Problem 41 - General Chemistry: Principles and Modern Applications

Question

A 1.620 g sample of naphthalene, \(C_{10} \mathrm{H}_{8}(\mathrm{s}),\) is completely burned in a bomb calorimeter assembly and a temperature increase of \(8.44^{\circ} \mathrm{C}\) is noted. If the heat of combustion of naphthalene is \(-5156 \mathrm{kJ} / \mathrm{mol} \mathrm{C}_{10} \mathrm{H}_{8}\) what is the heat capacity of the bomb calorimeter?

Step-by-Step Solution

Answer

The heat capacity of the bomb calorimeter is \(7.71 kJ/^{\circ} C\).

Step 1: Determine the moles of the sample

The amount of heat released by the naphthalene sample can be calculated by first finding the how many moles are in the sample. Since the molar mass of naphthalene \(C_{10}H_{8}\) is 128.17 g/mol, the number of moles \(n\) can be found by dividing the mass of the sample by the molar mass, i.e., \(n = \frac{1.620 g}{128.17 g/mol} = 0.01263 mol\).

Step 2: Calculate the heat released by the sample

The amount of heat \(\Delta{H}\) released by the naphthalene sample can be calculated by multiplying the number of moles \(n\) by the heat of combustion, i.e., \(\Delta{H} = n \times -5156 kJ/mol = -65.05 kJ\). Note that the heat of combustion is negative because it is an exothermic process (heat is released).

Step 3: Calculate the heat capacity of the bomb calorimeter

As per our initial analysis, the heat absorbed by the calorimeter equals the heat released by the burning sample. Hence, the heat capacity \(C\) of the calorimeter can be found by dividing the absolute value of the heat absorbed (equal to the heat released by the sample) by the change in temperature, i.e., \(C = \frac{\Delta{H}}{\Delta{T}} = \frac{65.05 kJ}{8.44^{\circ} C} = 7.71 kJ/^{\circ} C\).

Key Concepts

Heat of Combustion

When a substance burns, it releases energy in the form of heat. This is known as the heat of combustion. For naphthalene, the heat of combustion is given as \(-5156 \text{kJ/mol}\). This value tells us how much energy is released when one mole of naphthalene completely combusts. The negative sign indicates that this is an exothermic process, meaning energy is being released into the surroundings. Understanding this concept is essential when dealing with calorimetry, the science of measuring heat changes in physical and chemical processes.

In practical scenarios, such as with a bomb calorimeter, knowing the heat of combustion helps calculate the amount of heat released by your sample. This heat is then transferred to the calorimeter, causing a measurable temperature change. Through this, we can deduce properties about the sample or the calorimeter itself, making it a fundamental tool in quantitative chemistry.

Molar Mass

Molar mass is a crucial property of any chemical substance. It refers to the mass of one mole of a substance and is expressed in grams per mole \((g/mol)\). For naphthalene, \(C_{10}H_{8}\), the molar mass is calculated by adding together the atomic masses of all its constituent atoms.

The molar mass of naphthalene is \(128.17 \text{g/mol}\). This means that one mole of naphthalene weighs 128.17 grams. In calorimetry experiments such as our naphthalene example, accurately determining the number of moles of a reactant is crucial.

• To find the moles, divide the sample's mass by the molar mass.
• This calculation converts the mass into moles, which can then be used with the heat of combustion to find the total heat released.

Hence, molar mass serves as a bridge between the mass of a substance and its amount in moles, aiding in stoichiometric computations in reactions.

Heat Capacity

Heat capacity is a measure of the amount of heat needed to raise the temperature of an object or substance by one degree Celsius \((1^{\circ}C)\). It is typically expressed as \(\text{kJ}/^{\circ}C\) or \(\text{J}/^{\circ}C\). The heat capacity can tell us how much heat the calorimeter absorbs during the combustion reaction.

In our scenario with a bomb calorimeter, the heat capacity helps determine the calorimeter's effectiveness in absorbing the heat released by the combustion of the sample. The calculation of heat capacity involves dividing the total heat absorbed by the calorimeter by the change in temperature observed.

• This calculation reveals the calorimeter's ability to absorb heat, which is crucial for precise energy measurements.
• A higher heat capacity indicates a greater ability to absorb heat without a significant temperature change.

Understanding heat capacity is paramount in calorimetry, as it relates to the accuracy and sensitivity of the calorimetric measurements.