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, \(1.25 \mathrm{~g}\) 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 reaction 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 enthalpy change for the reaction is -25.69 kJ/mol.

Step-by-step solution

Determine the change in Temperature

The change in temperature (ΔT) is the final temperature minus the initial temperature. Calculate it using the given temperatures: ΔT = T_{final} - T_{initial} = 21.9°C - 25.8°C = -3.9°C.

Calculate the heat absorbed or released

Use the formula q = mcΔT to determine the amount of heat (q) absorbed or released by the solution. The mass (m) of the solution can be calculated assuming the density of water (since the solution is mostly water) is 1.0 g/mL, and the volume (V) of the solution is 25.0 mL. So, m = density × volume = 1.0 g/mL × 25.0 mL = 25.0 g. The specific heat capacity (c) is given as 4.18 J/g°C. So, q = (25.0 g)(4.18 J/g°C)(-3.9°C).

Calculate the enthalpy change

Enthalpy change (ΔH) for the reaction is the amount of heat absorbed or released per mole of the substance. First, find the number of moles of ammonium nitrate used by dividing the mass of ammonium nitrate by its molar mass (80.04 g/mol for NH4NO3). Moles = 1.25 g / 80.04 g/mol. Then calculate the enthalpy change (ΔH) by dividing the heat (q) by the number of moles. Remember to convert J to kJ by dividing by 1000.

Combine equations and solve

Combine all the steps above to calculate the enthalpy change. First, calculate the amount of heat in joules using the specific heat capacity equation, convert this to kilojoules, and then divide by the number of moles of NH4NO3 to get the enthalpy change ΔH in kJ/mol.

Key concepts

Understanding Endothermic Reactions

Endothermic reactions are chemical processes that absorb heat from their surroundings. This is opposed to exothermic reactions that release heat. Understanding how endothermic reactions work is fundamental in thermochemistry. In the case of cold packs, when ammonium nitrate (\r\(\r\mathrm{NH}_{4} \mathrm{NO}_{3}\r\)) dissolves in water, it absorbs heat, causing the solution's temperature to drop. This reaction stores energy in the form of chemical potential energy.

It's essential to understand that for endothermic reactions, the sign of the heat change (\r\(q\r\)) is positive, indicating absorption of heat. The temperature of the environment decreases as the reaction proceeds, which is why cold packs feel cooler upon activation.

Stoichiometry in Play

Stoichiometry is the portion of chemistry that relates to measuring and calculating the quantities of reactants and products involved in chemical reactions. It's based on the conservation of mass where the total mass of reactants equals the total mass of products. In the exercise concerning cold packs, stoichiometry is used to determine how much heat is absorbed per mole of ammonium nitrate that dissolves.

The dissolution reaction is balanced by nature, meaning that the number of atoms for each element in the reactants side is equal to that in the products side. We calculate the amount of substance using the molar mass and use it to find the enthalpy change per mole.

Fundamentals of Thermochemistry

Thermochemistry is a branch of physical chemistry concerned with the changes in energy, specifically thermal energy, that occur during chemical reactions and when substances change state. The core of thermochemistry lies in the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed.

In the context of the instant cold pack example, the focus is on the enthalpy change (\r\(\Delta H\r\)), which is the heat absorbed by the system at constant pressure. The calculation of this value is crucial to understanding the energy dynamics of the endothermic process. A negative \r\(\Delta H\r\) would signify an exothermic reaction, whereas a positive value indicates that the reaction is endothermic.

Heat Capacity and Its Role

Heat capacity is the amount of heat required to raise the temperature of an object by a certain temperature interval. It is often confused with specific heat capacity, which is the amount of heat per unit mass required to raise the temperature by one degree Celsius. The specific heat capacity of water (\r\(4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\r\circ} \mathrm{C}\r\)) is a key value in our cold pack exercise because it allows us to calculate how much energy is absorbed during the dissolution of ammonium nitrate.

The temperature drop in the water when the salt dissolves reveals the endothermic nature of the reaction and allows for a quantitative determination of the enthalpy change. Calculating the total amount of absorbed heat requires multiplying the mass of the water, its specific heat capacity, and the change in temperature.