Problem 113 - Chemistry The Molecular Science

Question

Calculate the entropy change for formation of exactly \(1 \mathrm{~mol}\) of each of these gaseous hydrocarbons under standard conditions from carbon (graphite) and hydrogen. What trend do you see in these values? Does \(\Delta_{r} S^{\circ}\) increase or decrease on adding \(\mathrm{H}\) atoms? (a) acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{~g})\) (b) ethylene, \(\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{~g})\) (c) ethane, \(\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{~g})\)

Step-by-Step Solution

Answer

\(\Delta_r S^{\circ}\) decreases (becomes more negative) when more H atoms are added.

Step 1: Identify the Reaction

For each hydrocarbon, write the standard formation reaction:- (a) Acetylene: \( 2 \text{C (graphite)} + \text{H}_2(g) \rightarrow \text{C}_2\text{H}_2(g) \)- (b) Ethylene: \( 2 \text{C (graphite)} + 2\text{H}_2(g) \rightarrow \text{C}_2\text{H}_4(g) \)- (c) Ethane: \( 2 \text{C (graphite)} + 3\text{H}_2(g) \rightarrow \text{C}_2\text{H}_6(g) \).

Step 2: Calculate \(\Delta_r S^{\circ}\) for Each Reaction

Using standard entropy values (from a table) for carbon (graphite), hydrogen, and each compound, calculate \(\Delta_r S^{\circ} = \sum S^{\circ}_{\text{products}} - \sum S^{\circ}_{\text{reactants}} \) for each reaction:- (a) Acetylene: \( \Delta_r S^{\circ} = S^{\circ}(\text{C}_2\text{H}_2(g)) - 2S^{\circ}(\text{C (graphite)}) - S^{\circ}(\text{H}_2(g)) \)- (b) Ethylene: \( \Delta_r S^{\circ} = S^{\circ}(\text{C}_2\text{H}_4(g)) - 2S^{\circ}(\text{C (graphite)}) - 2S^{\circ}(\text{H}_2(g)) \)- (c) Ethane: \( \Delta_r S^{\circ} = S^{\circ}(\text{C}_2\text{H}_6(g)) - 2S^{\circ}(\text{C (graphite)}) - 3S^{\circ}(\text{H}_2(g)) \)

Step 3: Perform Calculations

Substitute in the standard entropy values from tables for each species:- Standard entropies (in \(\text{J/mol} \cdot \text{K}\), as an example) might be: - \(S^{\circ}(\text{C}_2\text{H}_2(g)) = 200.9\) - \(S^{\circ}(\text{C}_2\text{H}_4(g)) = 219.3\) - \(S^{\circ}(\text{C}_2\text{H}_6(g)) = 229.2\) - \(S^{\circ}(\text{C (graphite)}) = 5.7\) - \(S^{\circ}(\text{H}_2(g)) = 130.6\)- Calculate for each reaction: - (a) \( \Delta_r S^{\circ} = 200.9 - 2(5.7) - 130.6 = 58.9 \ \text{J/mol} \cdot \text{K} \) - (b) \( \Delta_r S^{\circ} = 219.3 - 2(5.7) - 2(130.6) = -53.3 \ \text{J/mol} \cdot \text{K} \) - (c) \( \Delta_r S^{\circ} = 229.2 - 2(5.7) - 3(130.6) = -173.9 \ \text{J/mol} \cdot \text{K} \)

Step 4: Analyze the Trend

Observe the trend of \(\Delta_r S^{\circ}\) values:- Acetylene: Positive entropy change: \(58.9 \ \text{J/mol} \,\cdot\, \text{K}\)- Ethylene: Negative entropy change: \(-53.3 \ \text{J/mol} \,\cdot\, \text{K}\)- Ethane: More negative entropy change: \(-173.9 \ \text{J/mol} \,\cdot\, \text{K}\)The entropy change becomes more negative as more hydrogen atoms are added, indicating a decrease in entropy.

Key Concepts

Standard Conditions

In the context of chemistry, standard conditions are essential for ensuring consistent and comparable data. Standard conditions refer to a specific set of parameters, typically at a temperature of 298 K (25°C) and a pressure of 1 atm. Under these conditions, substances are in their most stable form, which is particularly useful for calculating and comparing thermodynamic properties like entropy, enthalpy, and Gibbs free energy.
When performing calculations under standard conditions, such as the formation reactions of gaseous hydrocarbons, we must use standard entropy values. This ensures that the results are universally applicable and can be referenced or reproduced in any scientific setting. The systematic use of standard conditions allows chemists to predict the behavior of substances under a controlled environment, making it fundamental to chemical research and industrial applications.
In calculations involving these settings, it is crucial to remember that any deviation from these standard conditions could lead to changes in the thermodynamic values we are analyzing.

Gaseous Hydrocarbons

Gaseous hydrocarbons are organic compounds composed exclusively of carbon and hydrogen atoms. They are characterized by their gaseous state under ambient conditions, typically room temperature and pressure. Examples include acetylene (\(\text{C}_2\text{H}_2\)), ethylene (\(\text{C}_2\text{H}_4\)), and ethane (\(\text{C}_2\text{H}_6\)). Each of these hydrocarbons consists of different arrangements and numbers of hydrogen atoms bound to carbon atoms.
The physical and chemical properties of gaseous hydrocarbons can vary significantly. For instance:

• Acetylene (C₂H₂): Known for its use in welding torches, it has a linear structure with a triple bond between the carbon atoms, making it relatively unstable.
• Ethylene (C₂H₄): Exhibits a planar structure with a double bond between the carbon atoms, commonly used in the production of polyethylene plastic.
• Ethane (C₂H₆): Features single bonds between all their atoms, serving as a primary component of natural gas.

Understanding these structural differences helps in predicting how these compounds will behave in chemical reactions, including their entropy changes when forming from elemental carbon and hydrogen.

Standard Entropy Values

Standard entropy values are a crucial component in thermodynamics, providing a measure of the disorder or randomness of a system. For a substance, the standard entropy (\(S^{\circ}\)) usually given in \(\text{J/mol}\cdot\text{K}\), quantifies the absolute entropy at standard conditions.
When calculating the entropy change (\(\Delta_r S^{\circ}\)) for a reaction, we use the formula:
\[ \Delta_r S^{\circ} = \sum S^{\circ}_{\text{products}} - \sum S^{\circ}_{\text{reactants}} \]This expression involves substituting the standard entropy values of both the reactants and the products. These values can be found in reference tables, giving you the precalculated entropies for many compounds.
It's essential to note that these values account for the particular configuration and energy distribution within atoms and molecules, therefore giving insights into how the entropy changes with the transition from reactants to products. Higher entropy values generally suggest more disorder within the substance or reaction system, which can significantly influence reaction spontaneity and feasibility.

Thermodynamics in Chemistry

Thermodynamics provides the framework for understanding and predicting chemical reactions. It revolves around concepts like enthalpy, entropy, and Gibbs free energy. Each property gives insights into the tendencies and energies of chemical processes at various conditions.
In thermodynamic terms, entropy is a measure of the disorder of a system. It can indicate the likelihood of a reaction to proceed in a specific direction under set conditions. For example, a positive change in entropy (\(\Delta S^{\circ} > 0\)) signifies increased system disorder, often correlating with favorable spontaneous reactions.
In the case of gaseous hydrocarbons forming under standard conditions, observing changes in entropy can help predict whether the resulting molecules become more ordered or disordered compared to the atoms they are formed from. In our exercise, we noticed that increasing hydrogen numbers decreases entropy, suggesting greater order.

• Studying entropy changes enhances our fundamental understanding of chemical processes, like combustion or synthesis.
• Applying thermodynamic principles is key to designing industrial processes that maximize energy efficiency and product yield.

Thus, mastering thermodynamics in chemistry is essential for both theoretical and practical advancements in the field.