/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 A liquid mixture contains \(N\) ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 61

A liquid mixture contains \(N\) components ( \(N\) may be any number from 2 to 10 ) at pressure \(P(\mathrm{mm} \mathrm{Hg})\). The mole fraction of the ith component is \(x_{i}(i=1,2, \ldots, N),\) and the vapor pressure of that component is given by the Antoine equation (see Table B.4) with constants \(A_{i}, B_{i},\) and \(C_{i}\). Raoult's law may be applied to each component.(a) Write the equations you would use to calculate the bubble-point temperature of the mixture, ending with an equation of the form \(f(T)=0 .\) (The value of \(T\) that satisfies this equation is the bubble-point temperature.) Then write the equations for the component mole fractions \(\left(y_{1}, y_{2}, \ldots, y_{N}\right)\) in the first bubble that forms, assuming that the temperature is now known.(b) Prepare a spreadsheet to perform the calculations of Part (a). The spreadsheet should include a title line for identification of the problem and a row that has entries for the given pressure and an estimate of the system temperature. Be sure to label these variables and show the units in which each is expressed. Adjacent columns should be headed Species, \(p_{i}^{*}, x_{i}, p_{i},\) and \(y_{i} .\) Values of vapor pressures at the estimated temperature should be calculated using the physical property database in APEx, and Raoult's law should be used to determine partial pressures. The final row in the table should have the sums of the vapor mole fractions and partial pressures. Place a convergence function \(f(T)=P-\Sigma p_{i}\) below the table so that Goal Seek can be used to vary the estimated \(T\) until \(f(T)=0 .\) Test the spreadsheet by calculating the bubble-point temperature for a liquid mixture containing 22.6 mole \(\%\) benzene, \(22.6 \%\) ethylbenzene, \(22.3 \%\) toluene, and the balance styrene at pressures of \(250 \mathrm{mm} \mathrm{Hg}, 760 \mathrm{mm} \mathrm{Hg},\) and \(7500 \mathrm{mm}\) Hg. Identify any concerns you may have about the calculated results.(c) It is determined that instead of styrene, the balance of the above mixture in Part (b) is propylbenzene. Upon entering the name "propylbenzene鈥� in the APEx AntoineP estimator, you probably get the error message #VALUE!, which means that this substance is not in the APEx database. Poling et al. (see Footnote 2, p. A.57) provide constants for the vapor pressure of propylbenzene corresponding to the following expression of the Antoine equation:$$\log _{10} p^{*}(\mathrm{bar})=A-B /\left[T\left(^{\circ} \mathrm{C}\right)+C\right]$$ where \(A=4.07664, B=1491.8,\) and \(C=207.25 ;\) the correlation is valid over the range \(324 \mathrm{K}-\) 461 K. Modify the spreadsheet to incorporate this expression, and estimate the bubble-point temperature of the mixture at a pressure of \(760 \mathrm{mm} \mathrm{Hg}\).

Short Answer

Expert verified
The bubble-point temperature will solve for when \(f(T) = P - \Sigma p_i = 0\). A customized spreadsheet would need to be set up that includes Antoine equation and Raoult's law calculations for each component. Further, the spreadsheet should be able to handle instances where the Antoine coefficients may not be available in the database and will need manual inputs.

Step by step solution

01

Identify Antoine equation and Raoult's law

The Antoine equation is given by \(\log_{10}(p_i^*) = A_i - \frac{B_i}{T+C_i}\), where \(p_i^*\) represents vapor pressure, \(A_i\), \(B_i\), and \(C_i\) are component-specific constants, and \(T\) is temperature. Raoult's law states that the partial pressure of each component in an ideal mixture of liquids is equal to the product of the vapor pressure of the pure component and its mole fraction in the mixture, represented by \(p_i = x_i*p_i^*\)
02

Formulate equation for bubble point temperature

By applying Raoult's law to each component and adding all the equations together, we get the equation \(\Sigma p_i = P\), where \(P\) is the total pressure. This leads us to \(f(T) = P - \Sigma p_i\). The bubble point temperature will be the value of \(T\) that makes \(f(T) = 0\).
03

Formulate equations for mole fractions

To find the equations of the mole fractions, we use the formula \(y_i = \frac{p_i}{P}\) for each \(i\). Now we substitue for \(p_i\) using Raoult's law resulting in \(y_i = \frac{x_i * p_i^*}{P}\).
04

Prepare the spreadsheet calculations

Starting with an estimated initial temperature \(T\), we can calculate all component's \(p_i^*\) using Antoine's equation, then \(p_i\) using Raoult's law and ultimately \(y_i\) using the mole fraction equation. The last row should sum all \(y_i\) and \(p_i\) and the function \(f(T) = P - \Sigma p_i\) should be placed below the table. The goal of the spreadsheet is to change the calculated temperature until \(f(T) = 0\).
05

Modifying the spreadsheet for propylbenzene

In part (c), we are given Antoine coefficients for one component (Propylbenzene) since it isn't available in the provided database. We should plug these values into the Antoine equation for this component and redo the spreadsheet calculations.

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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 is essential for understanding vapor-liquid equilibrium in ideal mixtures. It expresses how the partial pressure of each component in a liquid mix is calculated. According to Raoult's law, the partial pressure \( p_i \) of a component is the product of the vapor pressure \( p_i^* \) of the pure component and its mole fraction \( x_i \) in the mixture: \( p_i = x_i \cdot p_i^* \). This law is particularly useful when calculating the bubble-point temperature, which is the temperature at which a liquid mixture starts to boil and form the first bubbles of vapor. Raoult's law allows us to find out how much each component contributes to the total pressure \( P \) of the mixture. By applying Raoult's law to each component in the mixture and summing up partial pressures, we establish the equation \( \Sigma p_i = P \).In practice, this means you'll need to:
  • Calculate the vapor pressure for each component using the Antoine equation.
  • Use the mole fractions and calculated vapor pressures to compute partial pressures with Raoult's law.
This systematic approach is crucial for determining vapor-liquid equilibrium.
Antoine Equation
The Antoine equation is a formula that calculates the vapor pressure of pure components, a step integral to solving the bubble-point temperature problem. It has the form: \[ \log_{10}(p_i^*) = A_i - \frac{B_i}{T + C_i} \] Where:
  • \( p_i^* \) is the vapor pressure.
  • \( A_i \), \( B_i \), and \( C_i \) are component-specific constants.
  • \( T \) is the temperature in degrees Celsius.
These constants are derived from experimental data and are crucial for predicting vapor pressures across different temperatures. By inserting these constants and testing for various temperatures, you can estimate vapor pressures for each substance in your mixture.Understanding of this equation is essential when preparing a spreadsheet for calculating the bubble-point temperature. Once the Antoine equation calculates the vapor pressures, this data feeds into Raoult's law to figure out partial pressures. Each component's vapor pressure will vary with temperature, so it's used in conjunction to optimizing for \( f(T) = 0 \) in the bubble-point calculation.Remember, accuracy in the constants and correctly applying the equation ensures reliability in the overall phase equilibrium analysis.
Vapor-Liquid Equilibrium
Vapor-liquid equilibrium (VLE) defines a state where both liquid and vapor phases are in balance, meaning no net evaporation or condensation occurs. Understanding this concept is critical when solving for the bubble-point temperature of a mixture.In the context of calculating bubble-point temperature, VLE involves finding a temperature where the sum of component partial pressures equals the system鈥檚 total pressure. This is represented as \( \Sigma p_i = P \). When this condition holds true, the first tiny bubbles of vapor appear, indicating the onset of boiling.To reach VLE using the given mixture data:
  • Start by estimating an initial temperature and use the Antoine equation to find each component's vapor pressure.
  • Apply Raoult's law to calculate partial pressures and mole fractions \( y_i = \frac{x_i \cdot p_i^*}{P} \).
  • Adjust the temperature until the function \( f(T) = P - \Sigma p_i \) equals zero, indicating VLE has been reached.
Exploring vapor-liquid equilibrium further involves concepts like using spreadsheets to iterate temperature values, ensuring consistent calculation updates. Mastery in navigating VLE not only assists with theoretical understanding but also practical applications in fields like chemical engineering and process design.

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Most popular questions from this chapter

The sulfur dioxide content of a stack gas is monitored by passing a sample stream of the gas through an SO_ analyzer. The analyzer reading is \(1000 \mathrm{ppm} \mathrm{SO}_{2}\) (parts per million on a molar basis). The sample gas leaves the analyzer at a rate of \(1.50 \mathrm{L} / \mathrm{min}\) at \(30^{\circ} \mathrm{C}\) and \(10.0 \mathrm{mm}\) Hg gauge and is bubbled through a tank containing 140 liters of initially pure water. In the bubbler, \(S O_{2}\) is absorbed and water evaporates. The gas leaving the bubbler is in equilibrium with the liquid in the bubbler at \(30^{\circ} \mathrm{C}\) and 1 atm absolute. The \(\mathrm{SO}_{2}\) content of the gas leaving the bubbler is periodically monitored with the \(\mathrm{SO}_{2}\) analyzer, and when it reaches \(100 \mathrm{ppm} \mathrm{SO}_{2}\) the water in the bubbler is replaced with 140 liters of fresh water.(a) Speculate on why the sample gas is not just discharged directly into the atmosphere after leaving the analyzer. Assuming that the equilibrium between \(S O_{2}\) in the gas and dissolved \(S O_{2}\) is described by Henry's law, explain why the SO_ content of the gas leaving the bubbler increases with time. What value would it approach if the water were never replaced? Explain. (The word "solubility" should appear in your explanation.)(b) Use the following data for aqueous solutions of \(\mathrm{SO}_{2}\) at \(30^{\circ} \mathrm{C}^{14}\) to estimate the Henry's law constant in units of \(\mathrm{mm}\) Hg/mole fraction:$$\begin{array}{|l|c|c|c|c|c|}\hline \mathrm{g} \mathrm{SO}_{2} \text { dissolved/ } 100 \mathrm{g}\mathrm{H}_{2} \mathrm{O}(\mathrm{l}) & 0.0 & 0.5 & 1.0 & 1.5 & 2.0 \\\\\hline p_{\mathrm{SO}_{2}}(\mathrm{mm} \mathrm{Hg}) & 0.0 & 37.1 & 83.7 &132 & 183 \\\\\hline\end{array}.$$(c) Estimate the SO_concentration of the bubbler solution (mol SO_/liter), the total moles of SO_ dissolved, and the molar composition of the gas leaving the bubbler (mole fractions of air, \(\mathrm{SO}_{2}\), and water vapor) at the moment when the bubbler solution must be changed. Make the following assumptions: \bullet. The feed and outlet streams behave as ideal gases. \bullet Dissolved SO_ is uniformly distributed throughout the liquid. ? The liquid volume remains essentially constant at 140 liters. \- The water lost by evaporation is small enough for the total moles of water in the tank to be considered constant. \- The distribution of SO_ between the exiting gas and the liquid in the vessel at any instant of time is governed by Henry's law, and the distribution of water is governed by Raoult's law (assume \(\left.x_{\mathrm{H}_{2} \mathrm{O}} \approx 1\right)\).(d) Suggest changes in both scrubbing conditions and the scrubbing solution that might lead to an increased removal of \(\mathrm{SO}_{2}\) from the feed gas.

In an attempt to conserve water and to be awarded LEED (Leadership in Energy and Environmental Design) certification, a 20,000-liter cistem has been installed during construction of a new building. The cistem collects water from an HVAC (heating, ventilation, and air-conditioning) system designed to provide 2830 cubic meters of air per minute at \(22^{\circ} \mathrm{C}\) and \(50 \%\) relative humidity after converting it from ambient conditions \(\left(31^{\circ} \mathrm{C}, 70 \% \text { relative humidity }\right) .\) The collected condensate serves as the source of water for lawn maintenance. Estimate (a) the rate of intake of air at ambient conditions in cubic feet per minute and (b) the hours of operation required to fill the cistern.

Dehydration of natural gas is necessary to prevent the formation of gas hydrates, which can plug valves and other components of a gas pipeline, and also to reduce potential corrosion problems. Water removal can be accomplished as shown in the following schematic diagram: Natural gas containing \(80 \mathrm{lb}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} / 10^{6} \mathrm{SCF}\) gas \(\left[\mathrm{SCF}=\mathrm{ft}^{3}(\mathrm{STP})\right]\) enters the bottom of an absorber at a rate of \(4.0 \times 10^{6}\) SCF/day. A liquid stream containing triethylene glycol (TEG, molecular weight \(=150.2\) ) and a small amount of water is fed to the top of the absorber. The absorber operates at 500 psia and \(90^{\circ} \mathrm{F}\). The dried gas leaving the absorber contains \(10 \mathrm{lb}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} / 10^{6} \mathrm{SCF}\) gas. The solvent leaving the absorber, which contains all the TEG-water mixture fed to the column plus all the water absorbed from the natural gas, goes to a distillation column. The overhead product stream from the distillation column contains only liquid water. The bottoms product stream, which contains TEG and water, is the stream recycled to the absorber.(a) Draw and completely label a flowchart of this process. Calculate the mass flow rate ( \(\left(\mathrm{b}_{\mathrm{m}} / \mathrm{day}\right)\) and volumetric flow rate (ft \(^{3}\) /day) of the overhead product from the distillation column. (b) The greatest possible amount of dehydration is achieved if the gas leaving the absorption column is in equilibrium with the solvent entering the column. If the Henry's law constant for water in TEG at \(90^{\circ} \mathrm{F}\) is \(0.398 \mathrm{psia} / \mathrm{mol}\) fraction, what is the maximum allowable mole fraction of water in the solvent fed to the absorber?(c) A column of infinite height would be required to achieve equilibrium between the gas and liquid at the top of the absorber. For the desired separation to be achieved in practice, the mole fraction of water in the entering solvent must be less than the value calculated in Part (b). Suppose it is \(80 \%\) of that value and the flow rate of TEG in the recirculating solvent is 37 Ib \(_{\mathrm{m}}\) TEG/lb \(_{\mathrm{m}}\) water absorbed in the column. Calculate the flow rate ( \(\left(\mathrm{b}_{\mathrm{m}} / \mathrm{day}\right)\) of the solvent stream entering the absorber and the mole fraction of water in the solvent stream leaving the absorber. (d) What is the purpose of the distillation column in the process? (Hint: Think about how the process would operate without it.)

The following diagram shows a staged absorption column in which \(n\) -hexane (H) is absorbed from a gas into a heavy oil.A gas feed stream containing 5.0 mole \(\%\) hexane vapor and the balance nitrogen enters at the bottom of an absorption column at a basis rate of \(100 \mathrm{mol} / \mathrm{s}\), and a nonvolatile oil enters the top of the column in a ratio 2 mol oil fed/mol gas fed. The absorber consists of a series of ideal stages (see Problem 6.66), arranged so that gas flows upward and liquid flows downward. The liquid and gas streams leaving each stage are in equilibrium with each other (by the definition of an ideal stage), with compositions related by Raoult's law. The absorber operates at an approximately constant temperature \(T\left(^{\circ} \mathrm{C}\right)\) and \(760 \mathrm{mm}\) Hg. Of the hexane entering the column, \(99.5 \%\) is absorbed and leaves in the liquid column effluent. At the given conditions it may be assumed that \(\mathrm{N}_{2}\) is insoluble in the oil and that none of the oil vaporizes.(a) Calculate the molar flow rates and mole fractions of hexane in the gas and liquid streams leaving the column. Then calculate the average values of the liquid and gas molar flow rates in the column, \(\dot{n}_{\mathrm{L}}(\mathrm{mol} / \mathrm{s})\) and \(\dot{n}_{\mathrm{G}}(\mathrm{mol} / \mathrm{s}) .\) For simplicity, in subsequent calculations use the average values for liquid and gas molar flow rates within the column, but the actual values for the corresponding flow rates entering and leaving the column.(b) Considering the bottom stage to be ideal, estimate the mole fractions of hexane in the gas leaving that stage \(\left(y_{N}\right)\) and in the liquid entering it \(\left(x_{N-1}\right)\) if the column temperature is \(50^{\circ} \mathrm{C}\). (c) Suppose that \(x_{i}\) and \(y_{i}\) are the mole fractions of hexane in the liquid and gas streams leaving stage \(i\) Derive the following equations from an equilibrium relationship and a mass balance around a section of the column encompassing stage \(i\) and the bottom of the column:$$\begin{array}{c}y_{i}=x_{i} p_{i}^{*}(T) / P \\ x_{i-1}=\left(x_{N} n_{L, N}+y_{i} \dot{n}_{\mathrm{G}}-y_{N+1} \dot{n}_{\mathrm{G}+1}\right) / \dot{n}_{\mathrm{L}}\end{array}$$Verify that these equations yield the answers you calculated in Part (b). (d) Examine the effect of operating temperature on the column by estimating the number of ideal stages necessary to achieve the desired separation. In the calculations, which will be done using a spreadsheet, take the pressure in the column to be constant at 760 torr, but consider three different operating temperatures: \(30^{\circ} \mathrm{C}\) \(50^{\circ} \mathrm{C},\) and \(70^{\circ} \mathrm{C} .\) The calculations will follow a stage-to-stage strategy beginning at the bottom of the column and repeatedly applying Equations (1) and (2) until the mole fraction of hexane in the vapor leaving the column is less than or equal to that calculated in Part (a). You may use APEx or the Antoine equation and Table B.4 to estimate the hexane vapor pressure. The calculations for the case of \(T=30^{\circ} \mathrm{C}\) illustrate how to proceed; for this case, \(y_{N-1} < y_{1}=0.00263\) after only two stages.(e) You can see that the number of stages required increases as the column temperature increases. In fact, there is a maximum temperature beyond which the required separation cannot be achieved. At that temperature, the entering gas and leaving liquid are approximately in equilibrium, so that \(x_{N} p^{*}(T)=y_{N+1} P .\) Use either APEx or the Antoine equation to estimate the maximum temperature at which the separation can be achieved.

An aqueous solution of potassium hydroxide (KOH) is fed to an evaporative crystallizer at a rate of 875 kg/h. The crystallizer operates at 10^'C and produces crystals of KOH-2H_O. Water evaporated from the crystallizer flows to a condenser, and the resulting condensate is collected in a tank. During a 30-minute period, 73.8 kg of water is collected. Five-gram samples of the feed to the crystallizer and the liquid removed with the crystals are taken for analysis and subsequently titrated with \(0.85 \mathrm{M}\) \(\mathrm{H}_{2} \mathrm{SO}_{4} .\) It is found that \(22.4 \mathrm{mL}\) of the \(\mathrm{H}_{2} \mathrm{SO}_{4}\) solution is required for the feed and \(26.6 \mathrm{mL}\) is required for the product liquid.(a) What fraction of the KOH in the feed is crystallized? (b) Later you learn that a solution in equilibrium with KOH \(\cdot 2 \mathrm{H}_{2} \mathrm{O}\) crystals at \(10^{\circ} \mathrm{C}\) has a concentration of \(103 \mathrm{kg} \mathrm{KOH} / 100 \mathrm{kg} \mathrm{H}_{2} \mathrm{O} .\) How would this information cause you to reconsider the procedure by which a sample of the mother liquor was obtained? (Hint: Consider removing a slurry sample- -i.e., one containing both solution and KOH \(\cdot 2 \mathrm{H}_{2} \mathrm{O}\) crystals - that is maintained at \(10^{\circ} \mathrm{C},\) but that initially had a solute concentration of \(121 \mathrm{kg} \mathrm{KOH} / 100 \mathrm{kg} \mathrm{H}_{2} \mathrm{O} .\) What would that concentration be after the sample is stored for several hours?)
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