91Ó°ÊÓ

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.)

Step by step solution

01

Writing expressions for amount of each species

For the reaction \(A + B ⇌ C + D\), the changes in the number of moles of the substances during the reaction can be written as: \(n_A = n_{A0} - ξ\), \(n_B = n_{B0} - ξ\), \(n_C = n_{C0} + ξ\), \(n_D = n_{D0} + ξ\). Here, \(ξ\) 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 \(y_A = n_A / n_{Total} = (n_{A0} - ξ) / n_{Total}\), 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 \(ξ_c\).

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 Châtelier'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.