Question

Instant cold packs used to ice athletic injuries on the field contain ammonium nitrate and water separated by a thin plastic divider. When the divider is broken, the ammonium nitrate dissolves according to the endothermic reaction: $$\mathrm { NH } _ { 4 } \mathrm { NO } _ { 3 } ( s ) \longrightarrow \mathrm { NH } _ { 4 } ^ { + } ( a q ) + \mathrm { NO } _ { 3 } ^ { - } ( a q )$$ In order to measure the enthalpy change for this reaction, 1.25 g of \(\mathrm { NH } _ { 4 } \mathrm { NO } _ { 3 }\) is dissolved in enough water to make 25.0\(\mathrm { mL }\) of solution. The initial temperature is \(25.8 ^ { \circ } \mathrm { C }\) , and the final temperature (after the solid dissolves) is \(21.9 ^ { \circ } \mathrm { C }\) . Calculate the change in enthalpy for the re- action in kJ. (Use 1.0\(\mathrm { g } / \mathrm { mL }\) as the density of the solution and 4.18\(\mathrm { J } / \mathrm { g } \cdot ^ { \circ } \mathrm { C }\) as the specific heat capacity.)

Short answer

The change in enthalpy for the reaction is calculated to be the heat absorbed per mole of ammonium nitrate in kJ/mol.

Step-by-step solution

Identify Given Data

Extract and list all the relevant data from the exercise: mass of ammonium nitrate (m) is 1.25 g, volume of the water solution (V) is 25.0 mL, initial temperature (T_initial) is 25.8 °C, final temperature (T_final) is 21.9 °C, density of the solution (ρ) is 1.0 g/mL, and specific heat capacity (c) is 4.18 J/g·°C.

Calculate the Heat Absorbed by the Solution

Using the formula Q = m*c*ΔT, calculate the amount of heat (Q) absorbed by the solution, where ΔT = T_final - T_initial. First, convert the volume of the solution to mass by multiplying it by the density since mass (m_solution) = V * ρ. Then calculate Q using the mass of the solution, the specific heat capacity, and the change in temperature.

Calculate the Enthalpy Change (ΔH)

The enthalpy change for the reaction is equal to the heat absorbed by the solution, but since the reaction is endothermic, the sign must be positive. Divide the heat absorbed by the mass of ammonium nitrate used to find ΔH in J/g, then convert it to kJ/mol using the molar mass of ammonium nitrate (NH4NO3).

Final Calculation

Report the final value for the enthalpy change ΔH of the dissolution of ammonium nitrate in kJ/mol.

Key concepts

Endothermic Reaction

An endothermic reaction is a chemical process in which the system absorbs heat from its surroundings. This is often observed as a drop in temperature, which can be quantifiable and is a crucial aspect when studying thermal reactions. In an endothermic process, energy is required to break the bonds of the reactants, which is greater than the energy released when new bonds are formed in the products. For instance, the dissolution of ammonium nitrate in water is an endothermic reaction. As the solid salt dissolves, it withdraws heat from the water, causing the overall temperature of the solution to decrease. This phenomenon is the fundamental reason why instant cold packs feel cold when activated.

Endothermic reactions are not limited to dissolution processes; they occur in various situations, such as photosynthesis in plants, where energy from sunlight is absorbed to convert carbon dioxide and water into glucose and oxygen. Understanding this concept is imperative, as it helps explain why certain reactions require external energy sources to proceed and can help us manipulate these reactions for practical applications like cold therapy or energy storage.

Specific Heat Capacity

Specific heat capacity is a property that describes how much heat energy is needed to raise the temperature of a given mass of a substance by one degree Celsius (or one Kelvin). Mathematically, it is denoted by the symbol 'c' and is often measured in J/g·°C. The specific heat capacity is inherent to the material and can vary even among different phases (solid, liquid, gas) of the same substance. Water, for instance, has a high specific heat capacity of 4.18 J/g·°C, meaning that it can absorb a lot of heat before its temperature rises significantly.

This property is essential when calculating thermal energy changes in chemical reactions as it helps predict how much a substance will change temperature when it absorbs or releases heat. In the dissolution of ammonium nitrate, for example, the specific heat capacity of water is used to determine how much heat energy is absorbed by the solution, subsequently leading to a calculation of the overall enthalpy change of the reaction. A substance's specific heat capacity is also invaluable in many engineering applications, such as in designing heating and cooling systems.

Stoichiometry

Stoichiometry is a branch of chemistry that involves the calculation of the quantitative relationships or ratios between reactants and products in a chemical reaction. The principle of stoichiometry relies on the law of conservation of mass, which states that mass is neither created nor destroyed in a chemical reaction. Consequently, a balanced chemical equation allows predict the amounts of reactants and products that will be consumed or formed. For stoichiometric calculations, the relationships are usually expressed in moles, which is a standardized way to count particles in chemistry.

Understanding stoichiometry is fundamental when determining the quantity of reactants needed or predicting the yield of a reaction. In our scenario, it can be applied to figure out how much heat (enthalpy change) is associated with the dissolution of a specific amount of ammonium nitrate. By using stoichiometry, one can also determine the theoretical yield of a product or the limiting reactant in a given chemical process, making it an indispensable tool for chemical synthesis and industrial applications.

Enthalpy of Dissolution

The enthalpy of dissolution, also known as the heat of solution, is the enthalpy change that occurs when a solute dissolves in a solvent. This thermodynamic quantity reflects whether a dissolution process is endothermic or exothermic—absorbing or releasing heat, respectively. It's important not only for understanding basic principles of physical chemistry but also for practical applications like the design of pharmaceuticals, where the solubility of drugs in bodily fluids is critical.

In our example, when ammonium nitrate dissolves in water, the system absorbs heat, and the enthalpy change is endothermic. By measuring the change in temperature of the water and knowing its specific heat capacity, we can calculate the enthalpy change for the dissolution process. It's essential to ensure the reaction occurs at a constant pressure so that enthalpy is a true representation of heat change. The enthalpy of dissolution provides valuable information for predicting how temperature changes affect solubility, and it is crucial when considering safety and procedural protocols in laboratories and industry.