Problem 81 - Chemistry and Chemical Reactivity

Question

The heat evolved in the decomposition of \(7.647 \mathrm{g}\) of ammonium nitrate can be measured in a bomb calorimeter. The reaction that occurs is $$\mathrm{NH}_{4} \mathrm{NO}_{3}(\mathrm{s}) \longrightarrow \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$ The temperature of the calorimeter, which contains \(415 \mathrm{g}\) of water, increases from \(18.90^{\circ} \mathrm{C}\) to \(20.72^{\circ} \mathrm{C}\). The heat capacity of the bomb is \(155 \mathrm{J} / \mathrm{K}\). What quantity of heat is evolved in this reaction, in \(\mathrm{kJ} / \mathrm{mol}\) ?

Step-by-Step Solution

Answer

The heat evolved is \(-35.82 \text{kJ/mol}\) for the decomposition of ammonium nitrate.

Step 1: Calculate Temperature Change

First, determine the temperature change of the calorimeter. The initial temperature is \(18.90^{\circ}\text{C}\) and the final temperature is \(20.72^{\circ}\text{C}\). Thus, the change in temperature, \(\Delta T\), is calculated as follows: \[\Delta T = 20.72 \degree\text{C} - 18.90 \degree\text{C} = 1.82 \degree\text{C}\] or \(1.82\text{K}\) since the temperature scale in Celsius and Kelvin can be used interchangeably for changes.

Step 2: Calculate Heat Absorbed by Water

The heat absorbed by the water can be calculated using the formula: \[ q = m \cdot c \cdot \Delta T \]where \(m\) is the mass of the water \(415 \text{ g}\), \(c\) the specific heat capacity of water \(4.184 \text{ J/g}\cdot\text{K}\), and \(\Delta T\) the temperature change \(1.82 \text{K}\).\[ q = 415 \times 4.184 \times 1.82 = 3139.22 \text{ J} \]

Step 3: Calculate Heat Absorbed by the Bomb

The heat absorbed by the bomb calorimeter is given by:\[ q_{bomb} = C_{bomb} \cdot \Delta T \]where \(C_{bomb}\) is the calorimeter heat capacity \(155 \text{ J/K}\). Using the temperature change from Step 1:\[ q_{bomb} = 155 \times 1.82 = 282.1 \text{ J} \]

Step 4: Total Heat Evolvement

The total heat evolved in the reaction is the sum of the heat absorbed by the water and the bomb. Since the heat from the reaction is released, we take this total as negative:\[ q_{total} = - (q + q_{bomb}) = - (3139.22 + 282.1) = -3421.32 \text{ J} \]

Step 5: Convert Heat to kJ

Convert the heat evolved from Joules to kilojoules:\[ q_{total} = -3.42132 \text{ kJ} \]

Step 6: Calculate Heat per Mol of Ammonium Nitrate

First, calculate the moles of \(\text{NH}_4\text{NO}_3\) decomposed using its molar mass:\[ \text{Molecular Weight of NH}_4\text{NO}_3 = 14 + 4 + 14 + (16 \times 3) = 80.043 \text{ g/mol} \] \[ n = \frac{7.647}{80.043} = 0.0955 \text{ mol} \]Now, calculate the heat evolved per mole:\[ q_{molar} = \frac{-3.42132}{0.0955} = -35.82 \text{ kJ/mol} \]

Key Concepts

Heat Capacity

Heat capacity is an essential concept in understanding calorimetry, which is the study of measuring the amount of heat absorbed or released during a chemical reaction. It defines how much heat a substance needs to change its temperature by one degree Celsius or Kelvin.
The formula to calculate the heat involved in temperature changes is given by:

• \[ q = C \cdot \Delta T \]

where

• \( q \) is the heat absorbed or released,
• \( C \) is the heat capacity,
• and \( \Delta T \) is the temperature change.

In the context of the ammonium nitrate decomposition exercise, the calorimeter (a device used to measure changes in thermal energy) has a known heat capacity, which allows us to quantify the heat change associated with the chemical reaction.
The heat capacity of the calorimeter itself is separate from that of the water it contains, which is crucial in calculating the total thermal change. Understanding how to measure and apply heat capacity is vital when analyzing energy changes in chemical processes.

Ammonium Nitrate Decomposition

Ammonium nitrate decomposition is a process where ammonium nitrate (\( \text{NH}_4 ext{NO}_3 \)) breaks down into nitrous oxide (\( \text{N}_2 ext{O} \)) and water (\( \text{H}_2 ext{O} \)).
Here's the chemical equation representing the reaction:

• \[ \text{NH}_4\text{NO}_3(\text{s}) \rightarrow \text{N}_2\text{O}(\text{g}) + 2\text{H}_2\text{O}(\text{g}) \]

This reaction absorbs or releases energy, interchangeably described as either endothermic or exothermic in different contexts.
In the provided exercise, the decomposition of ammonium nitrate is an endothermic process, meaning that it requires or absorbs heat.
To measure how much heat is involved, calorimeters are used. They help us determine the amount of energy (calorimetry) that changes during the chemical reaction by observing temperature changes in the surroundings, such as water.During the exercise, measuring the weight of ammonium nitrate and determining the molar mass allowed us to calculate the moles involved. This then led to calculating the heat transfer per mole, revealing the energy dynamics of the decomposition process.

Chemical Thermodynamics

Chemical thermodynamics is a branch of chemistry that deals with the interplay of heat and energy in chemical processes. It allows chemists to predict the direction of reactions and the energy changes that occur.
Key concepts in chemical thermodynamics include:

• Systems and Surroundings: This defines the part of the universe we are focused on (system) and everything else (surroundings).
• Endothermic and Exothermic Processes: Endothermic processes absorb heat, while exothermic processes release heat.
• Enthalpy (\( H \)): A measure of total heat content in a system. The change in enthalpy (\( \Delta H \)) during a reaction indicates whether the reaction absorbs or releases heat.

In the ammonium nitrate decomposition exercise, the system under study is the chemical reaction happening inside the bomb calorimeter.
By analyzing temperature changes in the water and calorimeter, scientists deduce enthalpy changes to understand the energetics better.Measurements showed that the heat evolved during the decomposition amounted to \(-35.82 \text{ kJ/mol}\), highlighting an endothermic process where energy is absorbed for the decomposition of one mole of ammonium nitrate. This showcases how chemical thermodynamics can intricately explain the energy landscape of chemical reactions.