Question

When \(1.50 \mathrm{~g}\) of magnesium metal is allowed to react with \(200 \mathrm{~mL}\) of \(6.00 \mathrm{M}\) aqueous \(\mathrm{HCl}\), the temperature rises from \(25.0^{\circ} \mathrm{C}\) to \(42.9^{\circ} \mathrm{C} .\) Calculate \(\Delta H\) in kilojoules for the reaction, assuming that the heat capacity of the calorimeter is \(776 \mathrm{~J} /{ }^{\circ} \mathrm{C}\), that the specific heat of the final solution is the same as that of water \(\left[\left(4.18 \mathrm{~J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\right]\right.\), and that the density of the solution is \(1.00 \mathrm{~g} / \mathrm{mL}\).

Short answer

\(\Delta H \approx 469.61 \mathrm{~kJ/mol}\)

Step-by-step solution

Calculate Total Mass of Solution

Since the density of the solution is \(1.00 \mathrm{~g/mL}\), the volume of \(200 \mathrm{~mL}\) corresponds to a mass of \(200 \mathrm{~g}\). Adding the mass of magnesium, the total mass of the solution is: \[ m_{\text{total}} = 200 \mathrm{~g} + 1.50 \mathrm{~g} = 201.50 \mathrm{~g} \]

Calculate Temperature Change

The temperature change \((\Delta T)\) is calculated as: \[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 42.9^{\circ} \mathrm{C} - 25.0^{\circ} \mathrm{C} = 17.9^{\circ} \mathrm{C} \]

Calculate Heat Absorbed by Solution

Using the specific heat capacity \(c = 4.18 \mathrm{~J}/\mathrm{g}^{\circ} \mathrm{C}\), the mass of the solution \(m = 201.50 \mathrm{~g}\), and the temperature change \(\Delta T = 17.9^{\circ} \mathrm{C}\):\[ q_{\text{solution}} = m \times c \times \Delta T = 201.50 \times 4.18 \times 17.9 = 15084.37 \mathrm{~J} \]

Calculate Heat Absorbed by Calorimeter

Using the heat capacity of the calorimeter \(C = 776 \mathrm{~J}/^{\circ} \mathrm{C}\), and \(\Delta T = 17.9^{\circ} \mathrm{C}\):\[ q_{\text{calorimeter}} = C \times \Delta T = 776 \times 17.9 = 13890.4 \mathrm{~J} \]

Calculate Total Heat Absorbed by System

The total heat absorbed \(q_{\text{total}}\) by the solution and calorimeter is the sum of \(q_{\text{solution}}\) and \(q_{\text{calorimeter}}\):\[ q_{\text{total}} = q_{\text{solution}} + q_{\text{calorimeter}} = 15084.37 + 13890.4 = 28974.77 \mathrm{~J} \]

Calculate Moles of Magnesium

The molar mass of magnesium is \(24.31 \mathrm{~g/mol}\). The number of moles \(n\) is given by: \[ n = \frac{1.50 \mathrm{~g}}{24.31 \mathrm{~g/mol}} = 0.0617 \mathrm{~mol} \]

Calculate Enthalpy Change (\(\Delta H\))

The enthalpy change \(\Delta H\) for the reaction is the total heat absorbed divided by \(n\), converted to kilojoules: \[ \Delta H = \frac{28974.77 \mathrm{~J}}{0.0617 \mathrm{~mol}} \approx 469,612.80 \mathrm{~J/mol} = 469.61 \mathrm{~kJ/mol} \]

Key concepts

Calorimetry

Calorimetry is a method used to measure the amount of heat involved in chemical reactions or physical changes. It helps us understand how much energy is being absorbed or released during a reaction. In simple terms, calorimetry allows us to "take the temperature" of a reaction. There are different types of calorimeters, from simple coffee-cup calorimeters to more advanced bomb calorimeters. In the context of chemistry, calorimetry is very useful because it enables chemists to measure the heat of reaction directly.

In this exercise, the calorimeter used has a known heat capacity, which means it can absorb a certain amount of heat to raise its temperature by one degree Celsius. By tracking the temperature change ( ext{\( \Delta T \)}), we can determine the total energy involved in the reaction, assuming no heat is lost to the surroundings. Let's recall how the calorimeter absorbs part of that heat, just like how the solution absorbs its share, which makes calorimetry measurements precise.

While performing calorimetry, ensure the calorimeter setup is insulated to prevent any significant heat loss, as such losses could lead to inaccuracies in measurements.

Specific Heat Capacity

Specific heat capacity is a property that tells us how much heat a substance can absorb before its temperature changes by a certain amount. It is measured in units of Joules per gram per degree Celsius (\( \text{J/g}^{\circ}\text{C} \)). Different substances have different specific heat capacities.

In this problem, the solution's specific heat capacity is assumed to be the same as water, which is 4.18 J/g°C. This means for every gram of the solution, 4.18 Joules are needed to raise the temperature by one degree Celsius. This is a convenient assumption since water's specific heat capacity is well-known and is often used as a standard in calorimetry calculations.

Calculating the heat absorbed by the solution involves multiplying the specific heat capacity by the mass of the solution and the change in temperature. These steps allow us to quantify how much energy the solution itself absorbs during the chemical reaction.

Chemical Reaction

A chemical reaction involves the transformation of reactants into products, often accompanied by the release or absorption of energy. In this case, magnesium reacts with hydrochloric acid (HCl) to produce magnesium chloride and hydrogen gas. The reaction is:\[ \text{Mg (s) + 2 HCl (aq) } \rightarrow \text{MgCl}_2 \text{ (aq) + H}_2 \text{ (g)} \]This exercise problem focuses on calculating the enthalpy change (\(\Delta H \)) for this reaction, which is a measure of the total heat absorbed or released when the reaction occurs at constant pressure. Here, magnesium is the limiting reactant as we use its entire mass for our calculations.

Understanding chemical reaction dynamics helps us understand which products invite more heat and which ones release more heat during the process. For this reaction, the considerable rise in temperature indicated that a significant amount of heat was released, which is characteristic of exothermic reactions.

Recognizing that the chemical reaction's enthalpy change relates to the calculated total heat absorbed, it can be thought of as a bridge between measured calorimetric data and theoretical energy changes in the reaction.