Question

Lime is a term that includes calcium oxide \((\mathrm{CaO},\) also called quicklime) and calcium hydroxide \(\left[\mathrm{Ca}(\mathrm{OH})_{2}\right.\) also called slaked lime]. It is used in the steel industry to remove acidic impurities, in air-pollution control to remove acidic oxides such as \(\mathrm{SO}_{2}\), and in water treatment. Quicklime is made industrially by heating limestone \(\left(\mathrm{CaCO}_{3}\right)\) above \(2000^{\circ} \mathrm{C}:\) \(\begin{aligned} \mathrm{CaCO}_{3}(s) \longrightarrow \mathrm{CaO}(s)+\mathrm{CO}_{2}(g) & \Delta H^{\circ}=177.8 \mathrm{~kJ} / \mathrm{mol} \end{aligned}\) Slaked lime is produced by treating quicklime with water: \(\mathrm{CaO}(s)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Ca}(\mathrm{OH})_{2}(s)_{\Delta H^{\circ}}=-65.2 \mathrm{~kJ} / \mathrm{mol}\) The exothermic reaction of quicklime with water and the rather small specific heats of both quicklime \(\left[0.946 \mathrm{~J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\right]\) and slaked lime \(\left[1.20 \mathrm{~J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\right]\) make it hazardous to store and transport lime in vessels made of wood. Wooden sailing ships carrying lime would occasionally catch fire when water leaked into the hold. (a) If a 500.0 -g sample of water reacts with an equimolar amount of \(\mathrm{CaO}\) (both at an initial temperature of \(\left.25^{\circ} \mathrm{C}\right)\), what is the final temperature of the product, \(\mathrm{Ca}(\mathrm{OH})_{2} ?\) Assume that the product absorbs all the heat released in the reaction. (b) Given that the standard enthalpies of formation of \(\mathrm{CaO}\) and \(\mathrm{H}_{2} \mathrm{O}\) are -635.6 and \(-285.8 \mathrm{~kJ} / \mathrm{mol}\), respectively, calculate the standard enthalpy of formation of \(\mathrm{Ca}(\mathrm{OH})_{2}\).

Short answer

(a) \(758.30^{\circ}C\); (b) \(-986.6\ \mathrm{kJ/mol}\).

Step-by-step solution

Find the Moles of Water

The molar mass of water (\(\mathrm{H}_2\mathrm{O}\)) is 18.02 g/mol. Calculate the moles of water using the given 500.0 g sample.\[moles\ of\ \mathrm{H}_2\mathrm{O} = \frac{500.0\ g}{18.02\ g/mol} \approx 27.76\ mol\]

Equimolar Amount of \(\mathrm{CaO}\)

Since the reaction is equimolar, the number of moles of \(\mathrm{CaO}\) will be equal to the moles of \(\mathrm{H}_2\mathrm{O}\). Thus, we have 27.76 moles of \(\mathrm{CaO}\).

Calculate Total Heat Released

The reaction releases \(-65.2\ \mathrm{kJ/mol}\). Calculate the total heat released using the number of moles.\[\text{Total heat released} = 27.76\ mol \times (-65.2\ \mathrm{kJ/mol}) = -1809.95\ \mathrm{kJ}\]

Find the Moles of Product, \(\mathrm{Ca(OH)}_2\)

Again, using stoichiometry and the equimolar condition, the moles of \(\mathrm{Ca(OH)}_2\) will be 27.76 moles.

Calculate Final Temperature of \(\mathrm{Ca(OH)}_2\)

Assuming all released heat is absorbed in heating \(\mathrm{Ca(OH)}_2\), use the specific heat and mass to find final temperature. The molar mass of \(\mathrm{Ca(OH)}_2\) is 74.1 g/mol.Mass of \(\mathrm{Ca(OH)}_2\) = \(27.76\ \mathrm{mol} \times 74.1\ \mathrm{g/mol} = 2056.536\ \mathrm{g}\)Use the specific heat formula:\[q = mc\Delta T\]Rearranging to find \(\Delta T\):\[-1809.95\times 10^3 \text{ J} = 2056.536\ \mathrm{g}\times 1.20\ \mathrm{J/g^{\circ}C}\times \Delta T\]Solving for \(\Delta T\):\[\Delta T \approx 733.30^{\circ}C\]Since the initial temperature is \(25^{\circ} C\), the final temperature is:\[T_{final} = 25 + 733.30 = 758.30^{\circ}C\]

Use Hess's Law for Standard Enthalpy of Formation

The enthalpy of formation reaction is \(\mathrm{CaO}(s) + \mathrm{H}_2\mathrm{O}(l) \rightarrow \mathrm{Ca(OH)}_2(s)\). Given:- \(\Delta H_{rxn} = -65.2\ \mathrm{kJ/mol}\), - \(\Delta H^f_{\mathrm{CaO}} = -635.6\ \mathrm{kJ/mol}\),- \(\Delta H^f_{\mathrm{H_2O}} = -285.8\ \mathrm{kJ/mol}\).Using Hess's Law:\[\Delta H^f_{\mathrm{Ca(OH)}_2} = \left(\Delta H_{rxn} + \Delta H^f_{\mathrm{CaO}} + \Delta H^f_{\mathrm{H_2O}}\right)\]\[= \left(-65.2\ \mathrm{kJ/mol} - 635.6\ \mathrm{kJ/mol} - 285.8\ \mathrm{kJ/mol}\right) = -986.6\ \mathrm{kJ/mol}\]

Provide Final Answer

Based on calculated data:(a) The final temperature of the product, \(\mathrm{Ca(OH)}_2\), is approximately \(758.30^{\circ}C\).(b) The standard enthalpy of formation of \(\mathrm{Ca(OH)}_2\) is \(-986.6\ \mathrm{kJ/mol}\).

Key concepts

Thermochemistry

Thermochemistry is a branch of chemistry that studies the heat evolution or absorption in chemical reactions, which are primarily explored through enthalpy changes (\( \Delta H \). Reactions can be exothermic, releasing heat, or endothermic, absorbing heat. Enthalpy changes are crucial for understanding reaction energetics, influencing how substances react, especially in industrial applications.
For instance, in the exercise above, the reaction between calcium oxide (CaO) and water to form calcium hydroxide (Ca(OH)_2) is exothermic, as it releases -65.2 kJ/mol. This means that energy is transferred to the surroundings as heat. Accounting for energy changes is essential to predict reaction behavior, which informs processes such as safety measures in handling reactive substances.
Moreover, calculating the enthalpies of formation and the total heat involved helps in determining temperature changes in laboratory and industrial contexts. It allows chemists to ensure that equipment is designed to handle the resulting heat, preventing hazardous reactions, like fires on wooden ships historically carrying lime.

Stoichiometry

Stoichiometry involves understanding and calculating the relative quantities of reactants and products in chemical reactions. This concept is foundational in chemistry, allowing you to predict how much product a reaction will yield from given amounts of reactants.
To navigate stoichiometry, the balanced chemical equation serves as your guide. It reveals the molar ratios of each substance involved. For the exercise, equal moles of calcium oxide ( CaO) and water ( H_2O) react to produce calcium hydroxide ( Ca(OH)_2). These stoichiometric relationships help determine the mass amounts needed or produced, ensuring efficiency and accuracy in chemical processes.
By calculating moles, you relate mass and molar mass to other quantities, crucial for experiments, especially those in industrial settings where precision is vital, like producing lime for various applications. Practicing stoichiometry enhances your problem-solving skills in chemistry, emphasizing careful measurement and calculation to ensure desired chemical outcomes.

Calcium Compounds

Calcium compounds, like calcium oxide ( CaO) and calcium hydroxide ( Ca(OH)_2), are highly significant in various industries, including steel production, construction, and environmental management. These compounds play vital roles in processes such as removing impurities, neutralizing acids, and stabilizing biophysical properties in water treatment.
Calcium oxide, known as quicklime, is formed by heating calcium carbonate at high temperatures; this results in the release of carbon dioxide. Upon hydration, quicklime transforms into calcium hydroxide, or slaked lime, through an exothermic reaction that generates substantial heat.
Understanding properties like reactivity, solubility, and thermal behavior is critical when working with calcium compounds. For instance, their role in producing lime involves controlling conditions meticulously to prevent unwanted energy release or reactions, thereby ensuring safety in storage and transportation. Consequently, mastering the energy dynamics and chemical transformations of calcium compounds is crucial for their practical and efficient use in industrial applications.

Exothermic Reactions

Exothermic reactions are processes that release energy in the form of heat to their surroundings. These types of reactions are prevalent in both natural and engineered systems and can be highly beneficial or potentially hazardous.
During an exothermic reaction, like the formation of calcium hydroxide ( Ca(OH)_2) from calcium oxide ( CaO) and water, energy in the form of heat is produced. The released energy can cause significant temperature changes in the surrounding environment. In the exercise, this reaction releases -65.2 kJ/mol, leading to a large temperature increase of the product, highlighting the practical implications regarding safety.
Knowing how to manage and harness this energy is crucial in chemical engineering. Proper insulation and controlled environments are necessary to handle exothermic reactions safely. Additionally, understanding the specifics of such reactions helps develop safer processes for industries that rely on heat exchange, such as in manufacturing products like lime or in pollution control strategies.