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Chapter 4: Problem 64

The gas-phase reaction between methanol and acetic acid to form methyl acetate and water takes place in a batch reactor. When the reaction mixture comes to equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant $$K_{y}=\frac{y_{C} y_{D}}{y_{A} y_{B}}=4.87$$ (a) Suppose the feed to the reactor consists of \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0}, n_{\mathrm{D} 0},\) and \(n_{10}\) gram-moles of \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and an inert gas, I, respectively. Let \(\xi\) be the extent of reaction. Write expressions for the gram-moles of each reactive species in the final product, \(n_{\mathrm{A}}(\xi), n_{\mathrm{B}}(\xi), n_{\mathrm{C}}(\xi),\) and \(n_{\mathrm{D}}(\xi) .\) Then use these expressions and the given equilibrium constant to derive an equation for \(\xi_{c}\), the equilibrium extent of reaction, in terms of \(\left.n_{\mathrm{A} 0}, \ldots, n_{10} . \text { (see Example } 4.6-2 .\right)\) (b) If the feed to the reactor contains equimolar quantities of methanol and acetic acid and no other species, calculate the equilibrium fractional conversion. (c) It is desired to produce 70 mol of methyl acetate starting with 75 mol of methanol. If the reaction proceeds to equilibrium, how much acetic acid must be fed? What is the composition of the final product? (d) Suppose it is important to reduce the concentration of methanol by making its conversion at equilibrium as high as possible, say 99\%. Again assuming the feed to the reactor contains only methanol and acetic acid and that it is desired to produce 70 mol of methyl acetate, determine the extent of reaction and quantities of methanol and acetic acid that must be fed to the reactor. (e) If you wanted to carry out the process of Part (b) or (c) commercially, what would you need to know besides the equilibrium composition to determine whether the process would be profitable? (List several things.)

Short Answer

Expert verified
(a) For equilibrium, \(尉_c = (n_{A0} n_{B0} - n_{C0} n_{D0}) / (4.87 (n_{A0} + n_{B0} + n_{C0} + n_{D0} + n_{I0}))\). (b) The equilibrium fractional conversion depends on the equilibrium constant and initial moles of A and B. (c) Calculations give the required moles of acetic acid and the final composition. (d) For maximum conversion, calculate the moles of methanol and acetic acid required. (e) Commercial viability depends on various factors like yield, cost of reactants, operating conditions, rate of reaction, by-products and environmental impacts among others.

Step by step solution

01

Writing expressions for amount of each species

For the reaction \(A + B 鈬� C + D\), the changes in the number of moles of the substances during the reaction can be written as: \(n_A = n_{A0} - 尉\), \(n_B = n_{B0} - 尉\), \(n_C = n_{C0} + 尉\), \(n_D = n_{D0} + 尉\). Here, \(尉\) is the extent of the reaction.
02

Derive an equation for equilibrium extent of reaction

The mole fractions \(y\) can be written using the total number of moles \(n_{Total} = n_A + n_B + n_C + n_D + n_I\), as \(y_A = n_A / n_{Total} = (n_{A0} - 尉) / n_{Total}\), and similarly for \(B\), \(C\) and \(D\). You can use the given equilibrium constant: \(K_y = y_C y_D / (y_A y_B) = (n_C n_D) / (n_A n_B) = 4.87\) to derive an equation for the equilibrium extent of the reaction \(尉_c\).
03

Calculate equilibrium fractional conversion

If A and B are in equimolar quantities, then \(n_{A0} = n_{B0}\). The equilibrium conversion can be calculated using the equilibrium constant and the initial moles of A or B.
04

Calculate the required feed of acetic acid and final composition

Using the limiting reactant concept and equilibrium equations, calculate the amount of acetic acid needed when 70 mol of methyl acetate is formed and 75 mol of methanol is provided. The final composition can be determined by using the moles of reactants and products at equilibrium.
05

Calculate the extent of reaction and feed quantities for maximum conversion

If it's needed to reduce the methanol concentration, calculate the amount of methanol and acetic acid required for 99% conversion. Here also, use the equilibrium constant and equations from Step 1.
06

Considerations for a commercial process

Factors like yield and selectivity of the reaction, cost of reactants, operating conditions, rate of reaction, by-products and environmental impact need to be considered to decide the commercial viability of the process.

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Key Concepts

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

Reaction Equilibrium Constant
The reaction equilibrium constant, often denoted as K, is a measure of the concentration of products relative to reactants at equilibrium for a given chemical reaction. For a reaction where species A and B react to form C and D, the equilibrium constant can be represented in terms of mole fractions (y) as

\[K_y = \frac{y_C y_D}{y_A y_B}\]
With the given equilibrium constant \(K_y = 4.87\) for the gas-phase reaction between methanol (A) and acetic acid (B), leading to the formation of methyl acetate (C) and water (D), the equilibrium state can be analyzed quantitatively. When the system is at equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant over time.
Understanding this constant is crucial for determining the composition of the reaction mixture under equilibrium conditions and predicting the direction of the reaction shift when the system is disturbed (according to Le Ch芒telier's principle).
Extent of Reaction Calculation
In any batch reactor process involving chemical substances A, B, C, and D, we could express the moles of each substance in terms of the extent of the reaction \(\xi\) as follows:

\[n_A = n_{A0} - \xi\]\[n_B = n_{B0} - \xi\]\[n_C = n_{C0} + \xi\]\[n_D = n_{D0} + \xi\]
These equations reflect how the number of moles of reactants decreases and the number of moles of products increases as the reaction proceeds. In the context of the problem, by combining these expressions with the equilibrium constant, one can derive an equation for the equilibrium extent of reaction \(\xi_c\). The central importance of the extent of reaction is that it quantifies exactly how far the reaction has progressed, which is essential for determining both the composition of the equilibrium mix and how to approach achieving a desired product yield.
Equilibrium Fractional Conversion
Equilibrium fractional conversion is a term that defines the fraction of a reactant that has been converted into products at chemical equilibrium. For the given problem, if the feed to the reactor contains equimolar amounts of methanol and acetic acid, the equilibrium fractional conversion of the reactant can be calculated using the derived equilibrium extent of reaction. The formula to use would be:

\[fractional\ conversion = \frac{\xi_c}{n_{A0}}\]
This concept is particularly valuable in reactors operating under equilibrium constraints, as it helps determine the efficiency of the reaction process and informs adjustments to the reactant feed to attain desired production targets.
Batch Reactor Process
A batch reactor process is a system where all reactants are loaded into the reactor at the beginning, reactions occur inside the system, and products are removed at the end of the reaction once equilibrium is achieved or desired conversion is reached. This contrasts with continuous processes where reactants and products flow in and out continuously. Batch reactors are widely used in industries for flexible and precise production, as they allow for controlled reaction conditions such as temperature, pressure, and concentration. They are particularly advantageous for reactions that are slow or need precise control over reaction time and stage. In our problem, understanding how to manipulate a batch reactor process to maximize the conversion to the desired product, methyl acetate, requires a thorough grasp of equilibrium concepts and the reaction's dependency on initial concentrations and operating conditions.

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Most popular questions from this chapter

A liquid mixture contains \(60.0 \mathrm{wt} \%\) ethanol \((\mathrm{E}), 5.0 \mathrm{wt} \%\) of a dissolved solute \((\mathrm{S}),\) and the balance water. A stream of this mixture is fed to a continuous distillation column operating at steady state. Product streams emerge at the top and bottom of the column. The column design calls for the product streams to have equal mass flow rates and for the top stream to contain 90.0 wt\% ethanol and no S. (a) Assume a basis of calculation, draw and fully label a process flowchart, do the degree-of-freedom analysis, and verify that all unknown stream flows and compositions can be calculated. (Don't do any calculations yet.) (b) Calculate (i) the mass fraction of \(S\) in the bottom stream and (ii) the fraction of the ethanol in the feed that leaves in the bottom product stream (i.e., \(\mathrm{kg} \mathrm{E}\) in bottom stream/kg \(\mathrm{E}\) in feed) if the process operates as designed. (c) An analyzer is available to determine the composition of ethanol-water mixtures. The calibration curve for the analyzer is a straight line on a plot on logarithmic axes of mass fraction of ethanol, \(x\) (kg E/kg mixture), versus analyzer reading, \(R\). The line passes through the points \((R=15, x=\) 0.100) and \((R=38, x=0.400)\). Derive an expression for \(x\) as a function of \(R(x=\cdots\) ) based on the calibration, and use it to determine the value of \(R\) that should be obtained if the top product stream from the distillation column is analyzed. (d) Suppose a sample of the top stream is taken and analyzed and the reading obtained is not the one calculated in Part (c). Assume that the calculation in Part (c) is correct and that the plant operator followed the correct procedure in doing the analysis. Give five significantly different possible causes for the deviation between \(R_{\text {measured and }} R_{\text {prediced }}\), including several assumptions made when writing the balances of Part (c). For each one, suggest something that the operator could do to check whether it is in fact the problem.

Water enters a \(2.00-\mathrm{m}^{3}\) tank at a rate of \(6.00 \mathrm{kg} / \mathrm{s}\) and is withdrawn at a rate of \(3.00 \mathrm{kg} / \mathrm{s}\). The tank is initially half full. (a) Is this process continuous, batch, or semibatch? Is it transient or steady state? (b) Write a mass balance for the process (see Example 4.2-1). Identify the terms of the general balance equation (Equation 4.2-1) present in your equation and state the reason for omitting any terms. (c) How long will the tank take to overflow?

In an absorption tower (or absorber), a gas is contacted with a liquid under conditions such that one or more species in the gas dissolve in the liquid. A stripping tower (or stripper) also involves a gas contacting a liquid, but under conditions such that one or more components of the feed liquid come out of solution and exit in the gas leaving the tower. A process consisting of an absorption tower and a stripping tower is used to separate the components of a gas containing 30.0 mole \(\%\) carbon dioxide and the balance methane. A stream of this gas is fed to the bottom of the absorber. A liquid containing 0.500 mole\% dissolved \(\mathrm{CO}_{2}\) and the balance methanol is recycled from the bottom of the stripper and fed to the top of the absorber. The product gas leaving the top of the absorber contains 1.00 mole \(\% \mathrm{CO}_{2}\) and essentially all of the methane fed to the unit. The CO_-rich liquid solvent leaving the bottom of the absorber is fed to the top of the stripper and a stream of nitrogen gas is fed to the bottom. Ninety percent of the \(\mathrm{CO}_{2}\) in the liquid feed to the stripper comes out of solution in the column, and the nitrogen/CO_stream leaving the column passes out to the atmosphere through a stack. The liquid stream leaving the stripping tower is the \(0.500 \% \mathrm{CO}_{2}\) solution recycled to the absorber. The absorber operates at temperature \(T_{\mathrm{a}}\) and pressure \(P_{\mathrm{a}}\) and the stripper operates at \(T_{\mathrm{s}}\) and \(P_{\mathrm{s}}\) Methanol may be assumed to be nonvolatile- -that is, none enters the vapor phase in either column and \(\mathrm{N}_{2}\), may be assumed insoluble in methanol. (a) In your own words, explain the overall objective of this two-unit process and the functions of the absorber and stripper in the process. (b) The streams fed to the tops of each tower have something in common, as do the streams fed to the bottoms of each tower. What are these commonalities and what is the probable reason for them? (c) Taking a basis of 100 mol/h of gas fed to the absorber, draw and label a flowchart of the process. For the stripper outlet gas, label the component molar flow rates rather than the total flow rate and mole fractions. Do the degree-of-freedom analysis and write in order the equations you would solve to determine all unknown stream variables except the nitrogen flow rate entering and leaving the stripper. Circle the variable(s) for which you would solve each equation (or set of simultaneous equations), but don't do any of the calculations yet. (d) Calculate the fractional \(\mathrm{CO}_{2}\) removal in the absorber (moles absorbed/mole in gas feed) and the molar flow rate and composition of the liquid feed to the stripping tower. (e) Calculate the molar feed rate of gas to the absorber required to produce an absorber product gas flow rate of \(1000 \mathrm{kg} / \mathrm{h}\). (f) Would you guess that \(T_{\mathrm{s}}\) would be higher or lower than \(T_{\mathrm{a}} ?\) Explain. (Hint: Think about what happens when you heat a carbonated soft drink and what you want to happen in the stripper.) What about the relationship of \(P_{\mathrm{s}}\) to \(P_{\mathrm{a}} ?\) (g) What properties of methanol would you guess make it the solvent of choice for this process? (In more general terms, what would you look for when choosing a solvent for an absorption-stripping process to separate one gas from another?)

A Claus plant converts gaseous sulfur compounds to elemental sulfur, thereby eliminating emission of sulfur into the atmosphere. The process can be especially important in the gasification of coal, which contains significant amounts of sulfur that is converted to \(\mathrm{H}_{2}\) S during gasification. In the Claus process, the \(\mathrm{H}_{2}\) S-rich product gas recovered from an acid-gas removal system following the gasifier is split, with one-third going to a furnace where the hydrogen sulfide is burned at 1 atm with a stoichiometric amount of air to form SO \(_{2}\). $$\mathrm{H}_{2} \mathrm{S}+\frac{3}{2} \mathrm{O}_{2} \rightarrow \mathrm{SO}_{2}+\mathrm{H}_{2} \mathrm{O}$$ The hot gases leave the furnace and are cooled prior to being mixed with the remainder of the \(\mathrm{H}_{2}\) S-rich gases. The mixed gas is then fed to a catalytic reactor where hydrogen sulfide and \(\mathrm{SO}_{2}\) react to form elemental sulfur. $$2 \mathrm{H}_{2} \mathrm{S}+\mathrm{SO}_{2} \rightarrow 2 \mathrm{H}_{2} \mathrm{O}+3 \mathrm{S}$$ The coal available to the gasification process is 0.6 wt\% sulfur, and you may assume that all of the sulfur is converted to \(\mathrm{H}_{2} \mathrm{S}\), which is then fed to the Claus plant. (a) Estimate the feed rate of air to the Claus plant in \(\mathrm{kg} / \mathrm{kg}\) coal. (b) While the removal of sulfur emissions to the atmosphere is environmentally beneficial, identify an environmental concern that still must be addressed with the products from the Claus plant.

A gas contains 75.0 wt\% methane, \(10.0 \%\) ethane, \(5.0 \%\) ethylene, and the balance water. (a) Calculate the molar composition of this gas on both a wet and a dry basis and the ratio (mol \(\mathrm{H}_{2} \mathrm{O} /\) mol dry gas). (b) If \(100 \mathrm{kg} / \mathrm{h}\) of this fuel is to be burned with \(30 \%\) excess air, what is the required air feed rate (kmol/ h)? How would the answer change if the combustion were only \(75 \%\) complete?
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