Problem 142 - General Chemistry

Question

An industrial process for manufacturing sulfuric acid, \(\mathrm{H}_{2} \mathrm{SO}_{4}\), uses hydrogen sulfide, \(\mathrm{H}_{2} \mathrm{~S}\), from the purification of natural gas. In the first step of this process, the hydrogen sulfide is burned to obtain sulfur dioxide, \(\mathrm{SO}_{2}\). $$ \begin{aligned} 2 \mathrm{H}_{2} \mathrm{~S}(g)+3 \mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+2 \mathrm{SO}_{2}(g) \\ \Delta H^{\circ} &=-1124 \mathrm{~kJ} \end{aligned} $$ The density of sulfur dioxide at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) is \(2.62 \mathrm{~g} / \mathrm{L}\) and the molar heat capacity is \(30.2 \mathrm{~J} /\left(\mathrm{mol} \cdot{ }^{\circ} \mathrm{C}\right) .\) (a) How much heat would be evolved in producing \(1.00 \mathrm{~L}\) of \(\mathrm{SO}_{2}\) at \(25^{\circ} \mathrm{C}\) and \(1.00\) atm? (b) Suppose heat from this reaction is used to heat \(1.00 \mathrm{~L}\) of \(\mathrm{SO}_{2}\) from \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) to \(500^{\circ} \mathrm{C}\) for its use in the next step of the process. What percentage of the heat evolved is required for this?

Step-by-Step Solution

Answer

(a) 22.94 kJ; (b) 2.55% of the evolved heat is required.

Step 1: Understand the Reaction Enthalpy

The given reaction is \(2 \mathrm{H}_{2} \mathrm{~S}(g) + 3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) + 2 \mathrm{SO}_{2}(g)\) with \(\Delta H^{\circ} = -1124\) kJ. This implies that when 2 moles of \(\mathrm{H}_{2} \mathrm{~S}\) react to form 2 moles of \(\mathrm{SO}_{2}\), 1124 kJ of heat is released.

Step 2: Calculate Moles of SO2 in 1 L

Using the density \(2.62 \mathrm{~g/L}\), calculate the mass of \(\mathrm{SO}_{2}\) in 1 L: \[\text{mass} = 2.62 \text{ g/L} \times 1 \text{ L} = 2.62 \text{ g}\] The molar mass of \(\mathrm{SO}_{2}\) is \(32.07 + 2(16.00) = 64.07\text{ g/mol}\). Therefore, the moles of \(\mathrm{SO}_{2}\) are:\[\text{moles of } \mathrm{SO}_{2} = \frac{2.62 \text{ g}}{64.07 \text{ g/mol}} = 0.0409 \text{ moles}\]

Step 3: Heat Evolved for 0.0409 Moles of SO2

Since 1124 kJ is released for 2 moles of \(\mathrm{SO}_{2}\): \[\text{heat evolved for 0.0409 moles} = \frac{1124 \text{ kJ}}{2 \text{ moles}} \times 0.0409 \text{ moles} = 22.94 \text{ kJ}\]

Step 4: Calculate Heat Required to Heat 1 L of SO2

The formula for heat required is: \[ q = nC\Delta T \]where, \( n = 0.0409 \text{ moles} \), \( C = 30.2 \mathrm{~J/mol}/^{\circ} \mathrm{C} \), \( \Delta T = 500^{\circ} \mathrm{C} - 25^{\circ} \mathrm{C} = 475^{\circ} \mathrm{C} \).Convert \(C\) from J to kJ:\[C = 30.2 \frac{\mathrm{J}}{\mathrm{mol} \cdot ^{\circ} \mathrm{C}} \times \frac{1 \text{ kJ}}{1000 \text{ J}} = 0.0302 \text{ kJ/mol}/^{\circ} \mathrm{C} \]Now calculate \(q\):\[q = 0.0409 \text{ moles} \times 0.0302 \text{ kJ/mol}/^{\circ} \mathrm{C} \times 475^{\circ} \mathrm{C} = 0.585 \text{ kJ}\]

Step 5: Calculate Percentage of Heat Required

Calculate the percentage of heat required with respect to heat evolved:\[\text{Percentage} = \frac{0.585 \text{ kJ}}{22.94 \text{ kJ}} \times 100\% = 2.55\%\]

Step 6: Conclusion

(a) The heat evolved when producing \(1.00 \mathrm{~L}\) of \(\mathrm{SO}_{2}\) is \(22.94 \text{ kJ}\). (b) Approximately \(2.55\%\) of this heat is required to raise the temperature of this volume of \(\mathrm{SO}_{2}\) from \(25^{\circ} \mathrm{C}\) to \(500^{\circ} \mathrm{C}\).

Key Concepts

Reaction Enthalpy

Reaction enthalpy is a key concept in understanding how energy changes during a chemical reaction. It is the difference in energy between the reactants and the products. When a reaction releases heat, like the one with hydrogen sulfide and oxygen to create sulfur dioxide and water, the reaction is exothermic, meaning it has a negative enthalpy change, denoted as \( \Delta H^{\circ} = -1124 \text{ kJ} \). This tells us that 1124 kJ of energy is released when two moles of hydrogen sulfide react. This release of energy can further be harnessed to do useful work or heat substances in subsequent steps of a manufacturing process, such as sulfuric acid production.

Molar Heat Capacity

Molar heat capacity is the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius. In our example, sulfur dioxide \((\mathrm{SO}_2)\) has a molar heat capacity of 30.2 J/mol/°C. This value is crucial to determine how much energy is needed to change the temperature of a mole of \(\mathrm{SO}_2\).
The unit J/mol/°C indicates how much heat one mole of a substance can store per degree temperature change. When scaling up to practical quantities, it's essential to multiply this by the number of moles and the desired temperature change to find the total energy needed, which is shown by the formula \( q = nC\Delta T \), where \( q \) is the heat energy in joules or kilojoules.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the quantitative relationships that govern chemical reactions. It helps us figure out how much reactants we need to produce a certain amount of product. In the provided reaction, stoichiometry tells us that 2 moles of hydrogen sulfide react with 3 moles of oxygen to produce 2 moles of sulfur dioxide and water.
Using stoichiometry, we can relate the quantities of the reactants and the resulting products, not only balancing the equation but also helping us calculate the moles of \(\mathrm{SO}_2\) produced, which is needed to determine how much heat is evolved or required for further steps.

Temperature Change in Chemical Reactions

Temperature change in chemical reactions often involves calculating the energy needed to change the temperature of a substance. This relates directly to the heat capacity of the substance and the amount the temperature is changed.
In the exercise, sulfur dioxide is heated from 25°C to 500°C. Using its molar heat capacity and the calculated moles of \(\mathrm{SO}_2\), you can find the total energy required for this change by the equation \( q = nC\Delta T \). This demonstrates not only the energy management in an industrial chemical process but also showcases how the heat evolved from one stage of a reaction can be used effectively in another, optimizing energy usage.