Question

Suppose you burned 0.300 g of \(\mathrm{C}(\mathrm{s})\) in an excess of \(\mathbf{O}_{2}(g)\) in a constant-volume calorimeter to give \(\mathrm{CO}_{2}(\mathrm{g})\) $$ \mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g}) $$ The temperature of the calorimeter, which contained 775 g of water, increased from \(25.00^{\circ} \mathrm{C}\) to \(27.38^{\circ} \mathrm{C} .\) The heat capacity of the bomb is \(893 \mathrm{J} / \mathrm{K} .\) Calculate \(\Delta U\) per mole of carbon.

Short answer

The change in internal energy, \(\Delta U\), is approximately 393.9 kJ/mol.

Step-by-step solution

Calculate Heat Absorbed by the Water

To find the heat absorbed by the water, we use the formula: \[ q = m imes c imes \Delta T \]where \( m = 775 \ \text{g} \) is the mass of the water, \( c = 4.18 \ \text{J/g°C} \) is the specific heat capacity of water, and \( \Delta T = 27.38 - 25.00 = 2.38 \ \text{°C} \) is the change in temperature.Plug in the values:\[ q = 775 \times 4.18 \times 2.38 = 7713.605 \ \text{J} \]

Calculate Heat Absorbed by the Bomb Calorimeter

The heat absorbed by the calorimeter itself is given by:\[ q_{\text{calorimeter}} = C_{\text{calorimeter}} \times \Delta T \]where \( C_{\text{calorimeter}} = 893 \ \text{J/K} \).Plug in the values:\[ q_{\text{calorimeter}} = 893 \times 2.38 = 2125.34 \ \text{J} \]

Calculate Total Heat Evolved

The total heat evolved in the reaction is the sum of the heat absorbed by the water and the calorimeter:\[ q_{\text{total}} = q_{\text{water}} + q_{\text{calorimeter}} \]\[ q_{\text{total}} = 7713.605 + 2125.34 = 9838.945 \ \text{J} \]

Calculate Moles of Carbon Burned

Determine the moles of carbon burned using its molar mass. The atomic mass of carbon, \( \text{C} \), is approximately \(12.01 \ \text{g/mol}\).\[ n = \frac{0.300 \ \text{g}}{12.01 \ \text{g/mol}} = 0.02498 \ \text{mol} \]

Calculate \(\Delta U\) per Mole of Carbon

Finally, to find \( \Delta U \) (the change in internal energy) per mole of carbon, use the equation:\[ \Delta U = \frac{q_{\text{total}}}{n} \]Plug in the values:\[ \Delta U = \frac{9838.945 \ \text{J}}{0.02498 \ \text{mol}} = 393,892.28 \ \text{J/mol} \]Convert this to kilojoules per mole:\[ \Delta U = 393.9 \ \text{kJ/mol} \]

Key concepts

Heat Capacity

The term "heat capacity" is fundamental in the study of calorimetry and chemical thermodynamics. Heat capacity refers to the amount of heat required to change the temperature of a substance by one degree Celsius (or kelvin). For our purposes, it is important to differentiate between specific heat capacity and heat capacity. Specific heat capacity is defined as the heat required to raise the temperature of one gram of a substance by one degree Celsius. On the other hand, heat capacity is the heat required to increase the entire system's temperature by one degree, regardless of the mass.

In the context of calorimetry, understanding heat capacity helps us measure how much energy is released or absorbed during a chemical reaction. By using the known heat capacity of water and the calorimeter, we can calculate the total heat energy exchanged in the system. In our problem, the bomb calorimeter has a heat capacity of 893 J/K, meaning it can absorb 893 joules of heat to change its temperature by one kelvin.

This concept is critical because it allows us to accurately measure reactions' thermal effects, informing our understanding of the enthalpies and internal energy changes involved. Without knowing the heat capacity, it would be challenging to quantify these changes accurately.

Internal Energy Change

Internal energy change (\(\Delta U\)) is a core concept in thermodynamics and is closely linked with calorimetry studies. It represents the total change in the internal energy of a system when a chemical reaction occurs. In the case of constant-volume processes like those measured in a bomb calorimeter, this change in internal energy is directly related to the heat exchanged.

The equation \(\Delta U = q_{\text{v}}\) captures this relationship, where \(q_{\text{v}\)} is the heat exchanged at constant volume. For reactions occurring in a calorimeter, \(\Delta U\) can be calculated using the total heat absorbed by the reaction mixture, including the calorimeter and the substances inside it.

In the exercise, the \(\Delta U\) value is derived from the sum of the heat absorbed by the water and the calorimeter divided by the moles of reactant involved. The value of \(393.9\) kJ/mol reflects the internal energy change when 1 mole of carbon is completely burned in oxygen under specified conditions. This energy change is critical for understanding how much energy is released or consumed during a chemical transformation.

Chemical Thermodynamics

Chemical thermodynamics provides a framework for understanding the energy changes that accompany chemical reactions. It involves the study of changes in energy, specifically heat and work, that occur during chemical processes. Central to chemical thermodynamics are the laws governing energy conservation, enthalpy changes, and internal energy changes.

In the context of the exercise, chemical thermodynamics helps us interpret the interaction between carbon and oxygen. The process converts chemical potential energy into thermal energy, illustrated through the increase in temperature of water and the calorimeter surrounding the reaction vessel.

Thermodynamics helps us predict and calculate the energy exchanged in a reaction. By using indicators like \(\Delta U\) (internal energy change), calorimetry experiments can deduce valuable insights about the nature of the reaction and potential applications. Understanding these principles is essential for fields like chemical engineering, material science, and even environmental science, where managing energy and reacting substances play crucial roles.