Question

Consider the dissolution of \(\mathrm{CaCl}_{2}\) : $$\mathrm{CaCl}_{2}(s) \longrightarrow \mathrm{Ca}^{2+}(a q)+2 \mathrm{Cl}^{-}(a q) \quad \Delta H=-81.5 \mathrm{kJ}$$ An \(11.0-\mathrm{g}\) sample of \(\mathrm{CaCl}_{2}\) is dissolved in 125 g water, with both substances at \(25.0^{\circ} \mathrm{C}\). Calculate the final temperature of the solution assuming no heat loss to the surroundings and assuming the solution has a specific heat capacity of \(4.18 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\).

Short answer

The final temperature of the \(\mathrm{CaCl}_{2}\) solution after dissolution is \(12.0^{\circ} \mathrm{C}\), assuming there is no heat loss to the surroundings and the specific heat capacity of the solution is 4.18 J/g°C.

Step-by-step solution

Calculate the moles of \(\mathrm{CaCl}_{2}\) dissolved in water

First, we need to figure out how many moles of \(\mathrm{CaCl}_{2}\) are present in 11.0 g of the substance. To do this, we'll need the molar mass of \(\mathrm{CaCl}_{2}\): Molar mass of \(\mathrm{CaCl}_{2} = 40.08 + 2 \times 35.45 = 110.98 \, \mathrm{g/mol}\) Now, we can calculate the moles of \(\mathrm{CaCl}_{2}\): moles of \(\mathrm{CaCl}_{2} = \frac{11.0 \, \mathrm{g}}{110.98 \, \mathrm{g/mol}} = 0.099 \, \mathrm{mol}\)

Calculate the heat released during the dissolution of \(\mathrm{CaCl}_{2}\)

We know the enthalpy of dissolution per mole is -81.5 kJ/mol. To find the total heat released during this dissolution, we'll multiply the enthalpy by the moles of \(\mathrm{CaCl}_{2}\): Total heat released, \(q = 0.099 \, \mathrm{mol} \times (-81.5 \, \mathrm{kJ/mol}) = -8.069 \, \mathrm{kJ}\) Since 1 kJ = 1000 J, the total heat released in joules is: \(q = -8.069 \, \mathrm{kJ} \times \frac{1000 \, \mathrm{J}}{1 \, \mathrm{kJ}} = -8069 \, \mathrm{J}\)

Calculate the final temperature of the solution

Now, we'll use the formula for specific heat capacity, which relates the heat gained or lost by a substance to its mass, specific heat capacity, and change in temperature: \(q = m \times c \times \Delta T\) Given the specific heat capacity of the solution is 4.18 J/g°C, we have: \(-8069 \, \mathrm{J} = (11.0 \, \mathrm{g} + 125 \, \mathrm{g}) \times 4.18 \, \mathrm{J/g°C} \times \Delta T\) Solve for \(\Delta T\): \(\Delta T = \frac{-8069 \, \mathrm{J}}{(11.0+125) \, \mathrm{g} \times 4.18 \, \mathrm{J/g°C}} = -13.0 \, ^{\circ} \mathrm{C}\) Now, we just need to add the change in temperature to the initial temperature: \(T_{final} = T_{initial} + \Delta T = 25.0^{\circ} \mathrm{C} - 13.0^{\circ} \mathrm{C} = 12.0^{\circ} \mathrm{C}\)

Report the final temperature of the solution

The final temperature of the \(\mathrm{CaCl}_{2}\) solution after dissolution is 12.0°C, assuming there is no heat loss to the surroundings and the specific heat capacity of the solution is 4.18 J/g°C.

Key concepts

Enthalpy Change

In the study of thermochemistry, **enthalpy change** refers to the heat change occurring when a chemical process takes place at constant pressure.
For the dissolution of calcium chloride (\(\mathrm{CaCl}_{2}\)), the enthalpy change is given as -81.5 kJ/mol. This value indicates that the process is exothermic, meaning heat is released.
### Types of Enthalpy ChangesUnderstanding enthalpy changes is crucial because it helps in predicting whether a particular reaction is endothermic or exothermic:- **Exothermic**: Heat is released. The enthalpy change (\(\Delta H\)) is negative.- **Endothermic**: Heat is absorbed. The enthalpy change (\(\Delta H\)) is positive.
In this specific dissolution process, when \(0.099 \, \mathrm{mol}\) of \(\mathrm{CaCl}_{2}\) is dissolved, about \(-8.069 \, \mathrm{kJ}\) of heat is released. This release of heat subsequently influences the temperature of the surrounding water, reducing it.

Heat Capacity

**Heat capacity** is a physical property that represents the amount of heat needed to change a substance's temperature by a particular degree.
In this exercise, the specific heat capacity for the resulting solution is given as \(4.18 \, \mathrm{J/g^{\circ}C}\). This means that this amount of energy is required to raise 1 gram of the solution by 1 °C.### Calculating Heat TransferTo calculate the heat transferred during the dissolving process, the formula used is:\[ q = m \times c \times \Delta T \]Where:- \( q \) represents the heat change,- \( m \) is the total mass of the solution, including both solute and solvent,- \( c \) stands for the specific heat capacity,- \( \Delta T \) is the change in temperature.
In this scenario, the solution composed of \(11.0\, \mathrm{g}\) of \(\mathrm{CaCl}_{2}\) and \(125\, \mathrm{g}\) of water has a combined mass of \(136\, \mathrm{g}\). Using the formula, the heat released (\(-8069\, \mathrm{J}\)) helps determine the temperature change in the solution.

Dissolution Process

The **dissolution process** is a physical change where a solute dissolves into a solvent, forming a homogeneous mixture. In this exercise, calcium chloride (\(\mathrm{CaCl}_{2}\)) is the solute, and water is the solvent.### Steps in a Dissolution Process1. **Breaking Solute Bonds**: The first step involves breaking the ionic bonds between \(\mathrm{Ca}^{2+}\) and \(\mathrm{Cl}^{-}\) ions in the solid calcium chloride.2. **Solvation**: The separated ions are then surrounded by water molecules. This step releases energy, often surpassing the energy required to break the solute bonds.3. **Equilibrium**: Finally, the process reaches a state where the ions are evenly distributed throughout the solvent, forming a uniform solution.
### Impact on TemperatureThe heat released due to this exothermic dissolution significantly impacts the surrounding's thermal state. The release of \(-8069\, \mathrm{J}\) of energy into the water reduces its temperature from its initial \(25.0^{\circ} \mathrm{C}\) to a final \(12.0^{\circ} \mathrm{C}\), as calculated. This temperature drop is standard in exothermic dissolutions, where the surrounding environment absorbs the liberated energy.