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 \(125 \mathrm{~g}\) 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 \(4.18 \mathrm{~J} /{ }^{\circ} \mathrm{C} \cdot \mathrm{g}\).

Understanding how to calculate molar mass is essential for various chemistry problems, including calculating the amounts of materials required for a reaction. Molar mass is the mass of one mole of a substance, measured in grams per mole (\text{g/mol}). A mole is a fundamental unit in chemistry that stands for Avogadro's number of entities (atoms, molecules, etc.), which is approximately \(6.022 \times 10^{23}\).

To find the molar mass of a compound such as calcium chloride (\text{CaCl}_2), we add the atomic masses of calcium (\text{Ca}) and chlorine (\text{Cl}) found on the periodic table of elements. Calcium has an atomic mass of 40.08 g/mol, and chlorine has an atomic mass of about 35.45 g/mol. Since there are two chlorine atoms in calcium chloride, we multiply the mass of chlorine by two. The molar mass calculation becomes:\begin{align*}M &= 40.08 (\text{Ca}) + 2 \times 35.45 (\text{Cl}) \&= 110.98 \frac{\text{g}}{\text{mol}}\end{align*}
Thus, the molar mass of \text{CaCl}_2 is \(110.98 \text{g/mol}\). This molar mass is integral to converting grams of \text{CaCl}_2 to moles, a step needed to understand the subsequent behavior of the substance when it undergoes a chemical reaction or process such as dissolution.

Enthalpy change, denoted as \(\Delta H\), indicates the amount of heat released or absorbed during a chemical process, measured in joules per mole (\text{J/mol}) or kilojoules per mole (\text{kJ/mol}). A negative \(\Delta H\) signifies an exothermic process where heat is released to the surroundings while a positive \(\Delta H\) indicates an endothermic process where heat is absorbed from the surroundings.

In the dissolution of calcium chloride (\text{CaCl}_2), the enthalpy change is given as \(-81.5 \text{kJ/mol}\). This tells us that for every mole of \text{CaCl}_2 that dissolves, \(81.5 \text{kJ}\) of heat is released into the surroundings. To calculate the total heat change (\text{q}) when a known amount of \text{CaCl}_2 dissolves, we multiply the number of moles (\text{n}) by the enthalpy change:\(\text{q} = \text{n} \cdot \Delta \text{H}\).
The step-by-step calculation provided exemplifies how to use the molar mass and moles of \text{CaCl}_2 to find the total heat released upon dissolution, an essential part of determining the solution's final temperature.

Heat capacity is a property of a substance that indicates how much heat energy is required to raise the temperature of that substance by one degree Celsius. The specific heat capacity (\text{c}) is defined as the amount of heat per unit mass needed to raise the temperature by one degree Celsius, expressed in joules per gram degree Celsius (\text{J/g}^\circ\text{C}).

Water has a relatively high specific heat capacity, which is typically \(4.18 \text{J/g}^\circ\text{C}\). This means that it takes \(4.18 \text{J}\) of energy to raise one gram of water by one degree Celsius. This property is crucial in the dissolution exercise because the heat released by the dissolving \text{CaCl}_2 is assumed to be entirely absorbed by the water, leading to a change in the water's temperature.

By using the specific heat capacity of water, we can calculate the heat absorbed and thus the change in temperature when a substance like \text{CaCl}_2 dissolves in water. The relationship between heat (\text{q}), mass (\text{m}), specific heat capacity (\text{c}), and temperature change (\(\Delta T\)) is given by the formula: \(\text{q} = \text{m} \text{c} \Delta T\).

Calculating temperature change is a pivotal task that correlates the heat absorbed or released by a substance to the resultant temperature difference. By rearranging the heat capacity equation (\text{q} = \text{mc}\Delta T), you can solve for the temperature change:\(\Delta T = \frac{\text{q}}{\text{mc}}\).

In practical terms, if we know the specific heat capacity of the solvent (water in this case) and the amount of heat released during the dissolution of a compound like \text{CaCl}_2, we can determine how much the temperature of the solvent will change. Remember, the sign of the heat (\text{q}) should reflect exothermic (\text{q} is negative) or endothermic (\text{q} is positive) processes.

Using the heat absorbed by the water and its specific heat capacity, you can find the temperature change of the water. This change in temperature is critical to determine the final temperature of the solution. For example, if the heat released is sufficient to reduce the water temperature by \(15.4^\circ\text{C}\), and the initial temperature was \(25.0^\circ\text{C}\), the final temperature will be the initial temperature minus the temperature change:\(\text{T}_{\text{f}} = \text{T}_{\text{i}} + \Delta T\).
This equation yields the final outcome for the temperature of the solution after the dissolution process, which is central to understanding the thermal effects in chemical processes.