WEBVTT 01-01 00:00:00.180 --> 00:00:01.823 Welcome to Week Three. 01-02 00:00:02.003 --> 00:00:06.696 This lesson begins with a question raised by our work in Lab Eight. 01-03 00:00:06.876 --> 00:00:13.916 If a model fits the observations closely, how much confidence should we place in those observations? 01-04 00:00:14.096 --> 00:00:24.229 We will examine thermodynamic consistency first, then move to a different question: when does a liquid mixture separate into two stable liquids? 01-05 00:00:24.409 --> 00:00:28.676 The lesson replaces our classroom session during the holiday. 01-06 00:00:28.856 --> 00:00:38.627 Allow about three hours altogether, including watching, calculations, Lab investigations, a break, and the closing reflection. 01-07 00:00:38.807 --> 00:00:41.879 The recording is only one part of that time. 01-08 00:00:42.059 --> 00:00:46.773 Keep your own answers and pause whenever an activity is assigned. 02-01 00:00:49.303 --> 00:00:54.573 Start by opening the four-page practice worksheet and the three-page Lab guide. 02-02 00:00:54.753 --> 00:00:57.910 Their links are on the lesson page and this slide. 02-03 00:00:58.090 --> 00:01:05.339 You can print the worksheet, annotate the PDF, or write in a notebook using the same question labels. 02-04 00:01:05.519 --> 00:01:07.482 Have a calculator ready. 02-05 00:01:07.662 --> 00:01:12.953 Keep the slide deck available beside the video so that you can follow its links. 02-06 00:01:13.133 --> 00:01:16.098 The worked solutions are a separate document. 02-07 00:01:16.278 --> 00:01:21.846 Open the relevant section only after you have attempted the corresponding questions. 02-08 00:01:22.026 --> 00:01:28.831 The background reading is optional; you do not need to complete another reading assignment before starting. 02-09 00:01:29.011 --> 00:01:33.513 All preparation belongs inside today's first fifteen minutes. 02-10 00:01:33.693 --> 00:01:38.535 Pause here briefly to arrange the files, then continue to the schedule. 02-11 00:01:38.715 --> 00:01:42.129 You can work through the entire lesson independently. 03-01 00:01:44.659 --> 00:01:49.715 The times on the slides are elapsed study time, not video timestamps. 03-02 00:01:49.895 --> 00:01:53.159 They include both listening and doing the work. 03-03 00:01:53.339 --> 00:01:59.525 During the first fifteen minutes, review Lab Eight and calculate one activity coefficient. 03-04 00:01:59.705 --> 00:02:04.820 Next, spend twenty minutes on the Gibbs-Duhem constraint and its assumptions. 03-05 00:02:05.000 --> 00:02:09.309 A fifteen-minute hand calculation prepares you for Lab Three. 03-06 00:02:09.489 --> 00:02:14.439 The Lab comparison and interpretation take you to minute ninety. 03-07 00:02:14.619 --> 00:02:16.816 Then take a ten-minute break. 03-08 00:02:16.996 --> 00:02:24.399 From minute one hundred, examine stability and the lever rule before carrying out the Lab Four investigation. 03-09 00:02:24.579 --> 00:02:29.799 The final fifteen minutes are for self-review, the exit ticket, and feedback. 03-10 00:02:29.979 --> 00:02:32.987 Use these times as approximate guides. 03-11 00:02:33.167 --> 00:02:40.804 A pause for calculation uses the remaining time in its block; it is not added on top of three hours. 03-12 00:02:40.984 --> 00:02:44.781 Move forward when you have recorded the required evidence. 04-01 00:02:47.311 --> 00:02:50.895 Turn to page one, questions A one through A three. 04-02 00:02:51.075 --> 00:02:55.052 First, define x, y, z, and gamma in your own words. 04-03 00:02:55.232 --> 00:02:59.691 Keep phase composition separate from the overall mixture composition. 04-04 00:02:59.871 --> 00:03:05.476 Next, calculate gamma one from the supplied synthetic Lab Eight observation. 04-05 00:03:05.656 --> 00:03:14.381 Multiply the vapor mole fraction by total pressure, then divide by the liquid mole fraction times the saturation pressure. 04-06 00:03:14.561 --> 00:03:18.487 Both pressures are in kilopascals, so their units cancel. 04-07 00:03:18.667 --> 00:03:25.195 Use the full values printed on the worksheet rather than trying to copy every digit from the narration. 04-08 00:03:25.375 --> 00:03:29.279 Compare your answer with the exponential given in A two. 04-09 00:03:29.459 --> 00:03:35.112 Finally, answer whether a small fitting error proves thermodynamic consistency. 04-10 00:03:35.292 --> 00:03:41.884 If you need a reminder, use one relevant chapter of the Lab Eight walkthrough within this review block. 04-11 00:03:42.064 --> 00:03:47.013 Pause now and finish your attempt before the next slide reveals the check. 05-01 00:03:49.543 --> 00:03:52.317 Compare these statements with your answers. 05-02 00:03:52.497 --> 00:04:00.859 X one describes component one's mole fraction in the liquid, and y one describes its mole fraction in the vapor. 05-03 00:04:01.039 --> 00:04:04.090 Z one refers to the overall mixture. 05-04 00:04:04.270 --> 00:04:14.873 Gamma one is an activity coefficient: multiplying it by the liquid mole fraction gives liquid activity on the chosen pure-liquid standard state. 05-05 00:04:15.053 --> 00:04:20.471 The calculated gamma is about one point four nine one five four five. 05-06 00:04:20.651 --> 00:04:29.718 Taking the exponential of the rounded logarithm gives essentially the same answer; the final-digit difference comes from rounding. 05-07 00:04:29.898 --> 00:04:35.103 This inference uses ideal vapor and a negligible Poynting correction. 05-08 00:04:35.283 --> 00:04:42.259 For A three, a close fit shows that the selected model reproduces the observations under its assumptions. 05-09 00:04:42.439 --> 00:04:48.839 Consistency asks whether the inferred properties obey a thermodynamic relationship. 05-10 00:04:49.019 --> 00:04:53.499 Keep any correction beside your original answer, then continue. 06-01 00:04:56.029 --> 00:05:00.445 Now consider a binary liquid at fixed temperature and pressure. 06-02 00:05:00.625 --> 00:05:05.745 The Gibbs-Duhem equation connects changes in its two activity coefficients. 06-03 00:05:05.925 --> 00:05:12.005 If one coefficient changes with composition, the other cannot be chosen independently. 06-04 00:05:12.185 --> 00:05:16.943 Their mole-fraction-weighted logarithmic changes must sum to zero. 06-05 00:05:17.123 --> 00:05:25.827 This is a constraint on a collection of composition-dependent properties, rather than a test of one isolated gamma value. 06-06 00:05:26.007 --> 00:05:33.879 The second equation defines dimensionless excess Gibbs energy by weighting the two log activity coefficients. 06-07 00:05:34.059 --> 00:05:43.445 Under the same fixed conditions, the slope of that excess-energy curve equals the logarithm of gamma one divided by gamma two. 06-08 00:05:43.625 --> 00:05:47.935 That slope relationship will give us a local diagnostic. 06-09 00:05:48.115 --> 00:05:55.005 We can compare the slope inferred from the weighted properties with the log ratio at the same composition. 06-10 00:05:55.185 --> 00:05:57.660 Complete B one on page one. 06-11 00:05:57.840 --> 00:06:02.363 State the fixed conditions and explain the connection in one sentence. 07-01 00:06:04.893 --> 00:06:11.997 Experimental vapor-liquid data do not always follow a path with both temperature and pressure fixed. 07-02 00:06:12.177 --> 00:06:21.009 In an isothermal experiment, temperature stays constant while the equilibrium pressure generally changes with composition. 07-03 00:06:21.189 --> 00:06:28.442 The more general relationship therefore contains a pressure term involving molar excess volume. 07-04 00:06:28.622 --> 00:06:32.526 Our teaching calculation neglects that contribution. 07-05 00:06:32.706 --> 00:06:36.653 This approximation must accompany the interpretation. 07-06 00:06:36.833 --> 00:06:41.803 In an isobaric experiment, pressure is fixed but temperature changes. 07-07 00:06:41.983 --> 00:06:46.314 The local relationship then needs an excess-enthalpy correction. 07-08 00:06:46.494 --> 00:06:54.793 For real measurements, we need independent excess-enthalpy information before assigning that corrected local verdict. 07-09 00:06:54.973 --> 00:06:59.303 A missing correction is not evidence that the measurements failed. 07-10 00:06:59.483 --> 00:07:06.417 Lab Three also supplies synthetic excess enthalpy for its generated isobaric cases. 07-11 00:07:06.597 --> 00:07:12.847 That value comes from the generator; it is not an independent calorimetric measurement. 07-12 00:07:13.027 --> 00:07:20.942 Complete B two: identify the omitted pressure contribution and the property needed for the temperature correction. 08-01 00:07:23.472 --> 00:07:27.184 The four approaches ask related but different questions. 08-02 00:07:27.364 --> 00:07:34.425 The isothermal area method integrates the log-activity ratio across the full composition range. 08-03 00:07:34.605 --> 00:07:37.891 Contributions of opposite sign can cancel. 08-04 00:07:38.071 --> 00:07:49.299 The local differential method compares a slope relationship at individual interior compositions, but numerical differentiation is sensitive to noise and spacing. 08-05 00:07:49.479 --> 00:07:55.389 Herington supplies an empirical allowance for an isobaric boiling- temperature range. 08-06 00:07:55.569 --> 00:07:59.537 That allowance does not replace measured excess enthalpy. 08-07 00:07:59.717 --> 00:08:08.442 The Van Ness-type approach asks whether a selected activity model can reproduce the observed pressure and vapor composition. 08-08 00:08:08.622 --> 00:08:17.027 Before interpreting any result, check whether the method applies to the data path and whether the required information is available. 08-09 00:08:17.207 --> 00:08:22.221 Not applicable, not assessed, and not run are not synonyms for failed. 08-10 00:08:22.401 --> 00:08:25.003 They are also not successful tests. 08-11 00:08:25.183 --> 00:08:27.146 Read the stated reason. 08-12 00:08:27.326 --> 00:08:32.211 Complete B three on page one, using guide page two if needed. 08-13 00:08:32.391 --> 00:08:35.207 You do not need to operate the Lab yet. 09-01 00:08:37.737 --> 00:08:42.388 We will first make the area calculation small enough to check by hand. 09-02 00:08:42.568 --> 00:08:53.683 Page one, section C, gives five equally spaced compositions and five values of L, the natural logarithm of the activity-coefficient ratio. 09-03 00:08:53.863 --> 00:08:57.703 Treat neighboring points as the ends of a straight segment. 09-04 00:08:57.883 --> 00:09:03.237 For each interval, multiply its width by the average of the two endpoint values. 09-05 00:09:03.417 --> 00:09:08.665 Keep the sign: a segment below zero contributes a negative signed area. 09-06 00:09:08.845 --> 00:09:11.811 Add the four contributions for C one. 09-07 00:09:11.991 --> 00:09:19.116 For C two, increase every L value by zero point two and calculate the signed sum again. 09-08 00:09:19.296 --> 00:09:25.693 For C three, explain what this integral cannot establish about individual observations. 09-09 00:09:25.873 --> 00:09:31.164 These are synthetic values whose endpoint limits are supplied analytically. 09-10 00:09:31.344 --> 00:09:34.266 We are not inventing endpoint measurements. 09-11 00:09:34.446 --> 00:09:40.548 Stop the video here and complete C one through C three before opening the solution slide. 10-01 00:09:43.078 --> 00:09:56.076 The four baseline contributions are zero point three seven five, zero point one two five, minus zero point one two five, and minus zero point three seven five. 10-02 00:09:56.256 --> 00:10:03.872 The positive area is one half, the signed negative area is minus one half, and the total is zero. 10-03 00:10:04.052 --> 00:10:13.161 Raising the entire curve by zero point two adds a rectangle of height zero point two over a composition interval of one. 10-04 00:10:13.341 --> 00:10:17.096 The new signed integral is therefore zero point two. 10-05 00:10:17.276 --> 00:10:23.584 Lab Three calls the signed integral I and also reports a normalized score, D. 10-06 00:10:23.764 --> 00:10:31.166 D compares the magnitudes of the positive and negative areas; it is not the signed integral itself. 10-07 00:10:31.346 --> 00:10:35.762 The teaching area criterion is D below five percent. 10-08 00:10:35.942 --> 00:10:40.486 If both magnitudes vanish, that normalization is undefined. 10-09 00:10:40.666 --> 00:10:43.760 Check your work and keep the distinction visible. 10-10 00:10:43.940 --> 00:10:50.894 Real interior measurements still cannot establish missing endpoint intervals without an extra assumption. 11-01 00:10:53.424 --> 00:10:58.608 Open Lab Three and guide page one, with worksheet page two beside them. 11-02 00:10:58.788 --> 00:11:04.079 Before changing a control, understand what the local residual represents. 11-03 00:11:04.259 --> 00:11:10.766 It subtracts the log-activity ratio from the slope of dimensionless excess Gibbs energy. 11-04 00:11:10.946 --> 00:11:18.846 Under our isothermal pressure-term approximation, a consistent continuous description makes that difference vanish. 11-05 00:11:19.026 --> 00:11:22.760 The Lab estimates slopes numerically at interior points. 11-06 00:11:22.940 --> 00:11:30.001 Its teaching score is one hundred times the mean absolute residual, with a threshold of five. 11-07 00:11:30.181 --> 00:11:36.282 That scaled score is not a probability or a percentage of incorrect observations. 11-08 00:11:36.462 --> 00:11:43.225 Numerical differentiation can create a small residual even for clean generated data. 11-09 00:11:43.405 --> 00:11:46.699 Also, an average can dilute a narrow peak. 11-10 00:11:46.879 --> 00:11:49.482 Inspect the shape as well as the score. 11-11 00:11:49.662 --> 00:11:59.283 This finite-difference diagnostic is distinct from the fitted Padé slope used in the NIST and Kojima point-test implementation. 12-01 00:12:01.813 --> 00:12:05.376 For D one, make the initial state reproducible. 12-02 00:12:05.556 --> 00:12:09.934 Select Synthetic, isothermal, three hundred fifty kelvin. 12-03 00:12:10.114 --> 00:12:15.058 Choose Consistent reference, and explicitly select forty-one points. 12-04 00:12:15.238 --> 00:12:20.528 The page normally starts with eighty-one points, so do not skip that change. 12-05 00:12:20.708 --> 00:12:25.615 Under Inspect, choose Log-activity ratio and integrated area. 12-06 00:12:25.795 --> 00:12:30.894 Record the signed integral, the normalized D score, and the area status. 12-07 00:12:31.074 --> 00:12:38.410 Then switch Inspect to Local Gibbs-Duhem residual and record its score, status, and general shape. 12-08 00:12:38.590 --> 00:12:41.875 The results update when you change these controls. 12-09 00:12:42.055 --> 00:12:46.663 Amplitude is disabled for the clean reference, which is expected. 12-10 00:12:46.843 --> 00:12:52.923 These data use Lab Three's own generator, not the Lab Eight demonstration dataset. 12-11 00:12:53.103 --> 00:12:57.818 Save the reference evidence with Export all results or a screenshot. 12-12 00:12:57.998 --> 00:13:00.664 Rename the JSON clearly. 12-13 00:13:00.844 --> 00:13:04.023 Wait until D four to run the model fit. 12-14 00:13:04.203 --> 00:13:08.512 Pause now and complete D one before changing the perturbation. 13-01 00:13:11.042 --> 00:13:16.440 For D two, keep the temperature path and forty-one-point sampling unchanged. 13-02 00:13:16.620 --> 00:13:21.569 Change Controlled perturbation to Canceling errors, zero net area. 13-03 00:13:21.749 --> 00:13:25.226 Check that the amplitude is zero point two five. 13-04 00:13:25.406 --> 00:13:30.654 This construction adds a cosine-shaped term to the log-activity ratio. 13-05 00:13:30.834 --> 00:13:39.226 Its integral across the full composition range is exactly zero, although the term is not zero at every composition. 13-06 00:13:39.406 --> 00:13:46.019 The weighted changes to the two log activity coefficients cancel in the excess-energy value. 13-07 00:13:46.199 --> 00:13:51.447 This is designed to separate a global integral from a local relationship. 13-08 00:13:51.627 --> 00:13:55.766 Inspect the area plot and then the local residual plot. 13-09 00:13:55.946 --> 00:14:02.495 Record their scores and statuses without turning them into a single overall pass or fail. 13-10 00:14:02.675 --> 00:14:05.214 Explain what each actually tests. 13-11 00:14:05.394 --> 00:14:10.236 Save this case under a different filename so that you retain the reference. 13-12 00:14:10.416 --> 00:14:15.003 Pause here and finish D two before moving to the localized defect. 14-01 00:14:17.533 --> 00:14:25.384 For D three, change only the perturbation to Localized defect near x one equals zero point three five. 14-02 00:14:25.564 --> 00:14:29.852 Keep amplitude zero point two five and forty-one points. 14-03 00:14:30.032 --> 00:14:35.493 This time the discrepancy occupies a narrow part of the composition range. 14-04 00:14:35.673 --> 00:14:42.286 Locate it on the local residual plot, then compare its appearance with the single averaged score. 14-05 00:14:42.466 --> 00:14:51.064 For these exact settings, both aggregate criteria are met, even though the residual plot shows the localized defect. 14-06 00:14:51.244 --> 00:14:57.132 Do not assume that deliberately introducing a defect must make every badge change. 14-07 00:14:57.312 --> 00:15:02.240 Record what the Lab actually reports and explain the scope of the average. 14-08 00:15:02.420 --> 00:15:04.958 Save the localized case separately. 14-09 00:15:05.138 --> 00:15:13.443 If you have spare time inside this activity block, compare one hundred sixty-one points, then return to forty-one. 14-10 00:15:13.623 --> 00:15:21.538 That optional resolution check changes the numerical sampling; it does not create new experimental evidence. 14-11 00:15:21.718 --> 00:15:23.787 Pause and complete D three. 15-01 00:15:26.317 --> 00:15:34.616 For the one required modeling run, return to Consistent reference at three hundred fifty kelvin and forty-one points. 15-02 00:15:34.796 --> 00:15:39.340 Click Run modeling test once and wait for the convergence message. 15-03 00:15:39.520 --> 00:15:42.101 Then inspect Modeling residuals. 15-04 00:15:42.281 --> 00:15:48.041 Record the pressure error in percent and the vapor-composition error in percentage points. 15-05 00:15:48.221 --> 00:15:52.872 These are different quantities, even though both are scaled by one hundred. 15-06 00:15:53.052 --> 00:16:01.713 The Lab fits two constant N R T L parameters with the nonrandomness parameter fixed at zero point four seven. 15-07 00:16:01.893 --> 00:16:08.485 Its teaching criteria require both reported errors to be below one in their stated units. 15-08 00:16:08.665 --> 00:16:20.868 The clean fit should give near-zero errors because it reproduces its own generating model; that is a check of model reproduction, not a certificate for real measurements. 15-09 00:16:21.048 --> 00:16:25.570 Herington is not applicable because this case is isothermal. 15-10 00:16:25.750 --> 00:16:29.825 Its boiling-range allowance belongs to an isobaric path. 15-11 00:16:30.005 --> 00:16:32.224 Pause now to finish D four. 15-12 00:16:32.404 --> 00:16:38.249 If the fit does not converge, record no modeling verdict instead of guessing a result. 16-01 00:16:40.779 --> 00:16:45.067 Compare your D one and D two records with these two plots. 16-02 00:16:45.247 --> 00:16:54.122 The teal reference and amber canceling-error curves have different log-activity ratios, yet both meet the area criterion. 16-03 00:16:54.302 --> 00:16:58.590 Their positive and negative contributions nearly balance. 16-04 00:16:58.770 --> 00:17:01.735 The local residual tells a different story. 16-05 00:17:01.915 --> 00:17:09.062 The reference remains close to zero, while the canceling-error residual oscillates across the composition range. 16-06 00:17:09.242 --> 00:17:19.780 The reference D score is about zero point zero seven nine percent, and the canceling-case score is about zero point zero seven four percent. 16-07 00:17:19.960 --> 00:17:30.264 Those similarly small area scores conceal a large difference in the local score: roughly zero point one one eight versus fifteen point six. 16-08 00:17:30.444 --> 00:17:35.202 Read that contrast from the numerical records and the plotted shape. 16-09 00:17:35.382 --> 00:17:37.878 Complete E one in your own words. 16-10 00:17:38.058 --> 00:17:43.967 Describe how cancellation can occur globally while a local relationship is violated. 16-11 00:17:44.147 --> 00:17:47.688 Pause before continuing to the interpretation check. 17-01 00:17:50.218 --> 00:17:57.728 Your explanation should connect the two observations rather than rank the methods as universally good or bad. 17-02 00:17:57.908 --> 00:18:04.734 The cosine perturbation integrates to zero, so the area method misses that particular discrepancy. 17-03 00:18:04.914 --> 00:18:08.776 The local relationship retains a structured departure. 17-04 00:18:08.956 --> 00:18:16.444 For E two, write a supported claim, give the numerical or graphical evidence, and state one limitation. 17-05 00:18:16.624 --> 00:18:25.438 Include the dataset, forty-one-point sampling, and perturbation amplitude so another person could reproduce the comparison. 17-06 00:18:25.618 --> 00:18:32.530 Mention the assumptions behind inferred activities or the sensitivity of derivatives to sampling. 17-07 00:18:32.710 --> 00:18:38.832 Keep the localized case in mind: even a mean local score can hide a narrow feature. 17-08 00:18:39.012 --> 00:18:45.455 After writing your own explanation, compare it with sections D and E of the worked solutions. 17-09 00:18:45.635 --> 00:18:48.707 Save the baseline and changed-case evidence. 17-10 00:18:48.887 --> 00:18:57.911 This constructed exercise demonstrates limitations of diagnostics; it does not estimate experimental uncertainty. 18-01 00:19:00.441 --> 00:19:02.532 Take a ten-minute break now. 18-02 00:19:02.712 --> 00:19:05.442 Save your work and leave the files open. 18-03 00:19:05.622 --> 00:19:10.998 Step away from the screen, then return to this point with worksheet page three ready. 18-04 00:19:11.178 --> 00:19:14.162 After the break, we will change the question. 18-05 00:19:14.342 --> 00:19:19.782 A model can obey Gibbs-Duhem and still favor separation into two liquids. 18-06 00:19:19.962 --> 00:19:24.762 We will use Gibbs energy to determine which liquid state is stable. 19-01 00:19:27.292 --> 00:19:37.298 At fixed temperature and pressure, stable equilibrium is the admissible state with the lowest total Gibbs energy, subject to the material balance. 19-02 00:19:37.478 --> 00:19:45.008 For a binary mixture, the mixing-energy curve contains the ideal mixing terms plus the excess contribution. 19-03 00:19:45.188 --> 00:19:53.252 This curve can become nonconvex even when the activity model satisfies the Gibbs-Duhem identity exactly. 19-04 00:19:53.432 --> 00:19:58.616 That is why consistency does not imply a stable homogeneous liquid. 19-05 00:19:58.796 --> 00:20:04.514 Equality of chemical potentials identifies candidate coexisting phases. 19-06 00:20:04.694 --> 00:20:11.862 We must still ask whether that candidate provides the lowest energy when all allowed compositions are considered. 19-07 00:20:12.042 --> 00:20:17.780 In a binary plot, a globally supporting common tangent supplies that comparison. 19-08 00:20:17.960 --> 00:20:22.931 The line represents the energy of a mixture of the two endpoint liquids. 19-09 00:20:23.111 --> 00:20:32.284 Complete F one on page three: explain what remains to be checked after a solver has found equal chemical potentials. 19-10 00:20:32.464 --> 00:20:36.688 Pause briefly before continuing to the geometric condition. 20-01 00:20:39.218 --> 00:20:45.853 A common tangent touches the curve at two distinct compositions and has the same slope at both. 20-02 00:20:46.033 --> 00:20:53.777 Its slope must also equal the rise between the two energy values divided by the distance between the compositions. 20-03 00:20:53.957 --> 00:21:00.182 Those local equalities are necessary, but the word supporting adds a global requirement. 20-04 00:21:00.362 --> 00:21:06.976 Across the entire admitted composition range, the line must remain at or below the curve. 20-05 00:21:07.156 --> 00:21:15.817 If it crosses above the curve somewhere, another allowed state can lower the energy and the candidate is not globally stable. 20-06 00:21:15.997 --> 00:21:22.653 Similarly, a solution whose two endpoints coincide does not establish two distinct liquids. 20-07 00:21:22.833 --> 00:21:31.302 Positive curvature at one composition is only a local statement; it cannot by itself prove global stability. 20-08 00:21:31.482 --> 00:21:34.234 Keep this condition ready for F three. 20-09 00:21:34.414 --> 00:21:39.854 First, move to the next slide and attempt the lever-rule calculation in F two. 21-01 00:21:42.384 --> 00:21:53.264 For F two, suppose a stable tie line has already been established with liquid compositions zero point two zero and zero point eight zero. 21-02 00:21:53.444 --> 00:22:00.186 These values are supplied for a hand example; they are not the endpoints you should expect from Lab Four. 21-03 00:22:00.366 --> 00:22:12.206 Calculate the fraction of the component-one-rich liquid by subtracting the lower endpoint from the overall composition, then dividing by the distance between the endpoints. 21-04 00:22:12.386 --> 00:22:16.183 The other liquid fraction is one minus that result. 21-05 00:22:16.363 --> 00:22:24.064 Do this first for an overall composition of zero point five zero, then for zero point three five. 21-06 00:22:24.244 --> 00:22:31.668 Reconstruct each overall composition from the fractions and endpoint compositions to check your balance. 21-07 00:22:31.848 --> 00:22:36.072 Here beta names the richer liquid, not a vapor fraction. 21-08 00:22:36.252 --> 00:22:41.479 Pause now and calculate both feeds before the next slide shows the answers. 22-01 00:22:44.009 --> 00:22:48.020 For the central feed, the two liquid fractions are equal. 22-02 00:22:48.200 --> 00:22:57.458 For an overall composition of zero point three five, the rich-liquid fraction is one quarter and the other fraction is three quarters. 22-03 00:22:57.638 --> 00:23:06.918 The balance is three quarters times zero point two, plus one quarter times zero point eight, which returns zero point three five. 22-04 00:23:07.098 --> 00:23:16.869 The coexisting compositions remain the same because we have kept the coexistence conditions fixed and moved the feed within the same tie line. 22-05 00:23:17.049 --> 00:23:19.396 The phase amounts change instead. 22-06 00:23:19.576 --> 00:23:26.893 Correct your F two calculation, then answer F three using the global supporting-tangent condition. 22-07 00:23:27.073 --> 00:23:33.388 Touching the curve twice is not enough if the line rises above an admissible part of it. 22-08 00:23:33.568 --> 00:23:37.344 Open Lab Four and guide page three when you are ready. 22-09 00:23:37.524 --> 00:23:42.900 The next investigation uses the model's actual calculated endpoints. 23-01 00:23:45.430 --> 00:23:49.675 For G one, predict the phase state before reading the result. 23-02 00:23:49.855 --> 00:23:58.474 In Lab Four, open Explore a state and choose Unstable, A equals three, z one equals zero point five zero. 23-03 00:23:58.654 --> 00:24:08.278 Verify the controls: one-parameter Margules, interaction parameter three, and feed mole fraction zero point five zero. 23-04 00:24:08.458 --> 00:24:12.597 Choose Gibbs energy and common tangent in the View control. 23-05 00:24:12.777 --> 00:24:18.900 If you type values instead of selecting the preset, click Find equilibrium to apply them. 23-06 00:24:19.080 --> 00:24:25.181 Record the calculated coexisting compositions and both liquid fractions in the table. 23-07 00:24:25.361 --> 00:24:31.462 Write the Gibbs-energy decrease divided by R T beside G four or below the table. 23-08 00:24:31.642 --> 00:24:41.050 The parameters represent a hypothetical model at fixed conditions; this Lab has no physical temperature or pressure input boxes. 23-09 00:24:41.230 --> 00:24:44.494 Keep the same model throughout this comparison. 23-10 00:24:44.674 --> 00:24:52.589 Pause now to make your prediction, run the baseline, and complete G one using the actual displayed outputs. 24-01 00:24:55.119 --> 00:25:00.858 Before changing the feed, save the baseline with Export study JSON. 24-02 00:25:01.038 --> 00:25:06.328 Give the file a name that identifies the feed as zero point five zero. 24-03 00:25:06.508 --> 00:25:14.338 For G two, change only Feed mole fraction to zero point three five, then click Find equilibrium. 24-04 00:25:14.518 --> 00:25:18.294 Leave the model and interaction parameter unchanged. 24-05 00:25:18.474 --> 00:25:22.143 Record the same outputs beside your baseline values. 24-06 00:25:22.323 --> 00:25:26.376 Compare the endpoint compositions separately from the fractions. 24-07 00:25:26.556 --> 00:25:35.986 Ask whether each liquid's composition changed, whether its amount changed, and whether the result still reconstructs the overall feed. 24-08 00:25:36.166 --> 00:25:39.494 Use the endpoints calculated by this model. 24-09 00:25:39.674 --> 00:25:47.546 Substituting zero point two and zero point eight from the earlier hand example would test a different tie line. 24-10 00:25:47.726 --> 00:25:53.507 Pause here and record the changed-feed result before checking the balance independently. 25-01 00:25:56.037 --> 00:26:02.586 Now verify G three with your calculator, not simply by copying the Lab's residual. 25-02 00:26:02.766 --> 00:26:12.516 Use the actual lower and upper coexistence compositions to calculate the rich-liquid fraction for the feed of zero point three five. 25-03 00:26:12.696 --> 00:26:16.557 Then obtain the other fraction by subtraction from one. 25-04 00:26:16.737 --> 00:26:22.476 Reconstruct the feed by weighting each endpoint composition by its phase fraction. 25-05 00:26:22.656 --> 00:26:27.690 Check that both fractions lie between zero and one and that they sum to one. 25-06 00:26:27.870 --> 00:26:33.865 If the displayed endpoints are rounded, a small final-digit mismatch is expected. 25-07 00:26:34.045 --> 00:26:39.805 You can use the additional digits in the JSON export for a tighter check. 25-08 00:26:39.985 --> 00:26:46.108 A successful balance verifies that the reported amounts and compositions match the feed. 25-09 00:26:46.288 --> 00:26:49.829 It does not establish global stability by itself. 25-10 00:26:50.009 --> 00:26:54.425 Pause now and show the substituted calculation on page three. 26-01 00:26:56.955 --> 00:27:02.011 Complete G four by combining the numerical and graphical evidence. 26-02 00:27:02.191 --> 00:27:08.676 Open Verify the equilibrium, then inspect the full Gibbs-energy and common-tangent plot. 26-03 00:27:08.856 --> 00:27:21.352 Confirm that the liquids have distinct compositions, their chemical potentials agree within the reported numerical tolerance, and the common tangent supports the whole admitted curve. 26-04 00:27:21.532 --> 00:27:25.074 Add your independent feed balance to this evidence. 26-05 00:27:25.254 --> 00:27:34.214 The Gibbs-energy decrease shows how much the equilibrium split lowers the energy relative to the hypothetical homogeneous feed. 26-06 00:27:34.394 --> 00:27:40.175 Do not confuse the homogeneous-feed label with the final equilibrium phase count. 26-07 00:27:40.355 --> 00:27:47.139 An unstable homogeneous feed can produce a stable equilibrium consisting of two liquids. 26-08 00:27:47.319 --> 00:27:53.527 Explain which quantities stayed fixed when only the feed changed and which quantities changed. 26-09 00:27:53.707 --> 00:28:01.899 Pause to write this reasoning, using the tangent and the energy comparison rather than relying only on a phase badge. 27-01 00:28:04.429 --> 00:28:09.506 Compare G one through G four with section G of the worked solutions. 27-02 00:28:09.686 --> 00:28:20.801 Check that your record identifies one-parameter Margules with A equal to three, both overall feed compositions, and the actual coexistence endpoints. 27-03 00:28:20.981 --> 00:28:30.712 Keep phase composition separate from phase fraction: composition describes each liquid, while fraction tells us how much of that liquid is present. 27-04 00:28:30.892 --> 00:28:38.871 If a result differs, first compare the model settings, check that typed values were applied, and inspect any rounding. 27-05 00:28:39.051 --> 00:28:44.363 Save the changed-feed JSON and a comparison table or plot. 27-06 00:28:44.543 --> 00:28:50.068 Retain your balance calculation and stability explanation with those files. 27-07 00:28:50.248 --> 00:28:53.043 The required investigation ends here. 27-08 00:28:53.223 --> 00:29:00.562 LLE fitting and vapor-liquid-liquid equilibrium are optional extensions for later study. 27-09 00:29:00.742 --> 00:29:04.774 Open page four of the worksheet for the short self-review. 28-01 00:29:07.304 --> 00:29:11.933 Answer H one through H four without consulting the solutions. 28-02 00:29:12.113 --> 00:29:18.513 Use a sentence or two for each explanation and include the brief calculation in H four. 28-03 00:29:18.693 --> 00:29:25.008 First, decide whether meeting an area criterion guarantees all local relationships. 28-04 00:29:25.188 --> 00:29:31.844 Second, explain what you should check before blaming a differential discrepancy on experimental error. 28-05 00:29:32.024 --> 00:29:38.473 Third, state what a small fitting error can establish and whether that is predictive validation. 28-06 00:29:38.653 --> 00:29:48.851 Finally, return to the supplied illustrative endpoints zero point two and zero point eight, with feed composition zero point three five. 28-07 00:29:49.031 --> 00:29:55.921 Calculate the rich-liquid fraction and say what establishes stable coexistence beyond the balance. 28-08 00:29:56.101 --> 00:29:58.171 Preserve your first answers. 28-09 00:29:58.351 --> 00:30:03.065 Pause now and finish all four before continuing to the review check. 29-01 00:30:05.595 --> 00:30:13.894 For H one, opposite local errors can cancel in the integral, so a small area is not a guarantee about every point. 29-02 00:30:14.074 --> 00:30:25.173 For H two, examine the thermodynamic path, required excess-property corrections, phase and property assumptions, and the numerical differentiation. 29-03 00:30:25.353 --> 00:30:32.457 If essential information is missing, the appropriate result may be not assessed rather than a failed test. 29-04 00:30:32.637 --> 00:30:40.061 For H three, a small residual demonstrates reproduction by the selected model on those observations. 29-05 00:30:40.241 --> 00:30:44.551 It does not independently establish accuracy for new data. 29-06 00:30:44.731 --> 00:30:52.923 For H four, the rich-liquid fraction is zero point two five and the balance returns zero point three five. 29-07 00:30:53.103 --> 00:31:00.932 A stable two-liquid interpretation also needs distinct phases and a globally supporting common tangent. 29-08 00:31:01.112 --> 00:31:04.803 Correct your answers in a different color or add a note. 29-09 00:31:04.983 --> 00:31:08.780 Keep the original attempt so you can see what changed. 30-01 00:31:11.310 --> 00:31:18.201 Open the existing Session Three exit-ticket and feedback form using the link or QR code. 30-02 00:31:18.381 --> 00:31:22.626 Section I on worksheet page four gives the same instructions. 30-03 00:31:22.806 --> 00:31:34.299 In the concept response, answer the LLE question: explain when a calculated split represents stable equilibrium, using your Lab Four reasoning. 30-04 00:31:34.479 --> 00:31:42.373 In the numerical-evidence response, describe the Lab Three comparison between the area and local diagnostics. 30-05 00:31:42.553 --> 00:31:49.373 Identify the case, sampling and amplitude, and include the results that support your statement. 30-06 00:31:49.553 --> 00:31:52.753 You may add your Lab Four balance if useful. 30-07 00:31:52.933 --> 00:31:57.839 Use your own explanation rather than copying the example conclusion. 30-08 00:31:58.019 --> 00:32:03.246 This form is anonymous and does not identify individual attendance. 30-09 00:32:03.426 --> 00:32:10.146 Pause to enter the exit-ticket responses, then continue to the feedback fields in the same form. 31-01 00:32:12.676 --> 00:32:16.324 Finish the feedback fields and submit the form once. 31-02 00:32:16.504 --> 00:32:25.101 Tell us whether the pace and Lab access worked for independent study, and name one explanation or instruction that could be clearer. 31-03 00:32:25.281 --> 00:32:28.993 Wait for the message Response received before leaving. 31-04 00:32:29.173 --> 00:32:41.681 Then check your own saved work: the completed worksheet with corrections, the Lab Three comparison and exports or screenshots, and both Lab Four cases with the independent balance. 31-05 00:32:41.861 --> 00:32:48.389 Exporting a Lab study saves a local file; it does not submit your assignment to a server. 31-06 00:32:48.569 --> 00:32:54.094 Keep these files together and follow any separately announced submission arrangements. 31-07 00:32:54.274 --> 00:32:56.578 This completes the three-hour route. 31-08 00:32:56.758 --> 00:33:00.726 No additional compulsory homework follows this lesson. 31-09 00:33:00.906 --> 00:33:09.269 Your evidence should now distinguish model reproduction, thermodynamic consistency, and stable phase equilibrium.