Problem 103 - Chemistry and Chemical Reactivity

Question

When 0.850 g of Mg was burned in oxygen in a constant volume calorimeter, \(25.4 \mathrm{kJ}\) of energy as heat was evolved. The calorimeter was in an insulated container with \(750 . \mathrm{g}\) of water at an initial temperature of \(18 . \overline{6}^{\circ} \mathrm{C}\). The heat capacity of the bomb in the calorimeter is \(820 . \mathrm{J} / \mathrm{K}\) (a) Calculate \(\Delta U\) for the oxidation of \(\mathrm{Mg}\) (in \(\mathrm{k} \mathrm{J} / \mathrm{mol}\) \(\mathrm{Mg})\) (b) What will be the final temperature of the water and the bomb calorimeter in this experiment?

Step-by-Step Solution

Answer

(a) \( \Delta U = 726.6 \, \text{kJ/mol Mg} \); (b) Final temperature: \( 25.0° \text{C} \).

Step 1: Determine Moles of Mg

To calculate \( \Delta U \), we first need the amount of magnesium burned in moles. The molar mass of \( \text{Mg} \) is \( 24.305 \, \text{g/mol} \). Calculate it as follows:\[ \text{moles of Mg} = \frac{0.850 \, \text{g}}{24.305 \, \text{g/mol}} \approx 0.03496 \, \text{mol} \]

Step 2: Calculate Energy Change per Mole of Mg

The energy evolved, \( q \), is 25.4 kJ. To find \( \Delta U \) per mole of magnesium, divide the total energy by the moles of Mg burned:\[ \Delta U = \frac{25.4 \, \text{kJ}}{0.03496 \, \text{mol}} \approx 726.6 \, \text{kJ/mol} \]

Step 3: Calculate Total Heat Capacity of System

The total heat capacity of the system includes both the water and the calorimeter: - Water: \( C_{\text{water}} = m \times c = 750 \, \text{g} \times 4.184 \, \text{J/g°C} = 3138 \, \text{J/°C} \)- Bomb: \( C_{\text{bomb}} = 820 \, \text{J/K} \)The combined heat capacity \( C_{\text{total}} = 3138 + 820 = 3958 \, \text{J/°C} \).

Step 4: Calculate Temperature Change

Use the formula \( q = C \times \Delta T \) to calculate the temperature change \( \Delta T \). Convert 25.4 kJ to J (1 kJ = 1000 J): \[ 25.4 \, \text{kJ} = 25400 \, \text{J} \]Then, solve for \( \Delta T \):\[ 25400 = 3958 \times \Delta T \]\[ \Delta T = \frac{25400}{3958} \approx 6.42^ {\circ} \text{C} \]

Step 5: Calculate Final Temperature

Add the temperature change to the initial temperature to find the final temperature:\[ T_{\text{final}} = T_{\text{initial}} + \Delta T = 18.6° \text{C} + 6.42° \text{C} \approx 25.0° \text{C} \].

Key Concepts

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It is a critical technique in thermochemistry as it allows us to quantify energy changes during a process. In the context of this exercise, a constant volume calorimeter, also known as a bomb calorimeter, is used to measure the heat released from burning magnesium.

In a bomb calorimeter:

• Reactions occur in a sealed container insulated from the environment.
• The heat generated by the reaction raises the temperature of the water and calorimeter surrounding it.
• By measuring this temperature change, we can calculate the heat evolved in the reaction.

Understanding calorimetry is essential because it provides insights into energy changes during reactions. It reveals important details about reaction enthalpies, helping students grasp broad thermochemical concepts.

Heat Capacity

The Heat Capacity of a substance is a measure of how much energy it takes to raise the temperature of a given quantity of the substance by one degree Kelvin (or Celsius). In the problem, the total heat capacity includes both the calorimeter's bomb and the water.

The equation used is:\[ C_{\text{total}} = C_{\text{water}} + C_{\text{bomb}} \]

• Where \( C_{\text{water}} \) is calculated based on mass and specific heat (4.184 J/g°C for water).
• \( C_{\text{bomb}} \) represents the heat capacity of the calorimeter's bomb, given as 820 J/K.

By determining the total heat capacity, you can calculate how the entire system responds to the energy released through a chemical reaction. This calculation is crucial for finding the temperature change, which subsequently informs us how much energy was transferred in a specific process.

Energy Change

Understanding the concept of Energy Change is vital in thermochemistry. We measure energy change, often denoted as \( \Delta U \), to understand the dynamics of a given reaction. In this exercise, the change in energy is calculated from the heat emitted during the combustion of magnesium.

When a reaction occurs, energy can be absorbed or released. In our specific problem, burning magnesium evolves heat, indicating it is an exothermic reaction.

To determine \( \Delta U \):

• Calculate the moles of magnesium used in the reaction with its given mass and molar mass (24.305 g/mol).
• Divide the total heat energy (25.4 kJ) by the moles of magnesium.

The result will give you the energy change per mole, providing insight into the energy exchange in the reaction. Such calculations help predict reaction behaviors, essential for both academic studies and practical applications in various scientific fields.