circ=0 and K=1.
| Case | Q | E (mV) | \Delta_rG (kJ per mol of reaction as written) | Deposition if the cell runs |
|---|---|---|---|---|
| Original | 0.10 | +29.58 | -5.708 | Right electrode |
| Exchanged | 10 | -29.58 | +5.708 | Left electrode |
| Equal | 1 | 0 | 0 | No net deposition |
For the original case, RT\ln(0.10)=-5.708 kJ mol^{-1}. Write one sentence connecting that sign to the observed cell polarity.
Change one condition and exchange your result
Choose one case and predict the outcome before touching its controls.
| Choose one lab | Change this condition |
|---|---|
| Lab 01 · Flash | Raise P from 90 to 100 kPa at the same T and z_1. Which phase or phases remain? |
| Lab 11 · Adsorption | Keep 20 g of adsorbent and double the total inventory. Predict the direction of the pressure change. |
| Lab 13 · Cell | In the original case, double both ion activities. Does the activity ratio change? Predict the voltage. |
Allow 20 minutes to calculate and test. Use 10 minutes to exchange cases with another pair and reproduce their result. In the final 7 minutes, share one result and one check the class can repeat.
Week 7 exit ticket and after-midterm feedback
5 min · Exit ticket for today’s workshop
Choose one case from today. Give a result with its units, or state that it is dimensionless. Explain what it means and name an independent check or a limitation.
3 min · Feedback on all of Weeks 1–7
Rate the overall pace after the midterm. Which resource helped you most? Name one thing to keep or change.
Complete both parts in the same form.
