Question

A quantity of \(1.6 \mathrm{~g}\) sample of \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) is decomposed in a bomb calorimeter. The temperature of the calorimeter decreases by \(6.0 \mathrm{~K}\). The heat capacity of the calorimeter system is \(1.25 \mathrm{~kJ} / \mathrm{K}\). The molar heat of decomposition for \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) is (a) \(7.5 \mathrm{~kJ} / \mathrm{mol}\) (b) \(-600 \mathrm{~kJ} / \mathrm{mol}\) (c) \(-375 \mathrm{~kJ} / \mathrm{mol}\) (d) \(375 \mathrm{~kJ} / \mathrm{mol}\)

Short answer

The correct answer is (c) -375 \text{ kJ/mol}.

Step-by-step solution

Calculate the heat absorbed/released by the calorimeter

Use the formula Q = C * ΔT, where Q is the heat absorbed/released, C is the heat capacity, and ΔT is the temperature change. Since the temperature decreases, it indicates that the heat is released by the sample and absorbed by the calorimeter.

Determine the total heat change

To find out the total heat change, multiply the heat capacity of the calorimeter by the temperature change. In this case, Q = (1.25 kJ/K) * (6.0 K).

Convert grams to moles

To find the number of moles of NH4NO3, use the molar mass of NH4NO3 (M = 18+14+3(16)+14+3(16) = 80 g/mol) and divide the mass of the sample by the molar mass.

Determine the molar heat of decomposition

By dividing the total heat change by the number of moles of NH4NO3, the molar heat of decomposition is calculated. Take note that the sign of the molar heat of decomposition should be negative, as the decomposition is exothermic.

Select the correct answer

The calculated molar heat of decomposition should be compared with the given options to select the correct answer.

Key concepts

Bomb Calorimeter

A bomb calorimeter is an essential apparatus used in thermochemical experiments to measure the heat changes of chemical reactions under constant volume conditions. It consists of a sturdy container called a bomb, where the sample is placed and an oxygen atmosphere is provided for combustion. The bomb is submerged in a water bath to absorb the heat released or absorbed by the reaction.

To accurately gauge the heat change, the calorimeter has a known heat capacity – a measure of how much heat energy is needed to raise its temperature by one degree. When a chemical reaction occurs inside the bomb, the heat exchanged with the surrounding water causes a temperature change, which, when multiplied by the calorimeter's heat capacity, gives the total heat change in the system. It's crucial to understand that the magnitude of the temperature change and the heat capacity directly influence the accuracy of the enthalpy determination.

Thermochemistry

Thermochemistry is the study of energy and heat associated with chemical reactions. The central concept in thermochemistry is enthalpy (H), which is the total heat content of a system. Reactions can be exothermic, where they release heat, or endothermic, where they absorb heat.

In your problem, the decomposition of NH_4NO_3 is exothermic, indicated by the temperature decrease in the calorimeter. The transfer of heat from the reaction to the calorimeter's water bath results in measuring the amount of heat released during the reaction – this is essentially the molar heat of decomposition. Understanding these enthalpy changes is crucial for various applications, including material science, biochemistry, and environmental science.

Stoichiometry of Reactions

Stoichiometry deals with the quantitative aspect of chemical reactions, focusing on the relationships between reactants and products. To comprehend any given reaction's stoichiometry, you need to have a balanced chemical equation, which respects the law of conservation of mass.

In the exercise, stoichiometry is used when converting the mass of NH_4NO_3 to moles. This step is fundamental as it allows us to link the measurable quantity (mass) to the theoretical one (amount of substance in moles). By utilizing molar mass – moles per gram – you can establish the amount of substance that actually participated in the reaction. This quantitative detail enables calculations on the macroscopic scale, like predicting yields or determining reaction conditions.

Chemical Thermodynamics

Chemical thermodynamics encompasses the principles of energy transfer and conversion in chemical processes. It combines the concepts of thermochemistry with the laws of thermodynamics to predict the direction of processes, their spontaneity, and the equilibria involved.

In relation to the textbook problem, thermodynamics tells us that energy cannot be created or destroyed, only converted. Here, the chemical potential energy stored within the bonds of NH_4NO_3 is transformed into thermal energy, which emphasizes the importance of thermochemical equations, enthalpies of reactions, and the understanding of system surroundings. Thermodynamic concepts also provide an understanding as to why certain reactions are favorable under specific conditions and how altering these conditions can shift the reaction balance.