Question

Potassium perchlorate \(\left(\mathrm{KClO}_{4}\right)\) has a lattice energy of \(-599 \mathrm{kJ} / \mathrm{mol}\) and a heat of hydration of \(-548 \mathrm{k} / \mathrm{mol} .\) Find the heat of solution for potassium perchlorate and determine the temperature change that occurs when 10.0 \(\mathrm{g}\) of potassium perchlorate is dissolved with enough water to make 100.0 \(\mathrm{mL}\) of solution. (Assume a heat capacity of 4.05 \(\mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C}\) for the solution and a density of 1.05 \(\mathrm{g} / \mathrm{mL} . )\)

Short answer

The heat of solution for potassium perchlorate is -1147 kJ/mol. For 10.0 grams, the temperature change of the solution is calculated to be the heat released divided by the product of the solution's mass and heat capacity.

Step-by-step solution

Calculate the Heat of Solution

The heat of solution can be found by adding the lattice energy and the heat of hydration of potassium perchlorate. Use the formula: Heat of Solution = Lattice Energy + Heat of Hydration.

Apply the values to the formula

Substitute the given values into the formula from step 1: Heat of Solution = (-599 kJ/mol) + (-548 kJ/mol).

Simplify the calculation

Perform the arithmetic to find the heat of solution: Heat of Solution = -599 kJ/mol - 548 kJ/mol = -1147 kJ/mol.

Convert grams to moles

Convert the mass of potassium perchlorate to moles using its molar mass. The molar mass of KClO4 is approximately 138.55 g/mol. Moles = mass (g) / molar mass (g/mol).

Apply the mass to the formula

Calculate the moles of potassium perchlorate using the given mass: Moles = 10.0 g / 138.55 g/mol.

Calculate heat released for the given mass

Multiply the heat of solution per mole by the number of moles to find the total heat released when 10.0 g of potassium perchlorate dissolves.

Calculate the temperature change

The amount of heat absorbed or released will cause a change in temperature. Use the formula: Heat (q) = mass (m) * specific heat capacity (c) * change in temperature(ΔT). Solve for ΔT: ΔT = q / (m * c).

Apply the heat capacity and density to the formula

Substitute the specific heat capacity and mass of the solution (calculated from the solution's volume and density) into the formula.

Calculate the final temperature change

Perform the calculation to find the temperature change using the heat released and the mass and specific heat capacity of the solution.

Key concepts

Lattice Energy

When we think about salts such as potassium perchlorate (KClO_4), we have to consider the strong ionic bonds that hold their lattice structure together. Lattice energy is a measure of the strength of these ionic bonds. It represents the amount of energy released when the gaseous ions of an ionic compound come together to form a solid lattice, or the energy required to completely separate a solid ionic compound into gaseous ions.
Lattice energy is a crucial concept because it helps us understand several properties of ionic compounds, including their stability, solubility, and melting points. In the context of dissolving ionic compounds like KClO_4, the lattice must first break apart before the ions can interact with the water molecules. This process is endothermic and will absorb energy, contributing to the heat of solution.
For KClO_4 with a given lattice energy of -599 kJ/mol, this number tells us the amount of energy released when 1 mole of solid potassium perchlorate forms from its constituent ions in the gaseous state.

Heat of Hydration

After considering the energy needed to break the ionic bonds in the lattice (lattice energy), we must also understand the heat of hydration. This concept pertains to the energy exchange that occurs when the separate ions are surrounded by water molecules as they dissolve. The heat of hydration is exothermic, meaning it releases energy. This occurs because the forming of new interactions between the water molecules and the ions releases more energy than is needed to break apart the water molecules' hydrogen bonds.
Specifically, for our exercise, the heat of hydration for potassium perchlorate is given as -548 kJ/mol. This represents the energy released when the ions of 1 mole of KClO_4 are solvated by water. When we calculate the heat of solution, we're actually considering the total energy change resulting from both the endothermic breaking of the lattice and the exothermic hydration of ions.

Temperature Change in Solutions

When a solute like potassium perchlorate dissolves in a solvent like water, the temperature of the solution can change due to the energy released or absorbed in the process. The overall temperature change depends on the heat of the solution and the specific heat capacity of the solution—the amount of energy needed to raise the temperature of 1 gram of the solution by 1 degree Celsius. As seen in our given exercise, this value is 4.05 J/g°C for the resulting potassium perchlorate solution.
In practice, to find the precise temperature change, you multiply the total heat involved in the dissolving process by the mass of the solution and the specific heat capacity. With the knowledge that the density of the solution is 1.05 g/mL, we can calculate the mass of the solution from its volume and use this, along with the heat released from dissolving potassium perchlorate, to compute the temperature change. It's through these energy exchanges that concepts like lattice energy and heat of hydration directly impact the thermal properties of a solution after a solute has dissolved.