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Chapter 7: Problem 63

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

Expert verified
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掳C.

Step by step solution

01

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 \(\mathrm{CaCl}_{2} = \frac{11.0 \, \mathrm{g}}{110.98 \, \mathrm{g/mol}} = 0.099 \, \mathrm{mol}\)
02

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: \(q = -8.069 \, \mathrm{kJ} \times \frac{1000 \, \mathrm{J}}{1 \, \mathrm{kJ}} = -8069 \, \mathrm{J}\)
03

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掳C, we have: \(-8069 \, \mathrm{J} = (11.0 \, \mathrm{g} + 125 \, \mathrm{g}) \times 4.18 \, \mathrm{J/g掳C} \times \Delta T\) Solve for \(\Delta T\): \(\Delta T = \frac{-8069 \, \mathrm{J}}{(11.0+125) \, \mathrm{g} \times 4.18 \, \mathrm{J/g掳C}} = -13.0 \, ^{\circ} \mathrm{C}\) Now, we just need to add the change in temperature to the initial temperature: \(T_{final} = T_{initial} + \Delta T = 25.0^{\circ} \mathrm{C} - 13.0^{\circ} \mathrm{C} = 12.0^{\circ} \mathrm{C}\)
04

Report the final temperature of the solution

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 掳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.

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Most popular questions from this chapter

Consider the following reaction: $$2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) \quad \Delta H=-572 \mathrm{kJ}$$ a. How much heat is evolved for the production of 1.00 mole of \(\mathrm{H}_{2} \mathrm{O}(l) ?\) b. How much heat is evolved when 4.03 g hydrogen are reacted with excess oxygen? c. How much heat is evolved when \(186 \mathrm{g}\) oxygen are reacted with excess hydrogen?

A sample of an ideal gas at \(15.0 \mathrm{atm}\) and \(10.0 \mathrm{L}\) is allowed to expand against a constant external pressure of \(2.00 \mathrm{atm}\) to a volume of \(75.0 \mathrm{L}\). Calculate the work in units of \(\mathrm{kJ}\) for the gas expansion.

Consider 2.00 moles of an ideal gas that are taken from state \(A\) \(\left(P_{A}=2.00 \mathrm{atm}, V_{A}=10.0 \mathrm{L}\right)\) to state \(B\left(P_{B}=1.00 \mathrm{atm}, V_{B}=\right.\) \(30.0 \mathrm{L})\) by two different pathways: These pathways are summarized on the following graph of \(P\) versus \(V:\) Calculate the work (in units of J) associated with the two pathways. Is work a state function? Explain.

Write reactions for which the enthalpy change will be a. \(\Delta H_{\mathrm{f}}^{\circ}\) for solid aluminum oxide. b. the standard enthalpy of combustion of liquid ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\). c. the standard enthalpy of neutralization of sodium hydroxide solution by hydrochloric acid. d. \(\Delta H_{\mathrm{f}}^{\circ}\) for gaseous vinyl chloride, \(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{Cl}(g)\). e. the enthalpy of combustion of liquid benzene, \(C_{6} \mathrm{H}_{6}(l)\). f. the enthalpy of solution of solid ammonium bromide.

The specific heat capacity of silver is \(0.24 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\). a. Calculate the energy required to raise the temperature of \(150.0 \mathrm{g}\) Ag from \(273 \mathrm{K}\) to \(298 \mathrm{K}\). b. Calculate the energy required to raise the temperature of 1.0 mole of Ag by \(1.0^{\circ} \mathrm{C}\) (called the molar heat capacity of silver). c. It takes \(1.25 \mathrm{kJ}\) of energy to heat a sample of pure silver from \(12.0^{\circ} \mathrm{C}\) to \(15.2^{\circ} \mathrm{C}\). Calculate the mass of the sample silver.
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