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: $$\mathrm{Mg}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{Mg}(\mathrm{OH})_{2}(s)+2 \mathrm{H}_{2}(g)$$ (a) Calculate the standard enthalpy change for this reaction. (b) Calculate the number of grams of Mg needed for this reaction to release enougy energy to increase the temperature of 75 mL of water from 21 to \(79^{\circ} \mathrm{C}\) .

Short answer

(a) The standard enthalpy change (ΔH) for the reaction is approximately -352.82 kJ/mol. (b) To increase the temperature of 75 mL water from 21°C to 79°C, approximately 1.0 gram of Mg is needed.

Step-by-step solution

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

To calculate the standard enthalpy change, ΔH, for the reaction, we use the following formula: \(ΔH = ∑(standard \ enthalpy \ of \ formation \ of \ products) \ -∑(standard \ enthalpy \ of \ formation \ of \ reactants)\) The standard enthalpies of formation for the given species are: Mg (s): 0 kJ/mol (since it is an element in its standard state) \(H_{2}O (l)\): -285.8 kJ/mol \(Mg(OH)_{2} (s)\): -924.42 kJ/mol \(H_{2} (g)\): 0 kJ/mol (since it is an element in its standard state) Now, we'll plug in these values into the equation: \(ΔH = [(1 \times (-924.42)) + (2 \times 0)] - [(1 \times 0) + (2 \times (-285.8))]\)

Calculate the heat produced:

By computing the above expression, we find: \(\)ΔH = (-924.42) - (-571.6) = -352.82 kJ/mol This is the standard enthalpy change for the reaction.

(b) Determine the mass of Mg needed:

To find the mass of Mg needed to produce enough heat to increase the temperature of 75 mL of water by the given amount, we will use the following equations: 1) \(q = mcΔT\) (where q represents heat, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature) 2) \(\frac{ΔH}{ΔT} = \frac{mass \ of \ Mg \ used}{molar \ mass \ of \ Mg}\) (where ΔH is the standard enthalpy change, ΔT is the change in temperature, and the molar mass of Mg is 24.305 g/mol) We know that the specific heat capacity of water, c, is 4.18 J/g°C and that the density of water is approximately 1 g/mL. First, let's find the mass of the 75 mL of water: mass of water = (density of water) × (volume of water) = 1 g/mL × 75 mL = 75 g Now, calculate the heat required, q, to increase the temperature from 21°C to 79°C: q = mcΔT = (75 g)(4.18 J/g°C)(79-21) = \(75 \times 4.18 \times 58 \approx 14,433.2 \mathrm{J}\), or 14.4332 kJ Since \(ΔH \approx -352.82 \mathrm{kJ/mol}\) for the reaction, we have about 352.82 kJ of heat released for every mole of Mg reacted. Now we can set up a proportion to find the number of grams of Mg needed to release 14.4332 kJ of heat: \(\frac{1 \ mol \ of \ Mg}{-352.82 \ kJ \ heat} = \frac{x \ grams \ of \ Mg}{14.4332 \ kJ \ heat}\) Solve for x, the mass of Mg needed: \(x = Molar\ mass\ of\ Mg\times\frac{14.4332}{352.82} = \frac{24.305 \times 14.4332}{-352.82} \approx 1.0 \ g\) The mass of Mg required to release sufficient heat to increase the temperature of 75 mL of water from 21°C to 79°C is approximately 1.0 gram.

Key concepts

Understanding Enthalpy Change

In thermochemistry, the term enthalpy change, denoted as \( \Delta H \), refers to the heat absorbed or released during a chemical reaction at constant pressure. It is a central concept in understanding energy changes in reactions.

To compute \( \Delta H \), we look at the enthalpy of formation for the reactants and products, which are the energies required to form one mole of a compound from its elements in their standard states. The enthalpy change for a reaction is found using the formula:\[ \Delta H = \sum (\text{standard enthalpy of formation of products}) - \sum (\text{standard enthalpy of formation of reactants}) \]

For the given problem, calculating \( \Delta H \) informs us about the thermal energy transmitted during the hydration of magnesium. This value is essential for understanding how much heat is produced or consumed by the reaction and is used in real-life applications such as heating Meals-ready-to-eat (MREs).

The Heat of Reaction

The heat of reaction is the amount of heat released or absorbed when a reaction takes place at a given temperature and pressure. It's equivalent to the enthalpy change, but it specifically emphasizes the energy exchange with the surroundings. Calculating the heat of reaction is vital for processes that require precise temperature management, like the heat-generating reaction used in MREs.

For endothermic reactions (those that absorb heat), the heat of reaction is positive, indicating that the system absorbs energy. Conversely, for exothermic reactions (those that release heat), like the MRE heating process, it is negative. The conversion from enthalpy change to the amount of heat produced or absorbed is simply a matter of units, where \(1 kJ = 1000 J\).

Molar Mass and Its Role in Calculations

The term molar mass signifies the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a bridge between the mass of a substance and the number of moles, which directly relates to its chemical compositions, such as the number of atoms, molecules, or formula units.

To find out how much of a substance is involved in a reaction, we often have to convert between mass and moles using its molar mass. For example, with magnesium's molar mass of 24.305 g/mol, we translate the amount of heat energy required into the corresponding mass of magnesium needed to produce or absorb that energy. This step is crucial for precise and practical applications like calculating the required amount of a reactant in MRE heaters.

Specific Heat Capacity

The specific heat capacity (c), measured in joules per gram per degree Celsius (J/g°C), describes how much heat energy is needed to raise the temperature of one gram of a substance by one degree Celsius.

It's a unique property of each material and plays a prominent role when gauging the energy needed to change the temperature of a substance over a given interval. Water, for instance, has a high specific heat capacity (4.18 J/g°C), which makes it an excellent substance for absorbing the heat generated in the MRE reaction without undergoing a drastic temperature change.

In the context of our exercise, the specific heat capacity allows us to determine the heat quantity (q) needed to increase water's temperature, using the formula \(q = mc\Delta T\), where \(m\) is the mass of water and \(\Delta T\) is the change in temperature. This calculation is fundamental in ensuring that the MREs provide the necessary warmth for a satisfactory meal.