Question

You should use care when dissolving \(\mathrm{H}_{2} \mathrm{SO}_{4}\) in water because the process is highly exothermic. To measure the enthalpy change, \(5.2 \mathrm{g} \mathrm{H}_{2} \mathrm{SO}_{4}(\ell)\) was added (with stirring) to 135 g of water in a coffee-cup calorimeter. This resulted in an increase in temperature from \(20.2^{\circ} \mathrm{C}\) to \(28.8^{\circ} \mathrm{C}\) Calculate the enthalpy change for the process \(\mathrm{H}_{2} \mathrm{SO}_{4}(\ell) \longrightarrow \mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq}),\) in \(\mathrm{k} \mathrm{J} / \mathrm{mol}\).

Short answer

The enthalpy change is approximately -95.2 kJ/mol.

Step-by-step solution

Determine the Mass of Solution

Start by calculating the total mass of the solution, which consists of the mass of water and the mass of sulfuric acid. The mass of water is given as 135 g, and the mass of \(\mathrm{H}_{2}\mathrm{SO}_{4}\) added is 5.2 g, so the total mass is:\[ m_{\text{solution}} = 135 \,\text{g} + 5.2 \,\text{g} = 140.2 \,\text{g} \]

Calculate the Temperature Change

Next, calculate the change in temperature, \(\Delta T\), which is the final temperature minus the initial temperature:\[ \Delta T = 28.8^{\circ}\mathrm{C} - 20.2^{\circ}\mathrm{C} = 8.6^{\circ}\mathrm{C} \]

Calculate Heat Absorbed by Solution

Use the specific heat capacity formula to find the amount of heat absorbed by the solution. Assuming the specific heat capacity \(c\) of the solution is 4.18 J/g°C (similar to water), the heat \(q\) in joules absorbed is:\[ q = m \cdot c \cdot \Delta T = 140.2 \,\text{g} \times 4.18 \,\text{J/g°C} \times 8.6^{\circ}\mathrm{C} \]Calculate:\[ q = 5042.8 \,\text{J} \] which is approximately 5.043 kJ.

Determine Moles of Sulfuric Acid

Calculate the number of moles of \(\mathrm{H}_{2}\mathrm{SO}_{4}\) using its molar mass (98.079 g/mol).\[ \text{moles of } \mathrm{H}_{2}\mathrm{SO}_{4} = \frac{5.2 \,\text{g}}{98.079 \,\text{g/mol}} \approx 0.053 \,\text{mol} \]

Calculate Enthalpy Change per Mole

Find the enthalpy change \(\Delta H\) per mole by dividing the total heat absorbed by the number of moles of \(\mathrm{H}_{2}\mathrm{SO}_{4}\) solute:\[ \Delta H = \frac{q}{\text{moles of } \mathrm{H}_{2}\mathrm{SO}_{4}} = \frac{5.043 \,\text{kJ}}{0.053 \,\text{mol}} \approx 95.2 \,\text{kJ/mol} \]This value is negative because the process is exothermic, releasing heat to the surroundings.

Key concepts

Exothermic Reaction

In chemical terms, an exothermic reaction is one that releases energy, typically in the form of heat, to its surrounding environment. This type of reaction results in an increase in the temperature of the surroundings, indicating that the system lost energy.

A familiar example of this is the dissolving of sulfuric acid (\(\mathrm{H}_{2}\mathrm{SO}_{4}\)) in water, as in the exercise. The heat released in this process is due to the formation of new bonds between water molecules and sulfuric acid ions, which releases more energy than was initially required to break apart the original bonds.

The energy can be measured using calorimetry, leading to terms like enthalpy change which quantifies the heat content variation at constant pressure. For our exercise, the enthalpy change obtained was negative (\(-95.2 \, \text{kJ/mol}\)), confirming that the process is exothermic.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. Using devices called calorimeters, such as the coffee-cup calorimeter used in the exercise, one can assess the enthalpy change of a process.

The coffee-cup calorimeter is ideal for reactions that occur at constant pressure and are straightforward in its use, often involving a simple insulated cup with a lid and a thermometer. Even though basic, it allows researchers or students to track temperature changes precisely.

In the given problem, the temperature was observed to rise from 20.2°C to 28.8°C, signifying that heat was released during the reaction of sulfuric acid with water. The relationship between temperature change and heat exchange then provides a practical classroom demonstration of heat measurements in chemistry.

Molar Mass

Molar mass is a foundational concept in chemistry involving the mass of one mole of a substance. It is typically expressed in grams per mole (g/mol) and aids in converting mass into a more understandable number of moles, which is necessary for stoichiometric calculations.

In the original step-by-step solution, the molar mass of sulfuric acid was utilized to determine the number of moles from a given mass. Sulfuric acid has a molar mass of approximately 98.079 g/mol. Thus, for 5.2 g of \(\mathrm{H}_{2}\mathrm{SO}_{4}\), the corresponding moles are calculated as \(\frac{5.2 \, \text{g}}{98.079 \, \text{g/mol}} \approx 0.053 \, \text{mol}\), a crucial step to find the enthalpy change per mole and understand the extent of the reaction.

Specific Heat Capacity

Specific heat capacity is a property that defines the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. It is denoted by the letter 'c' and typically expressed in units of J/g°C.

Water, for example, has a specific heat capacity of about 4.18 J/g°C, which is relatively high, meaning it can absorb significant amounts of heat without a dramatic temperature increase.

This concept is critical in calorimetry, as it allows the heat (q) absorbed by the solution to be calculated using the formula: \(q = m \cdot c \cdot \Delta T\). In the exercise, the combined mass of the solution (\(140.2 \, \text{g}\)), the known heat capacity of water (\(4.18 \, \text{J/g°C}\)), and the temperature change (\(8.6^{\circ}\mathrm{C}\)) were used to find that 5043 J (or 5.043 kJ) of heat were absorbed.