Question

Meals-ready-to-eat (MREs) are military meals that can be heated on a flameless heater. The heat is produced by the following reaction: \(\mathrm{Mg}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow\) \(\mathrm{Mg}(\mathrm{OH})_{2}(s)+\mathrm{H}_{2}(g)\) (a) Calculate the standard enthalpy change for this reaction. (b) Calculate the number of grams of \(\mathrm{Mg}\) needed for this reaction to release enough energy to increase the temperature of \(25 \mathrm{~mL}\) of water from \(15^{\circ} \mathrm{C}\) to \(85^{\circ} \mathrm{C}\).

Short answer

(a) The standard enthalpy change for the given reaction is -924.7 kJ/mol. (b) 0.1926 grams of Mg are needed for the reaction to release enough energy to increase the temperature of 25 mL of water from 15°C to 85°C.

Step-by-step solution

(a) Calculate the standard enthalpy change for the reaction.

First, we need to find the standard enthalpies of formation for the reactants and products. In this case, we'll look up the standard enthalpy of formation for magnesium hydroxide and hydrogen gas. The values are as follows: - \(\Delta H_f^{\circ}[\mathrm{Mg}(\mathrm{OH})_{2}(s)] = -924.7\ \mathrm{kJ/mol}\) - \(\Delta H_f^{\circ}[\mathrm{H}_2(g)] = 0\ \mathrm{kJ/mol}\) (since it's a pure element in its natural state) Now, apply the formula to find the enthalpy change for the reaction: \(\Delta H_{reaction}^{\circ} = \sum[\Delta H_{products}^{\circ}] - \sum[\Delta H_{reactants}^{\circ}]\) Since there's no enthalpy of formation for pure elements like Mg and H2O, the enthalpy change for the reaction is: \(\Delta H_{reaction}^{\circ} = [\Delta H_f^{\circ}[\mathrm{Mg}(\mathrm{OH})_{2}(s)] + \Delta H_f^{\circ}[\mathrm{H}_2(g)]] - [\Delta H_f^{\circ}[\mathrm{Mg}(s)] + \Delta H_f^{\circ}[2\mathrm{H}_2\mathrm{O}(l)]]\) \(\Delta H_{reaction}^{\circ} = (-924.7\ \mathrm{kJ/mol} + 0\ \mathrm{kJ/mol}) - (0\ \mathrm{kJ/mol} + 0\ \mathrm{kJ/mol})\) \(\Delta H_{reaction}^{\circ} = -924.7\ \mathrm{kJ/mol}\) So the standard enthalpy change for the given reaction is -924.7 kJ/mol.

(b) Calculate the number of grams of Mg needed for the reaction.

Now that we have the enthalpy change for the reaction, we can determine the quantity of Mg needed to release enough energy to raise 25 mL of water from 15°C to 85°C. We will use the following equation: \(q = mc\Delta T\) where: - \(q\) is the heat (energy) released or absorbed, - \(m\) is the mass of the substance, - \(c\) is the specific heat capacity, and - \(\Delta T\) is the change in temperature. The specific heat capacity for water is \(4.18 \ \mathrm{J/g^{\circ}C}\). The mass of 25 mL of water is approximately 25 grams (since the density of water is nearly 1 g/mL). The temperature change is from 15°C to 85°C, making the change 70°C. Now, we can calculate the quantity of heat needed to change the temperature of the water: \(q = (25\ \mathrm{g})(4.18 \ \mathrm{J/g^{\circ}C})(70^{\circ}\mathrm{C}) = 7325\ \mathrm{J}\) To determine the number of grams of Mg needed, we will use the relationship between the heat and the enthalpy change: \(-\Delta H_{reaction} = \frac{q}{n}\) where \(n\) is the moles of Mg We have the enthalpy change and the heat required, and we need to find the moles of Mg: \(-(-924.7\ \mathrm{kJ/mol} ) = \frac{7325\ \mathrm{J}}{n }\) 924.7 kJ = 924700 J (converting kJ to J) \(n = \frac{7325\ \mathrm{J}}{924700\ \mathrm{J/mol}} = 0.00792\ \mathrm{mol}\) Now, we can convert moles of Mg to grams by multiplying by the molar mass of Mg: mass of Mg = moles of Mg × molar mass of Mg mass of Mg = (0.00792 mol) × (24.31 g/mol) mass of Mg = 0.1926 g So, 0.1926 grams of Mg are needed for the reaction to release enough energy to increase the temperature of 25 mL of water from 15°C to 85°C.

Key concepts

Understanding Enthalpy Change

Enthalpy change is a crucial concept in thermochemistry. It is the measure of heat exchanged in a chemical reaction at constant pressure. The symbol for enthalpy change is \( \Delta H \). You can think of it as the energy stored in the bonds of molecules. When a reaction occurs, bonds are broken and formed, resulting in either the release or absorption of energy.
A negative enthalpy change, like the \( \Delta H_{reaction}^{\circ} = -924.7\ \mathrm{kJ/mol} \) in our example, signifies an exothermic reaction, meaning it releases heat. Conversely, a positive value would indicate an endothermic reaction where heat is absorbed.
Calculating the enthalpy change involves using the standard enthalpies of formation for the products and reactants. Here's the formula used: \[ \Delta H_{reaction}^{\circ} = \sum[\Delta H_{products}^{\circ}] - \sum[\Delta H_{reactants}^{\circ}] \]
In essence, understanding whether a reaction is releasing or absorbing heat helps in predicting the behavior of the system and its surroundings.

Specific Heat Capacity Explained

Specific heat capacity is a term used to describe the amount of heat needed to raise the temperature of one gram of a substance by one degree Celsius. It is a property intrinsic to each material and is essential for calculating heat changes in chemical reactions. The specific heat capacity is represented by the symbol \( c \).
In the case of water, the specific heat capacity is \( 4.18\ \mathrm{J/g^{\circ}C} \). This value indicates water's ability to absorb significant amounts of heat without a large change in temperature, which is why water is such an effective coolant.
When you know the specific heat capacity, you can calculate the total heat energy required to change a substance's temperature using the equation: \[ q = mc\Delta T \]

• \( q \) is the total heat in Joules (J).
• \( m \) stands for mass in grams (g).
• \( \Delta T \) represents the change in temperature in degrees Celsius (\( ^{\circ}C \)).

This equation is vital for understanding energy exchanges in chemical processes and for conducting experiments that involve temperature changes.

Chemical Reactions and Energy Changes

Chemical reactions are the foundation of thermochemistry, providing insights into how materials interact and transform. Each reaction is characterized by the breaking and forming of chemical bonds, which involve energy transfers.
During a chemical reaction, bonds between atoms in the reactants must break before new bonds form in the products. This change in bonding is where the energy of the reaction comes in. If more energy is required to break the bonds than to form them, the reaction absorbs energy and is endothermic. Conversely, if forming new bonds releases more energy than breaking the old ones, it is exothermic.
In thermochemistry, we often focus on exothermic reactions because of their real-world applications, like heating MREs (Meals Ready to Eat) used by the military. For this kind of reaction, calculating the enthalpy change helps us understand how much energy is released and how it can be utilized efficiently.
Knowing these concepts allows scientists and engineers to design processes that strategically utilize or mitigate energy changes, ensuring safety and efficiency in various applications.