Problem 65 - Chemistry: An Atoms-Focused Approach

Question

Hormone Mimics Phthalates that are used to make plastics flexible are among the most abundant industrial contaminants in the environment. Several have been shown to act as hormone mimics in humans by activating the receptors for estrogen, a female sex hormone. Combustion of one mole of one of these compounds, dimethyl phthalate \(\left(\mathrm{C}_{10} \mathrm{H}_{10} \mathrm{O}_{4}\right),\) produces \(4685 \mathrm{kJ}\) of thermal energy. If \(1.00 \mathrm{g}\) of dimethyl phthalate is combusted in a bomb calorimeter whose heat capacity ( \(C_{\text {calorimeter }}\) ) is \(7.854 \mathrm{kJ} /^{\circ} \mathrm{C},\) what is the change in temperature of the calorimeter?

Step-by-Step Solution

Answer

Answer: The change in temperature of the calorimeter is approximately 3.070 °C.

Step 1: Calculate the molar mass of dimethyl phthalate.

To find the molar mass of dimethyl phthalate (C10H10O4), we need to add the molar masses of all the atoms present in its molecular formula: 10 carbon atoms, 10 hydrogen atoms, and 4 oxygen atoms. Molar mass = (10 × 12.01 g/mol (C)) + (10 × 1.008 g/mol (H)) + (4 × 16.00 g/mol (O)) = 194.18 g/mol

Step 2: Calculate the number of moles of dimethyl phthalate.

Now we can use the molar mass to find the number of moles in 1.00 g of dimethyl phthalate. Moles = (mass)/(molar mass) = (1.00 g)/(194.18 g/mol) = 0.00515 mol

Step 3: Calculate the thermal energy released from the combustion of 1.00 g of dimethyl phthalate.

We know that one mole of dimethyl phthalate produces 4685 kJ of thermal energy. To find the thermal energy released from 0.00515 mol, we can do a simple cross-multiplication. Thermal energy = (0.00515 mol) × (4685 kJ/mol) = 24.12975 kJ

Step 4: Calculate the change in temperature of the calorimeter.

Finally, we will use the heat capacity of the calorimeter (7.854 kJ/°C) and the thermal energy (24.12975 kJ) to find the change in temperature. Change in temperature = (thermal energy)/(heat capacity) = (24.12975 kJ)/(7.854 kJ/°C) = 3.070 °C So, the change in temperature of the calorimeter is approximately 3.070 °C.

Key Concepts

Calorimetry

Calorimetry is the science of measuring heat in chemical reactions or physical changes. We specifically focus on **bomb calorimetry** for reactions occurring at constant volume, like combustion reactions. Bomb calorimeters are extensively used because they can accurately measure the energy change even for explosive reactions.

This method involves a calorimeter with known heat capacity, making it possible to calculate the energy released by knowing the temperature change. The heat capacity (\(C_{ ext{calorimeter}} \)) is essential in this calculation. By observing how much the temperature of the calorimeter increases, the total energy change can be determined.

It's especially useful in quantifying how much energy is released or absorbed during a reaction, allowing chemists to explore reaction energetics under controlled conditions. This knowledge extends to applications like fuel evaluation and understanding the energy content of various substances.

Heat Capacity

Heat capacity is a property that describes how much heat a substance can absorb before its temperature changes by a given amount. The heat capacity of an object determines how it responds to added or removed heat, playing a crucial role in calorimetry.

For a calorimeter, the heat capacity is crucial in determining the temperature change from known energy input. In our example, we used a calorimeter with a heat capacity of \(7.854 \, ext{kJ}/° ext{C}\). This means for every kilojoule of energy added, the temperature of the calorimeter will rise inversely relative to the heat capacity.

This relationship is given by \[\Delta T = \frac{Q}{C_{\text{calorimeter}}}\]where \(Q\) is the thermal energy exchanged. Knowing the heat capacity thus allows you to understand energy transitions in reactions precisely. It is a critical parameter in thermal analysis and practical applications such as designing thermal management systems.

Combustion Reaction

Combustion reactions are exothermic, meaning they release energy, typically in the form of heat, when a substance reacts with oxygen. These reactions are fundamental in both industrial applications and everyday life, like in engines and energy production.

For dimethyl phthalate, a combustion reaction releases a significant amount of heat. With one mole producing \(4685 \, ext{kJ}\) of energy, this exothermic reaction exemplifies how chemical potential energy stored in molecular bonds is transformed into thermal energy.

This type of reaction not only aids in understanding the energy potentials of different substances but also impacts environmental studies, showing how some industrial compounds might influence both ecological and energy systems. Calculating these reactions helps us evaluate fuel efficiency and environmental impacts, promoting more sustainable industrial practices.

Molar Mass Calculation

The molar mass of a compound is crucial in converting between the mass of a substance and the number of moles, which stands at the heart of quantitative chemical analysis. It is calculated by summing the atomic masses of the constituent atoms in a compound.

In the step-by-step solution, we calculated the molar mass of dimethyl phthalate \(\left(\text{C}_{10}\text{H}_{10}\text{O}_{4}\right)\) by adding the atomic masses:

• 10 carbon atoms: \(10 \times 12.01 \, \text{g/mol}\)
• 10 hydrogen atoms: \(10 \times 1.008 \, \text{g/mol}\)
• 4 oxygen atoms: \(4 \times 16.00 \, \text{g/mol}\)

This results in a molar mass of \(194.18 \, \text{g/mol}\).

Understanding molar mass is essential because it bridges the macroscopic measurements (grams) with the microscopic world of molecules (moles), facilitating the calculation of moles in subsequent reaction steps, as showcased by determining the moles in 1.00 gram of dimethyl phthalate.