Question

When one mole of caffeine \(\left(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\right)\) is burned in air, \(4.96 \times 10^{3} \mathrm{~kJ}\) of heat is evolved. Five grams of caffeine is burned in a bomb calorimeter. The temperature is observed to increase by \(11.37^{\circ} \mathrm{C}\). What is the heat capacity of the calorimeter in \(\mathrm{J} /{ }^{\circ} \mathrm{C} ?\)

Short answer

Answer: The heat capacity of the calorimeter is approximately 11,212.48 J/℃.

Step-by-step solution

Calculate moles of caffeine

First, we need to find out how many moles of caffeine are in the 5 grams. To do this, we need the molecular weight of caffeine, which is given by \(C_8H_{10}N_4O_2\). The molecular weight of each element is \(C (12.01 g/mol), H (1.01 g/mol), N (14.01 g/mol)\), and \(O (16.00 g/mol)\). Thus, the molecular weight of caffeine: \(MW = 8(12.01) + 10(1.01) + 4(14.01) + 2(16.00) = 194.19 g/mol\). Then, we can calculate the number of moles of caffeine by dividing the mass by the molecular weight: \(n = \frac{5 g}{194.19 g/mol} = 0.0257 mol\).

Calculate the heat evolved by the burning caffeine

We are given the amount of heat that is evolved when one mole of caffeine is burned, which is \(4.96 \times 10^3 kJ\). We need to find the amount of heat evolved from burning \(0.0257 mol\). \(q_{caffeine} = (4.96 \times 10^3 kJ/mol) \times 0.0257 mol = 127.47 kJ\). To convert this value to Joules, we multiply by \(1000\): \(q_{caffeine} = 127.47 kJ \times 1000 J/kJ = 127,470 J\).

Calculate the heat capacity of the calorimeter

We know the amount of heat evolved, \(q_{caffeine} = 127,470 J\), and the change in temperature in the calorimeter, \(ΔT = 11.37^{\circ} C\). We can use the equation \(q = CΔT\) to find the heat capacity of the calorimeter, \(C\): \(C = \frac{q}{ΔT} = \frac{127,470 J}{11.37^{\circ} C} = 11,212.48 J/ ^{\circ} C -\). Thus, the heat capacity of the calorimeter is approximately \(11,212.48 J/ ^{\circ} C\).

Key concepts

Bomb Calorimetry

Bomb calorimetry is an experimental technique used to measure the heat of combustion of a substance. The substance is placed in a bomb calorimeter, which is an isolated system designed to withstand high pressures and is filled with excess oxygen to ensure complete combustion. During the experiment, the substance inside the calorimeter is ignited, and the resulting combustion generates heat that is transferred to a known volume of water surrounding the bomb.

The temperature change of the water, measured accurately with a thermometer, reflects the amount of heat produced. This value, known as the change in enthalpy (ΔH), is then used to calculate the heat capacity of the calorimeter and other thermodynamic properties of the reaction. Notably, the heat capacity of a calorimeter is a crucial factor, as it allows the calculation of the heat involved in the chemical reaction for different amounts of substances.

Thermochemistry

Thermochemistry is the study of energy and heat associated with chemical reactions and physical transformations. Central to this field is the concept of enthalpy (ΔH), which represents the heat content of a system at constant pressure. Thermochemistry is grounded on the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred or converted from one form to another.

Within the context of calorimetry, thermochemistry involves the careful measurement of temperature changes due to chemical processes to determine the enthalpy change. By studying how substances exchange heat with their surroundings, scientists can draw conclusions about properties like stability and reaction spontaneity.

Enthalpy of Combustion

The enthalpy of combustion (ΔH_combustion) is the heat released when a substance fully reacts with oxygen under standard conditions to form water and carbon dioxide. This value is generally reported in terms of energy (usually kilojoules or joules) per mole of the burned substance and is a critical parameter in thermochemical calculations.

The enthalpy of combustion is crucial in bomb calorimetry as it helps to determine the energy content of various fuels and potential energy sources. For example, when a known amount of a substance like caffeine burns in a bomb calorimeter, the resulting heat generated increases the water temperature, and from this temperature rise, the ΔH_combustion can be calculated.

Mole Concept

The mole concept is a foundational principle in chemistry that provides a bridge between the atomic scale and the macroscopic world. A mole is defined as the amount of any substance that contains the same number of particles, specifically atoms, molecules, ions, or electrons, as there are atoms in 12 grams of carbon-12. This number, known as Avogadro's number, is approximately 6.022 x 10²³.

In practice, the mole concept allows chemists to count particles by weighing - we can measure out a known mass of a substance and calculate how many moles it contains based on its molar mass, the mass of one mole of that substance. The problem at hand demonstrates the use of the mole concept in determining the number of moles in a given mass of caffeine, which is then used to calculate the heat capacity of the calorimeter.