Problem 37 - Chemistry and Chemical Reactivity

Question

Suppose you burned \(1.500 \mathrm{g}\) of benzoic acid, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CO}_{2} \mathrm{H},\) in a constant volume calorimeter and found that the temperature increased from \(22.50^{\circ} \mathrm{C}\) to \(31.69^{\circ} \mathrm{C} .\) The calorimeter contained \(775 \mathrm{g}\) of water, and the bomb had a heat capacity of \(893 \mathrm{J} / \mathrm{K}\). Calculate \(\Delta U\) per mole of benzoic acid. (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

The change in internal energy, \( \Delta U \), per mole of benzoic acid is approximately \( 3.09 \times 10^6 \; \mathrm{J/mol} \).

Step 1: Calculate the Temperature Change

First, calculate the temperature change, \( \Delta T \), experienced by the system. \[ \Delta T = 31.69^{\circ} \mathrm{C} - 22.50^{\circ} \mathrm{C} = 9.19^{\circ} \mathrm{C} \]

Step 2: Calculate the Heat Absorbed by Water

The heat absorbed by the water is calculated using the formula: \( q = m \cdot c \cdot \Delta T \), where \( m \) is the mass of water, \( c \) is the specific heat capacity of water (4.18 J/g°C), and \( \Delta T \) is the temperature change. \[ q_{\text{water}} = 775 \; \mathrm{g} \times 4.18 \; \mathrm{J/g}^\circ \mathrm{C} \times 9.19^{\circ} \mathrm{C} \] \[ q_{\text{water}} = 29809.65 \; \mathrm{J} \]

Step 3: Calculate the Heat Absorbed by the Calorimeter

Calculate the heat absorbed by the calorimeter bomb using its heat capacity and the temperature change: \[ q_{\text{calorimeter}} = C_{\text{bomb}} \times \Delta T \] \[ q_{\text{calorimeter}} = 893 \, \mathrm{J/K} \times 9.19 \, \mathrm{K} \] \[ q_{\text{calorimeter}} = 8210.47 \, \mathrm{J} \]

Step 4: Total Heat Released by Combustion

Add the heat absorbed by water and the calorimeter to find the total heat released by the combustion. \[ q_{\text{total}} = q_{\text{water}} + q_{\text{calorimeter}} \] \[ q_{\text{total}} = 29809.65 \, \mathrm{J} + 8210.47 \, \mathrm{J} \] \[ q_{\text{total}} = 38020.12 \, \mathrm{J} \]

Step 5: Convert Heat to Internal Energy Change per Gram

Since the heat released represents \( \Delta U \) for 1.500 g of benzoic acid, calculate \( \Delta U \) per gram. \[ \Delta U_{\text{per gram}} = \frac{q_{\text{total}}}{\text{mass of benzoic acid}} \] \[ \Delta U_{\text{per gram}} = \frac{38020.12 \; \mathrm{J}}{1.500 \; \mathrm{g}} \] \[ \Delta U_{\text{per gram}} = 25346.75 \; \mathrm{J/g} \]

Step 6: Calculate Molar Internal Energy Change

Finally, convert \( \Delta U_{\text{per gram}} \) to \( \Delta U \) per mole using the molar mass of benzoic acid (122.12 g/mol). \[ \Delta U_{\text{per mole}} = \Delta U_{\text{per gram}} \times \text{molar mass} \] \[ \Delta U_{\text{per mole}} = 25346.75 \; \mathrm{J/g} \times 122.12 \; \mathrm{g/mol} \] \[ \Delta U_{\text{per mole}} = 3092529.51 \; \mathrm{J/mol} \] or approximately \( 3.09 \times 10^6 \; \mathrm{J/mol} \)

Key Concepts

Internal Energy Change

When a reaction occurs, the internal energy change \( \Delta U \) is a vital concept. It represents the total change in energy of a system during a reaction. In terms of constant volume calorimetry, \( \Delta U \) can be equated to the total heat absorbed or released by the system. This heat corresponds to the sum of heat absorbed by any materials, such as water, and any equipment used, like a calorimeter. In the case of the combustion reaction within the calorimeter, the total heat exchange provided the measure of the reaction's energy change. It is important for understanding energy relationships in reactions and is frequently measured in terms of joules. If the process is exothermic, as with combustion, the system releases heat causing an increase in the surroundings' temperature. The internal energy change for benzoic acid during the reaction was calculated to correctly reflect the overall energy dynamics involved.

Heat Capacity

Heat capacity is a measure of a substance or device's ability to absorb heat with a change in temperature. Within the context of constant volume calorimetry, it is critical to account for both the specific heat capacity of substances like water and the overall heat capacity of the apparatus such as the calorimeter.

• Specific Heat Capacity: For water, this is known to be \(4.18\) J/g°C, signifying the energy required to raise one gram of water by one degree Celsius. This is crucial in calculating the energy absorbed by the water during the reaction.
• Total Heat Capacity: For devices like a calorimeter, this capacity is often provided directly, like the \(893\) J/K for the bomb calorimeter in this example. It ensures we can account for energy absorbed by the calorimeter itself during the temperature increase.

This knowledge allows us to ascertain the energy changes in a system, by summing the heat absorbed by different components, crucial for determining \( \Delta U \).

Temperature Change

Temperature change \( (\Delta T) \) plays a significant role in calorimetry calculations. It represents the difference between the final and initial temperatures observed during the reaction. This change is directly linked to the heat produced or absorbed by the system. Analyzing \( \Delta T \) provides insights into the reaction's energy dynamics. For the combustion of benzoic acid, the temperature rose from \(22.50^{\circ} \mathrm{C}\) to \(31.69^{\circ} \mathrm{C}\), resulting in a \( \Delta T \) of \(9.19^{\circ} \mathrm{C}\). This change informs how much heat was absorbed by both the water and the calorimeter, thus enabling us to calculate the total energy change in the system. Understanding \( \Delta T \) helps map the energy flow, ensuring precise calculations for energy exchanges in chemical processes.

Combustion Reaction

A combustion reaction is a type of chemical process where a substance combines with oxygen, releasing energy in the form of heat and light. These reactions are typically exothermic, meaning they release energy to the surroundings, noticeable through a significant temperature increase. When benzoic acid is combusted in a calorimeter, it burns with oxygen. The heat release from this exothermic reaction leads to a rise in temperature, which is recorded to analyze the energy change. Calculating this energy change, especially in a constant volume setting, allows us to determine the internal energy change per mole. Such reactions are not only key in understanding chemical energy releases but are also pivotal in various real-world applications, such as energy generation and understanding metabolic processes. The careful analysis of such reactions helps us understand the precise energy shifts that occur, facilitating the study and design of energy-efficient processes.