Question

When a 9.55-g sample of solid sodium hydroxide dissolves in \(100.0 \mathrm{~g}\) of water in a coffee-cup calorimeter (Figure 5.17), the temperature rises from \(23.6^{\circ} \mathrm{C}\) to \(47.4^{\circ} \mathrm{C}\). Calculate \(\Delta H\) (in \(\mathrm{kJ} / \mathrm{mol} \mathrm{NaOH}\) ) for the solution process $$ \mathrm{NaOH}(s) \stackrel{-\cdots}{\mathrm{Na}}^{+}(a q)+\mathrm{OH}^{-}(a q) $$ Assume that the specific heat of the solution is the same as that of pure water.

Short answer

First, calculate the heat change, q, using the formula \(q = m \times c \times \Delta T\), where m is the mass of the solution, c is the specific heat, and ΔT is the temperature change. Next, determine the moles of NaOH in the solution using its molecular weight. Finally, calculate ΔH for the dissolution of NaOH by dividing the heat change by the number of moles and converting the units to \(kJ/mol\).

Step-by-step solution

Calculate the heat change

To calculate the heat change, q, we use the formula: \[q = m \times c \times \Delta T\] where m is the mass of the solution (mass of NaOH + mass of water), c is the specific heat of the solution (which is the same as that of water), and ΔT is the temperature change. Since the specific heat of water is \(4.18 \mathrm{J/g°C}\), we can write: \[q = (9.55\mathrm{g} + 100.0\mathrm{g}) \times 4.18\mathrm{J/g°C} \times (47.4°C - 23.6°C)\]

Determine moles of NaOH

Now, we want to find the number of moles of NaOH dissolved in the solution. To do this, we will use the molecular weight of NaOH which is approximately \(40\mathrm{g/mol}\). Then, we can write: \[moles\, of\, NaOH = \frac{9.55\mathrm{g}}{40\mathrm{g/mol}}\]

Calculate ΔH for dissolution of NaOH

Finally, we can calculate ΔH for the dissolution of NaOH by dividing the heat change by the number of moles. Note that we need to convert the heat change units to kJ instead of J. The equation is: \[ΔH=\frac{q}{moles\,of\,NaOH}\] By substituting the values calculated in steps 1 and 2, we can solve for ΔH in \(kJ/mol\).

Key concepts

Calorimetry

Calorimetry is an experimental technique that measures the energy exchanged in the form of heat during chemical reactions or physical changes. This method is widely used to study the thermal properties of substances and to determine the changes in enthalpy (ΔH), an essential thermodynamic quantity.

In a laboratory setting, calorimeters, such as the coffee-cup calorimeter mentioned in the exercise, are used to measure heat changes. The coffee-cup calorimeter is a simple and popular instrument in chemistry that measures heat transfer in reactions occurring at constant pressure. This is particularly practical for solution reactions like the dissolution of sodium hydroxide (NaOH) in water.

When NaOH dissolves, it exerts an exothermic reaction, meaning heat is released into the surroundings, particularly the water. As the calorimeter insulates the reaction mixture, the temperature change measured can be directly attributed to the energy change of the dissolution process.

Enthalpy (ΔH)

Enthalpy change (ΔH) is a thermodynamic quantity that describes the heat released or absorbed at constant pressure. It's an intrinsic property of a substance undergoing a transformation. A negative ΔH indicates an exothermic process (release of heat), while a positive ΔH suggests an endothermic process (absorption of heat).

The change in enthalpy is a crucial factor in understanding chemical reactions and can be calculated by measuring the temperature change of a reaction system with known heat capacity. In the given problem, the dissolution of NaOH in water results in a temperature rise, which is why we are looking for a negative enthalpy change, signaling that the solution process is exothermic.

The enthalpy change of the solution is not only a reflection of the strength of the interactions between the solute (NaOH) and solvent (water) molecules but also provides insights into the energetics of the dissolution process. This is why understanding ΔH is essential for scientists and engineers who design and control chemical processes.

Molar Heat of Solution

The molar heat of solution, also known as the molar enthalpy of solution, is the enthalpy change associated with dissolving one mole of a substance in a solvent at constant pressure. It represents the energy required to break intermolecular forces in the solute and solvent and the energy released upon forming new interactions between the solute and solvent molecules.

For a given solute like NaOH, the molar heat of solution is a unique value that can be experimentally determined using calorimetry. By dividing the total heat exchanged (calculated from the temperature change of the solution) by the number of moles of solute, we obtain the molar heat of solution, which, in our exercise case, is expressed as kJ/mol.

An understanding of the molar heat of solution is vital for various applications, such as designing industrial solvation processes, predicting the temperature changes in a given system, and understanding the energetics of reactions.

Specific Heat Capacity

Specific heat capacity is a property of a substance that indicates how much heat energy is required to raise the temperature of one gram of a substance by one degree Celsius. It is denoted by the symbol 'c' and is expressed in units of J/g°C. Water, for instance, has a relatively high specific heat capacity of approximately 4.18 J/g°C, which means it takes 4.18 joules of energy to raise one gram of water by one degree Celsius.

The specific heat capacity is important in calorimetry because it helps determine the amount of heat absorbed or released by a substance during temperature changes. In the sodium hydroxide dissolution exercise, we assume the solution has the specific heat capacity of pure water, allowing us to calculate the total heat change using the mass of the solution, its specific heat capacity, and the observed temperature change.

Knowledge of specific heat capacity is applied in various fields such as meteorology, cooking, and engineering because it affects how substances respond to heating and cooling. A substance with a low specific heat capacity heats up and cools down quickly, while one with a high specific heat capacity changes temperature more gradually.