Problem 37 - Chemistry and Chemical Reactivity

Question

Sulfur \((2.56 \mathrm{g})\) is burned in a constant volume calorimeter with excess \(\mathrm{O}_{2}(\mathrm{g}) .\) The temperature increases from \(21.25^{\circ} \mathrm{C}\) to \(26.72^{\circ} \mathrm{C} .\) The bomb has a heat capacity of \(923 \mathrm{J} / \mathrm{K},\) and the calorimeter contains \(815 \mathrm{g}\) of water. Calculate the heat evolved, per mole of \(\mathrm{SO}_{2}\) formed, for the reaction $$\mathrm{S}_{8}(\mathrm{s})+8 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 8 \mathrm{SO}_{2}(\mathrm{g})$$

Step-by-Step Solution

Answer

296,705.4 J/mol

Step 1: Calculate Temperature Change

The temperature change of the system, denoted as \( \Delta T \), is calculated by subtracting the initial temperature from the final temperature.\[ \Delta T = 26.72 - 21.25 = 5.47 \, \text{°C (or K)} \]since changes in Celsius and Kelvin are the same.

Step 2: Calculate Heat Absorbed by the Calorimeter

The heat absorbed by the bomb calorimeter is calculated using the formula \( q_\text{bomb} = C_\text{bomb} \times \Delta T \), where \( C_\text{bomb} \) is the heat capacity of the bomb and \( \Delta T \) is the temperature change.\[ q_\text{bomb} = 923 \, \text{J/K} \times 5.47 \, \text{K} = 5046.01 \, \text{J} \]

Step 3: Calculate Heat Absorbed by the Water

The heat absorbed by the water is calculated using the formula \( q_\text{water} = m_\text{water} \times c_\text{water} \times \Delta T \), where \( m_\text{water} \) is the mass of water, \( c_\text{water} \) is the specific heat capacity of water (4.18 \( \text{J/g°C} \)), and \( \Delta T \) is the temperature change.\[ q_\text{water} = 815 \, \text{g} \times 4.18 \, \text{J/g°C} \times 5.47 \, \text{°C} = 18578.49 \, \text{J} \]

Step 4: Calculate Total Heat Evolved in Reaction

The total heat evolved by the reaction is the sum of the heat absorbed by both the water and the bomb calorimeter. It can be found by: \[ q_\text{total} = q_\text{bomb} + q_\text{water} = 5046.01 \, \text{J} + 18578.49 \, \text{J} = 23624.5 \, \text{J} \]

Step 5: Calculate Moles of Sulfur Reacted

The moles of sulfur burned can be calculated by dividing the mass of sulfur by its molar mass (32.06 \( \text{g/mol} \) for one atom, but sulfur is \( \text{S}_8 \) so molar mass is 32.06 \( \text{g/mol} \times 8 \)).\[ \text{Molar mass of } \text{S}_8 = 256.48 \, \text{g/mol} \] \[ n = \frac{2.56 \, \text{g}}{256.48 \, \text{g/mol}} = 0.00998\, \text{mol} \]

Step 6: Calculate Heat Evolved per Mole of Sulfur Burned

The heat evolved per mole of \( \text{SO}_2 \) formed is calculated by dividing the total heat by the moles of sulfur reacted and by 8, since 1 mole of \( \text{S}_8 \) forms 8 moles of \( \text{SO}_2 \).\[ \text{Heat per mole of } \text{SO}_2 = \frac{23624.5 \, \text{J}}{0.00998 \, \text{mol} \times 8} = 296,705.4 \, \text{J/mol} \]

Key Concepts

Heat Capacity

Heat capacity is an important concept in calorimetry. It tells us how much heat is needed to change the temperature of an object or system by 1 Kelvin. Think of it as a "heat size" for materials. Just like different sizes of containers hold different amounts of water, different substances require different amounts of heat to change their temperature.

In this problem, the calorimeter is like a container with a known heat capacity of 923 J/K. This means that for every Kelvin increase or decrease, it requires or releases 923 Joules of heat. It acts as a buffer receiving heat from the reaction since it absorbs some of the energy released when the sulfur burns.

• When you know the heat capacity and the temperature change, you can calculate the total heat absorbed or released.
• The formula is: \( q = C_{heat} \times \Delta T \), where \( C_{heat} \) is the heat capacity, and \( \Delta T \) is the temperature change.

Understanding heat capacity helps in predicting how substances will behave when heated or cooled, making it crucial for reactions involving heat exchange.

Temperature Change

Temperature change is a measure of how much the temperature of a system increases or decreases. This change is often the primary indicator of a reaction's energy change. In our context, it provides a straightforward metric for quantifying the energy exchange in a chemical reaction.

For instance, to find the temperature change in the given exercise, we calculate \( \Delta T \) by subtracting the initial temperature from the final temperature: 26.72°C - 21.25°C = 5.47°C. This temperature change allows us to further determine the amount of heat transferred.

• Temperature change is often in Celsius or Kelvin since changes in these scales are equivalent.
• It's the driving force behind the heat transferred calculations, as it's a direct result of the energy released during the reaction.

Remember, the greater the temperature change, the more heat must have been transferred during the reaction.

Specific Heat

Specific heat is a property that indicates how much heat one gram of a substance must absorb to increase its temperature by one degree Celsius. It helps in calculating heat absorbed or released by substances. Water, for example, has a specific heat of 4.18 J/g°C, which is quite high, meaning it can store a lot of heat without a large temperature change.

Using the specific heat, you can determine the heat absorbed by substances like water in calorimetry. In the exercise, the heat absorbed by the water was calculated as follows:

• Formula: \( q_{water} = m_{water} \times c_{water} \times \Delta T \).
• Where \( m_{water} \) is the mass of the water, \( c_{water} \) is the specific heat capacity of water, and \( \Delta T \) is the temperature change.

Understanding how specific heat works is valuable for predicting and managing the thermal behavior of substances in reactions.

Mole Calculations

Mole calculations are essential when determining the amount of substances involved in a chemical reaction. The mole is a basic unit in chemistry that provides a bridge between the atomic world and the macroscopic amounts we see and measure in labs.

For example, to find the moles of sulfur burned in this exercise, you use the formula:

• \( n = \frac{m}{M} \)
• Where \( n \) is the number of moles, \( m \) is the mass, and \( M \) is the molar mass of the substance.

Sulfur in its molecular form \( S_8 \) has a molar mass of 256.48 g/mol. Knowing the weight of sulfur used (2.56g), you can calculate the moles involved.

These mole calculations are the backbone for determining reaction stoichiometry and further calculating the heat evolved per mole in this experiment. Understanding these calculations allows chemists to quantify and predict the outcomes of chemical reactions accurately.