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Chapter 9: Problem 53

A 12.0-molar solution of sodium hydroxide ( \(\mathrm{SG}=1.37\) ) is neutralized with \(75.0 \mathrm{mL}\) of a \(4.0 \mathrm{molar}\) solution of sulfuric acid ( \(\mathrm{SG}=1.23\) ) in a well-insulated container. (a) Estimate the volume of the sodium hydroxide solution and the final solution temperature if both feed solutions are at \(25^{\circ} \mathrm{C}\). The heat capacity of the product solution may be taken to be that of pure liquid water, the standard heat of solution of sodium sulfate is \(-1.17 \mathrm{kJ} / \mathrm{mol},\) and the energy balance reduces to \(Q=\Delta H\) for this constant-pressure batch process. (b) List several additional assumptions you made to arrive at your estimated volume and temperature.

Short Answer

Expert verified
The volume of the sodium hydroxide solution needed is estimated to be 50 mL. The estimated final temperature of the reaction is 693.4掳C.

Step by step solution

01

Identify Reaction Equations

First, write out the balanced chemical equation for the neutralization of sodium hydroxide (NaOH) with sulfuric acid (H2SO4). This yields:\[2NaOH(aq) + H_2SO_4(aq) \rightarrow Na_2SO_4(aq) + 2H_2O(l)\] Next, the reaction for dissolution of sodium sulfate is: \[ Na_2SO_4(s) \rightarrow Na_2SO_4(aq); \Delta H = -1.17 kJ/mol \]
02

Calculate Mols of Reactants

Use the volumes (\( V \)) and concentrations (\( M \)) of the reactants to calculate the amount of moles:For \( H_2SO_4 \): \( 75.0 mL * 4 mol/L = 0.3 mol \)For \( NaOH \), we know from the balanced equation that we need twice as many moles of \( NaOH \) as \( H_2SO_4 \), so we need \( 2*0.3 mol = 0.6 mol \) of \( NaOH \). Given this, and knowing the concentration of our \( NaOH \) is 12 mol/L, we can calculate the volume required: \( 0.6 mol / 12 mol/L = 0.05 L = 50 mL \). Thus, the estimated volume of the \( NaOH \) solution is 50 mL.
03

Calculate Final Temperature

With the molar enthalpies and amounts of reactants, the heat change (\( \Delta Q \)) of the reaction can be calculated. For each mol of reaction, we release 1.17 kJ of heat. Since we have 0.3 mol of reaction considering \( H_2SO_4 \), we have \( \Delta Q = 0.3 mol * -1.17 kJ/mol = -0.351 kJ \). This heat released increases the temperature of the solution.We can use the formula \( \Delta Q = mc\Delta T \), where \( m \) is the mass of the solution, \( c \) is the specific heat capacity of water (4.186 J/g掳C), and \( \Delta T \) is the change in temperature. We know that \( m = V \cdot \rho \), where \( V = 75 mL + 50 mL = 125 mL = 0.125 L \) is the total volume and \( \rho = 1 g/mL \) is the density of water. Therefore, \( \Delta T = -\Delta Q / mc = 351 J / (0.125 kg * 4.186 J/g掳C) = 668.4 掳C \).The final temperature is thus: \( 25掳C + 668.4掳C = 693.4掳C \).
04

List Assumptions

In arriving at these conclusions, several assumptions were made: frictionless, perfect conversion, the container perfectly insulates the heat, and the resulting solution has properties identical to those of pure water despite the dissolved salts (sodium sulfate). These assumptions simplify the problem while ignoring some real-world complexities.

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Key Concepts

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

Stoichiometry
Stoichiometry is a fundamental concept in chemistry that deals with the quantitative relationship between reactants and products in a chemical reaction. In our exercise, chemical equations serve as the roadmap for stoichiometry, allowing us to calculate the volume of a sodium hydroxide (NaOH) solution required to neutralize a given volume of sulfuric acid (H2SO4).

The stoichiometric calculations begin by balancing the chemical equation, ensuring the number of atoms for each element is equal on both sides of the reaction. We then use the molar concentration of each reactant to determine the number of moles, which in this case revealed a ratio of 2 moles of NaOH for every 1 mole of H2SO4. This ratio is critical as it tells us exactly how much of each reactant is needed to completely react with the other.
Enthalpy Change
The enthalpy change (\( \triangle H \)) in a chemical reaction is the amount of heat absorbed or released during the process. It's a measure of the energy change that occurs as reactants are transformed into products.

In our neutralization reaction, the dissolution of sodium sulfate (\( Na_2SO_4 \) is an exothermic process, meaning it releases heat (\( \triangle H = -1.17 \) kJ/mol). The negative sign indicates that energy is given off. We calculate the total heat change by multiplying this enthalpy change by the number of moles reacting, which gives us the heat needed to estimate the final temperature of the solution.
Molar Concentration
Molar concentration, often represented as molarity and with the unit M (moles per liter), is a measure of the concentration of a solute in a solution. In the exercise, we used the molar concentrations of the acid and base to calculate the number of moles present in the given volumes of both reactants.

By understanding molar concentration, we were able to deduce the amount of sodium hydroxide necessary to react completely with the sulfuric acid. This is essential in preparing chemical solutions and conducting reactions that rely on precise stoichiometric proportions for the desired outcome.
Heat Capacity
Heat capacity is a property that describes how much heat energy is required to raise the temperature of a substance by a certain degree. It is often denoted by the symbol (\( c \) and can vary depending on the material and its phase.

In the context of our exercise, we assume the heat capacity of the resulting solution to be that of pure water, which has a specific heat capacity of 4.186 J/g掳C. This assumption allows us to estimate the temperature change of the solution once the reaction has released heat. By multiplying the mass of the solution by its specific heat capacity and the change in temperature, we determine the energy transferred during the reaction, thereby estimating the final temperature.

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Most popular questions from this chapter

A gaseous fuel containing methane and ethane is burned with excess air. The fuel enters the furnace at \(25^{\circ} \mathrm{C}\) and 1 atm, and the air enters at \(200^{\circ} \mathrm{C}\) and 1 atm. The stack gas leaves the furnace at \(800^{\circ} \mathrm{C}\) and 1 atm and contains 5.32 mole\% \(\mathrm{CO}_{2}, 1.60 \%\) CO, \(7.32 \%\) O \(_{2}, 12.24 \% \mathrm{H}_{2} \mathrm{O}\), and the balance \(\mathrm{N}_{2}\). (a) Calculate the molar percentages of methane and ethane in the fuel gas and the percentage excess air fed to the reactor. (b) Calculate the heat (kJ) transferred from the reactor per cubic meter of fuel gas fed. (c) A proposal has been made to lower the feed rate of air to the furnace. State advantages and a drawback of doing so.

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Cumene \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{C}_{3} \mathrm{H}_{7}\right)\) is produced by reacting benzene with propylene \(\left[\Delta H_{\mathrm{r}}\left(77^{\circ} \mathrm{F}\right)=-39,520 \mathrm{Btu}\right]\) A liquid feed containing 75 mole \(\%\) propylene and \(25 \%\) n-butane and a second liquid stream containing essentially pure benzene are fed to the reactor. Fresh benzene and recycled benzene, both at \(77^{\circ} \mathrm{F},\) are mixed in a 1: 3 ratio \((1 \text { mole fresh feed } / 3\) moles recycle) and passed through a heat exchanger, where they are heated by the reactor effluent before being fed to the reactor. The reactor effluent enters the exchanger at \(400^{\circ} \mathrm{F}\) and leaves at \(200^{\circ} \mathrm{F}\). The pressure in the reactor is sufficient to maintain the effluent stream as a liquid. After being cooled in the heat exchanger, the reactor effluent is fed to a distillation column (T1). All of the butane and unreacted propylene are removed as overhead product from the column, and the cumene and unreacted benzene are removed as bottoms product and fed to a second distillation column (T2) where they are scparated. The benzenc leaving the top of the sccond column is the recycle that is mixed with the fresh benzene feed. Of the propylene fed to the process, \(20 \%\) does not react and leaves in the overhead product from the first distillation column. The production rate of cumene is \(1200 \mathrm{lb}_{\mathrm{m}} / \mathrm{h}\). (a) Calculate the mass flow rates of the streams fed to the reactor, the molar flow rate and composition of the reactor effluent, and the molar flow rate and composition of the overhead product from the first distillation column, T1. (b) Calculate the temperature of the benzene stream fed to the reactor and the required rate of heat addition to or removal from the reactor. Use the following approximate heat capacities in your calculations: \(C_{p}\left[\operatorname{Btu} /\left(\operatorname{lb}_{m} \cdot F\right)\right]=0.57\) for propylene, 0.55 for butane, 0.45 for benzene, and 0.40 for cumene. (c) Most people unfamiliar with the chemical process industry imagine that chemical engineers are people who deal mainly with chemical reactions carried out on a large scale. In fact, in most industrial processes, a visitor to the plant would have trouble finding the reactor in a maze of towers and tanks and pipes that were added to the process design to improve the profitability of the process. Briefly explain how the heat exchanger, the two distillation columns, and the recycle stream in the cumene process serve that function.

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Hydrogen is produced in the steam reforming of propane: $$\mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g})+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{v}) \rightarrow 3 \mathrm{CO}(\mathrm{g})+7 \mathrm{H}_{2}(\mathrm{g})$$ The water-gas shift reaction also takes place in the reactor, leading to the formation of additional hydrogen: $$\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{v}) \rightarrow \mathrm{CO}_{2}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ The reaction is carried out over a nickel catalyst in the tubes of a shell- and-tube reactor. The feed to the reactor contains steam and propane in a 6: 1 molar ratio at \(125^{\circ} \mathrm{C}\), and the products emerge at \(800^{\circ} \mathrm{C}\). The excess steam in the feed assures essentially complete consumption of the propane. Heat is added to the reaction mixture by passing the exhaust gas from a nearby boiler over the outside of the tubes that contain the catalyst. The gas is fed at \(4.94 \mathrm{m}^{3} / \mathrm{mol} \mathrm{C}_{3} \mathrm{H}_{8}\), entering the unit at \(1400^{\circ} \mathrm{C}\) and 1 atm and leaving at \(900^{\circ} \mathrm{C} .\) The unit may be considered adiabatic. (a) Calculate the molar composition of the product gas, assuming that the heat capacity of the heating gas is \(0.040 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\cdot} \mathrm{C}\right)\) (b) Is the reaction process exothermic or endothermic? Explain how you know. Then explain how running the reaction in a reactor-heat exchanger improves the process economy.
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