91Ó°ÊÓ

In this problem you will use a spreadsheet to create a \(T x y\) diagram for the benzene-chloroform system at 1 atm. Once the spreadsheet has been created, it can be used as a template for vapor-liquid equilibrium calculations for other species. The calculations will be based on Raoult's law (i.e., \(y_{i} P=x_{i} p_{i}^{*}\) ), although we recognize that this relationship may not produce accurate results for benzene-chloroform mixtures.(a) Begin by establishing bounds on the system behavior. Look up the normal boiling points of chloroform and benzene and, without performing any calculations, sketch the expected shape of a Txy diagram for these two species at 1 atm. (b) Using APEx or Table B.4, estimate the normal boiling points of the two species and compare them to the results in Part (a).(c) Prepare a spreadsheet that has a title row "Txy Diagram for Ideal Binary Solution of Chloroform and Benzene." In the first cell of Row 2, place the label " \(P(\mathrm{mm} \mathrm{Hg})="\) and in the adjacent cell enter the system pressure, which for this case is 760. In Row 3 place headings for columns: xC, xB, T, p^*C, p^* B, P, yC, yB, and yC + yB. Not all of these columns are essential, but when filled they will give a complete picture of the system and a final check of the calculations. Carry out the following procedures in each subsequent row: \(\bullet\) Enter values for the mole fraction of chloroform (the first entry should be 1.000 and the last should be 0.000).\(\bullet\) Calculate the mole fraction of benzene by subtracting the value in the previous cell from 1.000 .\(\bullet\)Enter an estimate of the equilibrium temperature that is between the two pure-component boiling points.\(\bullet\) Use APEx or Table B.4 to estimate \(p^{*} \mathrm{C}\) and \(p^{*} \mathrm{B}\) from the estimated temperature. \(\bullet\) Calculate \(p \mathbf{C}\) and \(p \mathbf{B}\) from Raoult's law.\(\bullet\) Calculate \(P=p_{\mathrm{C}}+p_{\mathrm{B}}\) and apply the Goal Seek tool to adjust the value of \(T\) until \(P=760 \mathrm{mm} \mathrm{Hg}\) \(\bullet\) Calculate \(y \mathbf{C}\) and \(y \mathbf{B}\) from the partial pressures and \(P\). \(\bullet\) Sum \(y \mathbf{C}\) and \(y \mathbf{B}\) to be sure they equal 1.000.Once you have completed a row for the first value of \(x \mathrm{C},\) you should be able to copy formulas into subsequent rows. When the calculation has been completed for all rows (i.e., \(x \mathrm{C}=0.0,0.2,0.4\) \(0.5, 0.6, 0.8, 1.0)\), draw the Txy diagram.(d) Explain what you did in the bulleted sequence of steps in Part (c) giving relevant relationships among system variables. The phrase "bubble point" should appear in your explanation. (e) The following vapor-liquid equilibrium data have been obtained for mixtures of chloroform (C) and benzene (B) at 1 atm.Plot these data on the graph generated in Part (c). Estimate the percentage errors in the Raoult's law values of the bubble-point temperature and vapor mole fraction for \(x_{\mathrm{C}}=0.44,\) taking the tabulated values to be correct. Why does Raoult's law give poor estimates for this system?

Step by step solution

01

Lookup the boiling points

Lookup the normal boiling points of chloroform and benzene. You can find these in a chemistry text book or reliable online database. These boiling points will serve as estimated bounds for the system behavior.

02

Comparison using APEx or Table B.4

Using APEx or Table B.4, further estimate the normal boiling points of both species. Compare these values with the results obtained in step 1.

03

Prepare the Spreadsheet

Prepare a spreadsheet with the title 'Txy Diagram for Ideal Binary Solution of Chloroform and Benzene.' Include the following headers in the rows: xC, xB, T, p^*C, p^* B, P, yC, yB, and yC + yB. This will order and organize the required calculations.

04

Fill in Data

Enter required data for the mole fractions of chloroform and benzene, estimate the equilibrium temperature, estimate \(p^{*} \mathrm{C}\) and \(p^{*} \mathrm{B}\) from the estimated temperature, calculate \(p \mathbf{C}\) and \(p \mathbf{B}\) from Raoult's law, calculate \(P=p_{\mathrm{C}}+p_{\mathrm{B}}\), apply the Goal Seek tool to adjust the value of \(T\) until \(P=760 \mathrm{mm} \mathrm{Hg}\), and finally calculate \(y \mathbf{C}\) and \(y \mathbf{B}\) from the partial pressures and \(P\). Also, ensure that the sum of \(y \mathbf{C}\) and \(y \mathbf{B}\) equals 1.000.

05

Repeat Calculations

Repeat the calculations from step 4 for different values of \(x \mathrm{C}\) (0.0, 0.2, 0.4, 0.5, 0.6, 0.8, 1.0).

06

Draw the Txy Diagram

Using your spreadsheet, draw the Txy diagram. This visual representation will show the behavior of the benzene-chloroform system.

07

Explanation of Calculations

Provide an explanation of what you did in the steps taken in creating the spreadsheet and deriving the diagram. Refer to the bubble point to explain system behaviors.

08

Plotting Vapor-Liquid Equilibrium Data

Plot the given vapor-liquid equilibrium data on the graph generated in part c. This provides a real-time comparison between the obtained results and actual data.

09

Estimate Errors in Raoult's Law

Estimate the percentage errors in the Raoult’s law values of the bubble-point temperature and vapor mole fraction for \(x_{\mathrm{C}}=0.44\). Assess the accuracy of the law by comparing tabulated values with those obtained from the diagram.

10

Evaluation

Finally, evaluate why Raoult's law might give poor estimates for this system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Raoult's Law

Raoult's Law provides a foundational understanding of how substances interact in a mixture. It expresses the vapor pressure of an ideal mixture as the sum of the products of the mole fraction of each component and its respective pure substance vapor pressure. Mathematically, this can be represented as:

• For a component in the vapor phase, denoted as i, the pressure is expressed: \( y_i P = x_i p_i^{*} \), where \( y_i \) is the mole fraction in the vapor phase, \( P \) is the total pressure, \( x_i \) is the mole fraction in the liquid phase, and \( p_i^{*} \) is the pure component vapor pressure.

Raoult's law works best when the substances in the mixture behave ideally, which means solvent-solvent, solute-solute, and solvent-solute interactions are similar. However, it can lead to inaccurate results for systems like benzene-chloroform where interactions deviate from ideal behavior. Understanding Raoult’s Law is crucial for estimating vapor-liquid equilibrium conditions, especially in simple mixtures.

Binary Mixtures

Binary mixtures involve two different components, often denoted as A and B. These mixtures can be analyzed using vapor-liquid equilibrium concepts to predict phase behavior at given conditions. In the context of mixtures like benzene-chloroform, each component exhibits distinct vapor pressures and boiling points.
When working with binary mixtures, it's essential to grasp how the components' interactions affect the overall phase equilibrium. When Raoult's Law is applied to a binary mixture, each component's contribution to the total vapor pressure must be considered:

• Component A: \( P_A = x_A p_A^{*} \)
• Component B: \( P_B = x_B p_B^{*} \)

The total pressure of the system is then \( P = P_A + P_B \). Binary mixtures often do not behave ideally, so discrepancies can arise between predicted and actual behavior. Understanding these interactions helps in creating a reliable Txy diagram as it helps in tracking how components transition between phases.

Txy Diagram

A Txy Diagram is a valuable tool for visualizing the phase behavior of a binary mixture in equilibrium. It shows the relationship between the temperature and the composition of liquid and vapor phases.
The Txy diagram typically consists of two curves:

• The lower curve represents the bubble point line, which indicates the temperature at which the liquid mixture starts to boil.
• The upper curve represents the dew point line, which shows the conditions where the vapor begins to condense.

In an ideal scenario, the lines coincide at various points as predicted by Raoult's Law. However, real mixtures may exhibit deviations. Plotting experimental data, like the benzene-chloroform system, next to a Raoult's Law-predicted curve helps in understanding the accuracy and limitations of the model. These diagrams provide an easier comprehension of complex equilibrium calculations and are indispensable in separating processes.

Bubble Point

The bubble point is a key concept in understanding vapor-liquid equilibrium. It defines the temperature at any given pressure where the first bubble of vapor forms from a liquid mixture.
The bubble point is a unique point for a specified liquid composition, which is visualized as the start of the boiling curve in the Txy diagram. At the bubble point, the liquid composition is just starting to produce vapor but has not yet fully transitioned into a gaseous state.
Calculating the bubble point involves using mole fractions of the components and their respective vapor pressures, often assuming ideal conditions with Raoult's Law. In practice, discrepancies in real systems might occur, as seen in benzene and chloroform mixtures.

• For each mole fraction, ensure the total pressure (sum of partial pressures from all components) equals the system pressure to find the bubble point.

This point helps elucidate at what condition each part of the mixture transitions between liquid and vapor, a fundamental part of designing separation processes in chemical engineering.