Problem 55 - Chemistry The Central Science In Si Units

Question

When an 18.6 -g sample of solid potassium hydroxide dissolves in \(200.0 \mathrm{~g}\) of water in a coffee-cup calorimeter (Figure 5.18), the temperature rises from 23.7 to \(44.5^{\circ}\) C. (a) Calculate the quantity of heat (in kJ) released in the reaction. (b) Using your result from part (a), calculate \(\Delta H\) (in k]/mol KOH) for the solution process. Assume that the specific heat of the solution is the same as that of pure water.

Step-by-Step Solution

Answer

The heat released is approximately 19.07 kJ, and \(\Delta H\) is about 57.56 kJ/mol.

Step 1: Calculate Temperature Change

Find the change in temperature, \(\Delta T\), by subtracting the initial temperature from the final temperature:\[\Delta T = 44.5^\circ C - 23.7^\circ C = 20.8^\circ C\]

Step 2: Determine Total Mass of Solution

The total mass of the solution is the sum of the mass of potassium hydroxide and the water. Given that the mass of KOH is \(18.6\) g and water is \(200.0\) g, the total mass is:\[m_{total} = m_{\text{KOH}} + m_{\text{water}} = 18.6 \text{ g} + 200.0 \text{ g} = 218.6 \text{ g}\]

Step 3: Calculate Quantity of Heat (q) Released

Use the formula for heat absorbed or released: \(q = m \cdot C \cdot \Delta T\), where \(C\) is the specific heat capacity of water (\(4.18 \, \text{J/g}^\circ\text{C}\)). Convert \(q\) to kilojoules: \[q = 218.6 \text{ g} \times 4.18 \, \text{J/g}^\circ\text{C} \times 20.8^\circ\text{C} = 19074.784 \, \text{J} = 19.074784 \, \text{kJ}\]The heat released is therefore approximately \(19.07 \, \text{kJ}\).

Step 4: Calculate Moles of KOH

Find the number of moles of KOH using its molar mass (\(56.11 \, \text{g/mol}\)): \[n_{\text{KOH}} = \frac{18.6 \, \text{g}}{56.11 \, \text{g/mol}} \approx 0.3313 \, \text{mol}\]

Step 5: Calculate \(\Delta H\) for the Solution Process

\(\Delta H\) is calculated using the formula: \[\Delta H = \frac{q}{n_{\text{KOH}}} = \frac{19.07 \, \text{kJ}}{0.3313 \, \text{mol}} \approx 57.56 \, \text{kJ/mol}\]So, \(\Delta H\) for the process is approximately \(57.56 \, \text{kJ/mol}\).

Key Concepts

Heat Capacity

Heat capacity is an essential concept in understanding how substances respond to the addition or loss of heat. It is defined as the amount of heat required to change the temperature of a substance by one degree Celsius. There are two types of heat capacity to consider:

• Specific heat capacity
• Molar heat capacity

The specific heat capacity, denoted by the symbol \( C \), is usually given in units of \( \, ext{J/g}^ ext{°C} \). It describes the heat capacity per gram of material and is extremely useful in calorimetry. In this exercise, we used the specific heat capacity of water, which is \( 4.18 \, ext{J/g}^ ext{°C} \). This means that it takes 4.18 joules of energy to raise 1 gram of water by 1 °C.
When performing calculations involving heat, it's crucial to ensure consistency in units, especially when converting different energy measures (like joules to kilojoules). Understanding specific heat capacity helps us predict how much energy is necessary to achieve the desired temperature change.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It is a vital tool in thermodynamics to understand the enthalpy changes in different reactions.
In this exercise, we used a coffee-cup calorimeter, a common device for such measurements because it's simple and effective in capturing the heat exchange with its surroundings. In this setup:

• We assume the calorimeter only loses a negligible amount of heat to the environment.
• The heat gained by the solution equals the heat released by the solute during dissolution.
• The specific heat of the solution is treated as if it's pure water.

The difference in temperature, \( \Delta T \), allows us to calculate the heat \( q \) using the formula \( q = m \cdot C \cdot \Delta T \). This gives us insight into the energy changes associated with dissolving potassium hydroxide in water. Accurate calorimetry relies heavily on correct measurements and assumptions, such as the heat capacity approximation and calculation accuracy.

Moles Calculation

When interpreting chemical reactions, calculating moles is crucial. It helps quantify substances involved and calculate other essential properties like enthalpy change.
Moles, represented by the symbol \( n \), measure the amount of a chemical substance and are calculated using the formula:

• \( n = \frac{m}{M} \)

Here, \( m \) is the mass of the substance, and \( M \) is its molar mass. For potassium hydroxide (KOH), with a molar mass of 56.11 g/mol, we found it has approximately 0.3313 moles in our exercise sample of 18.6 g.
Calculating moles correctly is foundational for further thermodynamic calculations, such as determining the enthalpy change per mole of reactant. It allows the transition from a macroscopic scale to the microscopic scale of molecules and atoms, making chemical calculations both meaningful and manageable.