Question

(a) When a 4.25-g sample of solid ammonium nitrate dissolves in \(60.0 \mathrm{~g}\) of water in a coffee-cup calorimeter (Figure \(5.17)\), the temperature drops from \(22.0\) to \(16.9^{\circ} \mathrm{C}\). Calculate \(\Delta H\left(\right.\) in \(\mathrm{kJ} / \mathrm{mol} \mathrm{} \mathrm{NH}_{4} \mathrm{NO}_{3}\) ) for the solution process: $$ \mathrm{NH}_{4} \mathrm{NO}_{3}(s) \longrightarrow \mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q) $$ Assume that the specific heat of the solution is the same as that of pure water. (b) Is this process endothermic or exothermic?

Short answer

The ΔH for the solution process of NH4NO3 is 25.76 kJ/mol, and the process is endothermic.

Step-by-step solution

Calculate the heat change for the solution in the calorimeter

First, we will use the formula for heat change, which is: \(q= mc\Delta{T}\). Here, q is the heat change, m is the mass of the solution, c is the specific heat of the solution, and ΔT is the change in temperature. Given that for this problem, we assume that the specific heat of the solution is the same as that of pure water, then c will be equal to 4.18 J/g°C. Now, the change in temperature, ΔT, is given as \(22.0 - 16.9 = 5.1\)°C. First, we find the mass of the solution, which will be the mass of water (60.0g) plus the mass of ammonium nitrate (4.25g), giving us 64.25g. Next, we plug our values into the formula, \(q = (64.25g)(4.18 \frac{\text{J}}{\text{g} \cdot \text{°C}})(5.1 \text{°C})\), and calculate the heat change(q).

Calculating the heat change (q)

Calculating the heat change (q) given our values: \(q = (64.25g)(4.18 \frac{\text{J}}{\text{g} \cdot \text{°C}})(5.1 \text{°C}) = 1367.36 \text{J}\).

Convert heat change to ΔH per mole of NH4NO3

First, let's find the molar mass of NH4NO3. The molar mass of NH4NO3 = (14.01 (N) + 4(1.01) (H) + 14.01 (N) + 3(16.00) (O)) = 80.05 g/mol. Now, let's calculate the moles of NH4NO3 in the 4.25 g sample. Moles of NH4NO3 = 4.25 g / 80.05 g/mol = 0.0531 mol. Finally, we will divide the heat change found in step 2 by the moles of NH4NO3 to calculate ΔH: ΔH = 1367.36 J/0.0531 mol = 25761 J/mol.

Convert ΔH from J/mol to kJ/mol

To convert ΔH from J/mol to kJ/mol, divide the value by 1000: ΔH = 25761 J/mol ÷ 1000 = 25.76 kJ/mol.

Determine if the process is endothermic or exothermic

Since the temperature dropped during the reaction and ΔH is positive (25.76 kJ/mol), the process is endothermic. In conclusion, the ΔH for the given reaction is 25.76 kJ/mol, and it is an endothermic process.

Key concepts

Calorimetry

Calorimetry is a technique used to measure the amount of heat energy either absorbed or released during a chemical reaction, physical change, or phase transition. A device called a calorimeter is employed in this process. It can be as simple as a coffee-cup calorimeter for reactions occurring in solution at constant pressure, which captures heat changes of the contents within.

In a calorimetry experiment, the specific heat capacity of the substances involved plays a crucial role. It helps to determine the total heat change by the formula:
\[ q = mc\Delta T \]
where q is the heat change, m is the mass, c is specific heat capacity, and \(\Delta T\) is the temperature change. This formula relates the thermal energy exchanged with the physical properties of the material and allows us to quantify interconvertible heat energy and work.

Endothermic Process

An endothermic process is characterized by the absorption of heat from the surroundings. This means the system gains energy during the reaction. Endothermic reactions result in a decrease in the temperature of the reaction mixture, as heat is absorbed to break bonds in the reactants and drive the reaction forward.

Examples of endothermic processes include photosynthesis, melting ice, and evaporating liquid water. In the context of the exercise, the dissolution of ammonium nitrate in water leads to a cooler solution, signaling an endothermic reaction; where the system (solution) absorbs heat from the surroundings (water and calorimeter), leading to a temperature drop.

Molar Enthalpy

Molar enthalpy, also known as enthalpy change per mole, represents the heat change during a process for one mole of a substance. It is a key term in thermodynamics and is denoted by the symbol \( \Delta H \). Molar enthalpy can be either positive or negative, corresponding to endothermic and exothermic processes, respectively.

The calculation of molar enthalpy involves measuring the total energy change and then dividing by the number of moles of substance involved in the process. In our exercise, after calculating the heat change using calorimetry, the molar enthalpy of the solution process is found by dividing this heat change by the number of moles of ammonium nitrate dissolved. Calculating the molar enthalpy is crucial for predicting the energy requirements and releases of reactions.

Specific Heat Capacity

Specific heat capacity, symbolized as c, is the amount of heat energy required to raise the temperature of one gram of a substance by one degree Celsius (or one Kelvin). It is a material-specific value that describes how a substance responds to the addition or removal of heat.

For instance, water has a relatively high specific heat capacity, which means it requires a considerable amount of heat energy to change its temperature compared to other substances. This characteristic is integral to the calculations in calorimetry. The assumption in our textbook exercise is that the specific heat of the solution is the same as that of pure water; hence, we use \(4.18 \frac{J}{g\cdot\degree C}\) in our heat change calculation. Understanding specific heat capacity is essential for various practical applications, including climate science, cooking, and material design.