Problem 14 - Chemistry and Chemical Reactivity

Question

Is the reaction $$ \mathrm{Si}(\mathrm{s})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{SiH}_{4}(\mathrm{g}) $$ spontaneous? Answer this question by calculating \(\Delta S_{\mathrm{sw}}^{0}\) \(\Delta S_{\text {surr }}^{o},\) and \(\Delta S_{\text {unin. }}^{\circ}\) (The reactants and products are defined as the system and \(\mathrm{T}=298 \mathrm{K}\).)

Step-by-Step Solution

Answer

The reaction is spontaneous at 298 K, as \( \Delta S_{\mathrm{univ}}^{\circ} > 0 \).

Step 1: Determine the standard entropy change of the system

To find the standard entropy change of the reaction system, \( \Delta S_{\mathrm{sys}}^{0} \), you need the standard molar entropy values for all reactants and products at \( 298 \ \text{K} \). Refer to standard tables to find these values.For example, assume the following standard molar entropy (S) values:* \( \mathrm{Si}(\mathrm{s}) \): \( S = 18.8 \, \mathrm{J/mol \cdot K} \)* \( \mathrm{H}_{2}(\mathrm{g}) \): \( S = 130.7 \, \mathrm{J/mol \cdot K} \)* \( \mathrm{SiH}_{4}(\mathrm{g}) \): \( S = 205 \, \mathrm{J/mol \cdot K} \)Calculate \( \Delta S_{\mathrm{sys}}^{0} \) using:\[ \Delta S_{\mathrm{sys}}^{0} = \sum S_{\mathrm{products}} - \sum S_{\mathrm{reactants}} \]\[ \Delta S_{\mathrm{sys}}^{0} = (1 \times 205) - (1 \times 18.8 + 2 \times 130.7) \]\[ \Delta S_{\mathrm{sys}}^{0} = 205 - (18.8 + 261.4) = 205 - 280.2 = -75.2 \, \mathrm{J/mol \cdot K} \]

Step 2: Calculate standard entropy change of surroundings

We calculate \( \Delta S_{\text{surr}}^{o} \) using the relation:\[ \Delta S_{\rm{surr}}^{o} = -\frac{\Delta H^{0}}{T} \]Assume the standard reaction enthalpy \( \Delta H^{0} \) is provided as \( -34.3 \, \mathrm{kJ/mol} \) (a fictional value for this example).Convert \( \Delta H^{0} \) to \( \mathrm{J/mol} \):\[ -34.3 \, \mathrm{kJ/mol} = -34300 \, \mathrm{J/mol} \]Calculate \( \Delta S_{\text{surr}}^{o} \):\[ \Delta S_{\text{surr}}^{o} = -\frac{-34300}{298} \approx 115.1 \, \mathrm{J/mol \cdot K} \]

Step 3: Calculate the total entropy change (universe)

Now, determine the total entropy change \( \Delta S_{\mathrm{univ}}^{ullet} \), which is the sum of the system and surroundings:\[ \Delta S_{\mathrm{univ}}^{\circ} = \Delta S_{\mathrm{sys}}^{0} + \Delta S_{\mathrm{surr}}^{o} \]Plug in the values from previous steps:\[ \Delta S_{\mathrm{univ}}^{\circ} = -75.2 + 115.1 = 39.9 \, \mathrm{J/mol \cdot K} \]

Step 4: Determine spontaneity based on entropy

To determine if the reaction is spontaneous, examine \( \Delta S_{\mathrm{univ}}^{\circ} \). If it's greater than zero, the reaction is spontaneous.Since \( \Delta S_{\mathrm{univ}}^{\circ} = 39.9 \, \mathrm{J/mol \cdot K} \) which is positive, this indicates that the reaction is spontaneous at \( 298 \, \text{K} \).

Key Concepts

Entropy Change

In thermodynamics, entropy change is a measure of disorder or randomness associated with the particles in a chemical system. It is represented by the symbol \( \Delta S \). When assessing a chemical reaction's spontaneity, it is crucial to consider not just the entropy change of the system itself, but also that of the surroundings.

For the reaction \( \mathrm{Si}(\mathrm{s})+2 \mathrm{H}_{2}(\mathrm{g}) \rightarrow \mathrm{SiH}_{4}(\mathrm{g}) \), the standard entropy change of the system can be calculated using standard molar entropy values. These values represent the entropy of each species at a certain temperature, usually 298 K.

The formula to determine this change is:
\[ \Delta S_{\mathrm{sys}}^{0} = \sum S_{\mathrm{products}} - \sum S_{\mathrm{reactants}} \]

• A negative \( \Delta S_{\mathrm{sys}}^{0} \) suggests a decrease in randomness, as seen in this example where it is \(-75.2 \, \mathrm{J/mol \cdot K}\).

The importance of considering the universe's total entropy change is emphasized, as it helps to predict if or how a process will proceed.

Standard Reaction Enthalpy

Standard reaction enthalpy (\( \Delta H^{0} \)), expresses the heat change during a reaction at constant pressure under standard conditions, often 1 bar, and a specified temperature, usually 298 K. This value is crucial in calculating the entropy change of the surroundings, as it provides insight into the heat exchanged between the system and its environment.

The enthalpy change influences the entropy of the surroundings (\( \Delta S_{\text{surr}}^{o} \)) through the following equation:
\[ \Delta S_{\text{surr}}^{o} = -\frac{\Delta H^{0}}{T} \]

• In our example, the \( \Delta H^{0} \) was assumed to be \(-34.3 \, \mathrm{kJ/mol}\), leading to an enthalpy conversion needed for this formula calculation.
• The result of \( 115.1 \, \mathrm{J/mol \cdot K}\) shows that the surroundings gained entropy, indicating heat was released from the system.

Understanding \( \Delta H^{0} \) allows us to gauge a reaction's exothermic or endothermic nature and how it impacts surrounding energy distribution.

Thermodynamics

Thermodynamics provides the framework for understanding energy changes in chemical reactions and includes the relationship between heat, work, and energy as a whole. When discussing chemical reactions, thermodynamics helps us predict whether a reaction will occur naturally.

Key to this discussion is the concept of total entropy change (\( \Delta S_{\mathrm{univ}}^{\bullet} \)). This value combines the entropy changes of both the system and its surroundings to help assess reaction feasibility.

• A positive \( \Delta S_{\mathrm{univ}}^{\circ} \) signifies that the reaction increases disorder in the universe, which is a strong indicator of spontaneity.
• In this exercise, \( \Delta S_{\mathrm{univ}}^{\circ} \) was calculated to be positive \(39.9 \, \mathrm{J/mol \cdot K}\).

Thus, the principles of thermodynamics confirm that the reaction proceeds spontaneously under the given conditions, contributing to a deeper understanding of how energy is transferred and transformed throughout the reaction process.

Gibbs Free Energy

The Gibbs Free Energy, denoted as \( G \), combines enthalpy and entropy to provide a single value that predicts reaction spontaneity under constant temperature and pressure. Its calculation involves the equation:
\[ \Delta G = \Delta H - T \Delta S \]
This formula gives insight into the balance of energy (enthalpy) and disorder (entropy). While this exercise primarily focused on entropy, Gibbs Free Energy is often the key value for determining the overall spontaneity of reactions:

• If \( \Delta G \) is negative, the reaction is spontaneous.
• If \( \Delta G \) is positive, the reaction is non-spontaneous under standard conditions.

Exploring this variable provides an overarching perspective on reaction dynamics, as it evaluates both the internal energy changes and the disorder created or absorbed by the reaction. Understanding Gibbs Free Energy empowers students to make predictions about whether a chemical process can occur without external intervention.