Problem 59 - Chemistry and Chemical Reactivity

Question

Almost 5 billion kilograms of benzene, \(\mathrm{C}_{6} \mathrm{H}_{6}\). are made each year. Benzene is used as a starting material for many other compounds and as a solvent (although it is also a carcinogen, and its use is restricted). One compound that can be made from benzenc is cyclohexanc, \(\mathbf{C}_{6} \mathbf{H}_{12}\) $$ \begin{aligned} \mathrm{C}_{6} \mathrm{H}_{6}(\ell)+3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow & \mathrm{C}_{6} \mathrm{H}_{12}(\ell) \\ \Delta H_{\mathrm{ran}}^{\circ}=-206.7 \mathrm{k} \mathrm{J} ; \quad \Delta S_{\mathrm{ran}}^{\circ} &=-361.5 \mathrm{J} / \mathrm{K} \end{aligned} $$ Is this reaction predicted to be spontancous under standard conditions at \(25^{\circ} \mathrm{C} ?\) Is the reaction enthalpy-or entropy-driven?

Step-by-Step Solution

Answer

The reaction is spontaneous and enthalpy-driven at 25°C.

Step 1: Identify Thermodynamic Quantities

We need to determine whether the reaction is spontaneous at standard conditions, which involves calculating the Gibbs free energy change (\( \Delta G^\circ \)). The values given for this reaction are \( \Delta H^\circ = -206.7 \) kJ and \( \Delta S^\circ = -361.5 \) J/K. Note that we need to convert \( \Delta S^\circ \) into kJ for consistency.

Step 2: Convert Entropy to Consistent Units

Convert \( \Delta S^\circ \) from J/K to kJ/K by dividing by 1000: \[ \Delta S^\circ = -361.5 \text{ J/K} = -0.3615 \text{ kJ/K} \]

Step 3: Calculate Gibbs Free Energy Change

Use the Gibbs free energy equation: \[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]Substitute \( T = 298 \text{ K} \):\[ \Delta G^\circ = -206.7 \text{ kJ} - 298 \text{ K} \times (-0.3615 \text{ kJ/K}) \]\[ \Delta G^\circ = -206.7 \text{ kJ} + 107.919 \text{ kJ} = -98.781 \text{ kJ} \]

Step 4: Determine Reaction Spontaneity

The reaction is spontaneous under standard conditions if \( \Delta G^\circ < 0 \). Our calculation shows \( \Delta G^\circ = -98.781 \text{ kJ} \), which is less than zero, indicating the reaction is spontaneous.

Step 5: Assess Reaction Driving Force

A negative \( \Delta H^\circ \) suggests that the reaction is exothermic, while a negative \( \Delta S^\circ \) indicates decreased randomness. Since \( \Delta G^\circ \) is negative and \( \Delta H^\circ \) is significantly negative, the reaction is primarily enthalpy-driven rather than entropy-driven.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships and conversions between heat and other forms of energy. One central concept in thermodynamics is Gibbs free energy, which helps determine whether a chemical reaction will proceed spontaneously. Gibbs free energy combines enthalpy (the total heat content of a system) and entropy (the measure of disorder within the system). By analyzing these properties, we can predict how a system behaves and if a reaction takes place naturally without external energy.

In the context of the given reaction, thermodynamics serves as the framework for understanding how benzene and hydrogen combine to form cyclohexane. By calculating Gibbs free energy (\( \Delta G \)), we can ascertain the spontaneity of the reaction - whether it will occur on its own under standard conditions (defined as 298 K and 1 atm).

To calculate \( \Delta G \), we use the equation: \( \Delta G = \Delta H - T \Delta S \). Here, \( \Delta H \) is the change in enthalpy, \( \Delta S \) is the change in entropy, and \( T \) is the temperature in Kelvin. Thermodynamics allows us to use these equations and concepts to predict the behavior of chemical reactions, ensuring a better understanding of the reaction dynamics.

Spontaneous Reactions

A spontaneous reaction is one that occurs naturally without any outside intervention. The spontaneity of a reaction heavily relies on the value of Gibbs free energy change (\( \Delta G \)). If \( \Delta G \)<0, the reaction is spontaneous.

The example reaction converts benzene (\( \mathrm{C}_{6} \mathrm{H}_{6} \)) into cyclohexane (\( \mathrm{C}_{6} \mathrm{H}_{12} \)) by interacting with hydrogen gas (\( \mathrm{H}_{2} \)). Calculating \( \Delta G \) shows us that this reaction has a value of \( \-98.781 \text{kJ} \), which is less than zero. Thus, the system spontaneously switches from reactants to products under standard conditions.

This spontaneity indicates that energy must be released for normal progress, revealing an exothermic tendency. Spontaneous reactions are crucial in many chemical and biological processes, contributing to energy transformations and other reactions.

By using the concept of spontaneity, you can determine if alterations to conditions, such as temperature or pressure changes, might affect whether a reaction will occur without energy input.

Enthalpy and Entropy

Enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)) are foundational concepts in thermodynamics. They help in understanding a reaction's characteristics and possible outcomes. Enthalpy represents the heat content, also reflecting whether the process is endothermic (absorbing heat) or exothermic (releasing heat).

In our reaction, \( \Delta H = -206.7 \text{kJ} \), indicating it is exothermic - releasing energy during the process.

On the other hand, entropy measures the disorder or randomness in the system. The entropy change \( \Delta S = -361.5 \text{J/K} \) suggests a decrease in randomness, as the system becomes more orderly.

The outcome of the reaction depends on the balance between these two properties. A highly favorable negative \( \Delta H \) often dominates an unfavorable \( \Delta S \), making the reaction enthalpy-driven. This means that the heat release significantly influences the progression of the reaction more than the change in disorder.

By understanding how entropy and enthalpy interplay, one can predict whether a reaction will proceed naturally. In reactions where \( \Delta G \) is negative due to a large negative \( \Delta H \), like ours, the reaction is predominantly enthalpy-driven.