Question

A \(3.52-\mathrm{g}\) sample of ammonium nitrate \(\left(\mathrm{NH}_{4} \mathrm{NO}_{3}\right)\) was added to \(80.0 \mathrm{~mL}\) of water in a constant-pressure calorimeter of negligible heat capacity. As a result, the temperature of the solution decreased from \(21.6^{\circ} \mathrm{C}\) to \(18.1^{\circ} \mathrm{C} .\) Calculate the heat of solution \(\left(\Delta H_{\mathrm{soln}}\right)\) in \(\mathrm{kJ} / \mathrm{mol}:\) \(\mathrm{NH}_{4} \mathrm{NO}_{3}(s) \longrightarrow \mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q)\) Assume the specific heat of the solution is the same as that of water.

Short answer

The heat of solution \(\Delta H_{soln}\) is \(-27.83 \text{ kJ/mol}\).

Step-by-step solution

Determine the Mass of the Solution

The mass of the solution consists of the mass of the water plus the mass of ammonium nitrate. Since 1 mL of water is equivalent to 1 g, the mass of water is 80.0 g. Therefore, the total mass of the solution is 80.0 g + 3.52 g = 83.52 g.

Calculate Temperature Change

The temperature change \(\Delta T\) is calculated by subtracting the final temperature from the initial temperature. \(\Delta T = 18.1^{\circ} \mathrm{C} - 21.6^{\circ} \mathrm{C} = -3.5^{\circ} \mathrm{C}\).

Calculate the Heat Change (q)

Use the formula \(q = mc\Delta T\) to find the change in heat. Here, \(m\) is the mass of the solution (83.52 g), \(c\) is the specific heat capacity of water (4.18 J/g°C), and \(\Delta T\) is the temperature change. \(q = 83.52 \times 4.18 \times (-3.5) = -1224.67 \text{ J}\).

Convert Heat Change to kJ

Since 1 kJ = 1000 J, convert the heat change into kJ: \(-1224.67 \text{ J} / 1000 = -1.22467 \text{ kJ}\).

Calculate Moles of Ammonium Nitrate

Calculate the number of moles of \(\mathrm{NH}_{4}\mathrm{NO}_{3}\) using its molar mass (80.04 g/mol). \(\text{Number of moles} = \frac{3.52 \text{ g}}{80.04 \text{ g/mol}} = 0.044 \text{ mol}\).

Calculate Heat of Solution

Using the heat change and moles calculated above, \(\Delta H_{soln} = \frac{-1.22467 \text{ kJ}}{0.044 \text{ mol}} = -27.83 \text{ kJ/mol}\).

Key concepts

Understanding Enthalpy Change

Enthalpy change, often denoted as \( \Delta H \), is a measure of heat change during a chemical reaction at constant pressure. It represents the energy absorbed or released. In the context of calorimetry, when a substance dissolves in water, an enthalpy change occurs. If the solution temperature decreases, the process absorbs heat and is endothermic, indicated by a positive \( \Delta H \).

Enthalpy changes aren't always easy to observe directly. Hence, calorimeters help measure the heat transferred. An enthalpy change can have significant implications for understanding a reaction's energetic nature. For instance, an exothermic reaction, where \( \Delta H \) is negative, indicates that energy is released to the surroundings. In our exercise, as the substance dissolves, heat is absorbed from the surrounding water, making \( \Delta H_{soln} \) positive.

Calculating Molar Mass

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It is calculated by adding the atomic masses of each element in a molecule. Knowing the molar mass allows us to convert between mass and the number of moles, a fundamental step in solving chemical problems.

In reactions and solutions, it's often necessary to calculate the moles of a reactant or product. For ammonium nitrate (\( \mathrm{NH}_{4}\mathrm{NO}_{3} \)), the molar masses of nitrogen, hydrogen, and oxygen are summed to yield a molar mass of 80.04 g/mol. This value is crucial as it connects mass with the concept of the mole, enabling further calculations like the number of moles used in reactions.

Understanding Specific Heat Capacity

Specific heat capacity, symbolized as \( c \), is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. It's a property intrinsic to materials, highlighting how different substances absorb heat.

In our problem, we use the specific heat capacity of water, which is known to be 4.18 J/g°C. This constant helps us calculate the heat change corresponding to temperature changes within the solution. By understanding specific heat capacity, students can better grasp how substances behave under thermal stresses. It explains why equal masses of different substances change temperature differently when the same amount of heat is applied.

Exploring Temperature Change

Temperature change, or \( \Delta T \), denotes the difference between the final and initial temperatures of a system. It is a direct indicator of heat exchange between the system and its surroundings. In calorimetry experiments, observing temperature change is crucial, as it forms the basis for further calculations such as determining heat energy change.

In the exercise, the temperature change was from 21.6°C to 18.1°C, calculated as -3.5°C. The negative sign signals a decrease in temperature, denoting an endothermic process where heat is absorbed. This shift is measured using the equation \( q = mc\Delta T \), linking temperature change to specific heat capacity and enabling the calculation of heat absorbed or released.