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A dilute aqueous solution of sulfuric acid at \(25^{\circ} \mathrm{C}\) is used to absorb ammonia in a continuous reactor, thereby producing ammonium sulfate, a fertilizer: $$2 \mathrm{NH}_{3}(\mathrm{g})+\mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq}) \rightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}(\mathrm{aq})$$ (a) If the ammonia enters the absorber at \(75^{\circ} \mathrm{C}\), the sulfuric acid enters at \(25^{\circ} \mathrm{C}\), and the product solution emerges at \(25^{\circ} \mathrm{C}\), how much heat must be withdrawn from the unit per mol of \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\) produced? (All needed physical property data may be found in Appendix B.) (b) Estimate the final temperature if the reactor of Part (a) is adiabatic and the product of the solution contains 1.00 mole \(\%\) ammonium sulfate. Take the heat capacity of the solution to be that of pure liquid water [4.184 kJ/(kg.'C)]. (c) In a real (imperfectly insulated) reactor, would the final solution temperature be less than, equal to, or greater than the value calculated in Part (b), or is there no way to tell without more information? Briefly explain your answer.

Step by step solution

01

Part a: Calculation of heat withdrawn

To determine the heat to be withdrawn, differential heat balances will be performed. During the absorption process, 1 mol of sulfuric acid reacts with 2 mol of ammonia to form 1 mol of ammonium sulfate. From Appendix B, we get values for enthalpy of reaction, heat capacities of all substances. We use \(\Delta H = \Delta H_{rxn} + \Delta H_{cooling}\), where \(\Delta H_{rxn}\) is heat due to reaction and \(\Delta H_{cooling}\) is heat of cooling to bring ammonia from 75°C to 25°C. Heat of cooling \(\Delta H_{cooling} = 2 * Cp_{NH_3} * (T_{initial} - T_{final})\). Plug in values to get the amount of heat withdrawn.

02

Part b: Final Temperature Calculation

An adiabatic reactor does not exchange heat with its surroundings. So, the heat produced in the reactor because of the reaction must be accounted for the increase in temperature. Initial temperature will be taken as 25°C (temperature of the entering sulfuric acid). Use the equation \(q_{absorption} = q_{reaction} + q_{heating}\) where \(q_{absorption}\) is enthalpy absorbed by absorber and \(q_{heating}\) is energy required to increase temperature from 25°C to T_final. As this is adiabatic conditions, \(q_{absorption} = 0\), so \(-\Delta H_{reaction} = Cp_{solution} * (T_{final} - T_{initial})\). Here \(\Delta H_{reaction}\) is enthalpy change due to reaction and \(Cp_{solution}\) is heat capacity of solution. Solve this to find \(T_{final}\).

03

Part c: Final solution temperature in real reactor

In an imperfectly insulated reactor, some heat will escape due to heat loss. As a result, not all heat generated by a reaction will be available to increase the temperature of the solution. Therefore, the temperature of the final solution in a real reactor would be lower than in an adiabatic reactor.

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Key Concepts

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

Heat Balances

When dealing with chemical processes, it's essential to consider the heat balances of reactions. This involves calculating the heat produced or absorbed during a chemical process. In the context of sulfuric acid absorbing ammonia to form ammonium sulfate, we perform a heat balance to find out how much heat needs to be withdrawn to maintain the product temperature at a desired level.

The heat balance equation often breaks down as follows:

• \(\Delta H = \Delta H_{rxn} + \Delta H_{cooling}\)

Here, \(\Delta H_{rxn}\) is the enthalpy change from the chemical reaction, and \(\Delta H_{cooling}\) is the heat removed to cool the substances to a desired final temperature. In this problem, to ensure that the final product temperature is 25°C, we subtract the heat generated in the reaction and cool the entering ammonia from 75°C to 25°C.

Understanding heat balances helps maintain control over reactions, both for safety and efficiency reasons.

Adiabatic Reactors

Adiabatic reactors are a type of reactor where no heat is exchanged with the environment. In these reactors, all heat generated by the reaction is used to change the temperature of the system.

In the case of our sulfuric acid and ammonia reaction, an adiabatic process means that the heat produced by the conversion to ammonium sulfate directly increases the temperature of the reaction mixture. We use the equation:

• \(-\Delta H_{reaction} = Cp_{solution} \times (T_{final} - T_{initial})\)

This equation implies that any heat from the reaction raises the temperature of the solution from the initial 25°C. Understanding adiabatic reactions is crucial because they give insight into how much the temperature will change without outside heat transfer.

Such knowledge is useful for designing reactors and predicting behavior under various conditions.

Enthalpy of Reaction

The enthalpy of reaction, \(\Delta H_{rxn}\), is a measure of the heat absorbed or released during a chemical reaction at constant pressure. In our exercise, the enthalpy of reaction tells us how much heat is generated or consumed when ammonia reacts with sulfuric acid.

This value is crucial for calculating overall heat balance. For example, if \(\Delta H_{rxn}\) is negative, the reaction is exothermic, releasing heat. Conversely, if it's positive, the reaction is endothermic, absorbing heat. Knowing \(\Delta H_{rxn}\) helps determine how much external heat or cooling is required to achieve desired product conditions.

Understanding this concept allows engineers and scientists to safely and effectively design and operate chemical processes, ensuring the desired outcomes and maintaining safety standards.