Calculate Standard Reaction Free Energy For TiCl4(g) + 2 H2O(g) → TiO2(s) + 4 HCl(g)

Introduction

In the realm of chemical thermodynamics, understanding the spontaneity of a reaction is paramount. Thermodynamics provides us with the tools to predict whether a reaction will occur spontaneously under a given set of conditions. A key concept in this prediction is the standard Gibbs free energy change, denoted as ΔG°. This thermodynamic parameter combines enthalpy and entropy changes to determine the spontaneity of a reaction at standard conditions (298 K and 1 atm pressure). This article delves into calculating the standard reaction free energy (ΔG°) for the chemical reaction:

$$TiCl_4(g) + 2 H_2 O(g) \rightarrow TiO_2(s) + 4 HCl(g)$$

We will utilize thermodynamic data, often found in resources like the ALEKS Data tab, to perform this calculation. This article aims to provide a comprehensive guide, ensuring clarity and precision in determining the spontaneity of chemical reactions. By understanding how to calculate ΔG°, we gain valuable insights into the feasibility and equilibrium of chemical processes, which is crucial in various scientific and industrial applications.

Understanding Standard Gibbs Free Energy Change (ΔG°)

The standard Gibbs free energy change (ΔG°) is a fundamental concept in chemical thermodynamics. It serves as a crucial indicator of the spontaneity of a chemical reaction under standard conditions, which are typically defined as 298 K (25 °C) and 1 atm pressure. ΔG° essentially quantifies the amount of energy available in a chemical reaction to do useful work. A negative ΔG° indicates that the reaction will occur spontaneously in the forward direction, meaning it is thermodynamically favorable. Conversely, a positive ΔG° suggests that the reaction is non-spontaneous and requires an input of energy to proceed. A ΔG° of zero implies that the reaction is at equilibrium, where the rates of the forward and reverse reactions are equal.

The Gibbs Free Energy Equation

The Gibbs free energy change is mathematically defined by the following equation:

$$ΔG = ΔH - TΔS$$

Where:

• ΔG is the Gibbs free energy change.
• ΔH is the enthalpy change, representing the heat absorbed or released during the reaction.
• T is the absolute temperature in Kelvin.
• ΔS is the entropy change, measuring the degree of disorder or randomness in the system.

At standard conditions, this equation becomes:

$$ΔG° = ΔH° - TΔS°$$

Where ΔH° and ΔS° are the standard enthalpy change and standard entropy change, respectively.

Significance of ΔG°

The significance of ΔG° extends beyond simply predicting spontaneity. It also provides insights into the equilibrium constant (K) of a reaction, which quantifies the extent to which a reaction will proceed to completion. The relationship between ΔG° and K is given by:

$$ΔG° = -RTlnK$$

Where:

• R is the ideal gas constant (8.314 J/(mol·K)).
• T is the absolute temperature in Kelvin.
• lnK is the natural logarithm of the equilibrium constant.

This equation highlights the direct correlation between the standard free energy change and the equilibrium constant. A large negative ΔG° corresponds to a large K, indicating that the reaction will proceed far towards the products at equilibrium. Conversely, a large positive ΔG° corresponds to a small K, suggesting that the reaction will favor the reactants at equilibrium.

Understanding ΔG° is crucial for various applications, including designing chemical reactions, predicting reaction yields, and assessing the stability of chemical compounds. By carefully considering the thermodynamic parameters, scientists and engineers can optimize chemical processes and develop new technologies.

Data Collection from Thermodynamic Tables

To calculate the standard reaction free energy (ΔG°) for the given chemical reaction, we need to gather standard thermodynamic data for each reactant and product involved. This data typically includes the standard Gibbs free energy of formation (ΔGf°), standard enthalpy of formation (ΔHf°), and standard entropy (S°) for each substance. These values are usually found in thermodynamic tables, such as those available in textbooks, online databases, or resources like the ALEKS Data tab, which is specified in this case.

The standard Gibbs free energy of formation (ΔGf°) is the change in Gibbs free energy when one mole of a substance is formed from its constituent elements in their standard states (usually at 298 K and 1 atm). Standard enthalpy of formation (ΔHf°) is the change in enthalpy when one mole of a substance is formed from its elements in their standard states. Standard entropy (S°) is the absolute entropy of a substance at standard conditions.

Identifying the Necessary Data

For the reaction:

$$TiCl_4(g) + 2 H_2 O(g) \rightarrow TiO_2(s) + 4 HCl(g)$$

We need the following standard Gibbs free energies of formation (ΔGf°) from the ALEKS Data tab or a similar thermodynamic data source:

• ΔGf° [TiCl₄(g)]
• ΔGf° [H₂O(g)]
• ΔGf° [TiO₂(s)]
• ΔGf° [HCl(g)]

Example Data (Illustrative)

Note: The following values are illustrative and may not be the exact values found in the ALEKS Data tab. Always refer to the specific data source for accurate values.

Let's assume we have the following standard Gibbs free energies of formation:

• ΔGf° [TiCl₄(g)] = -737.2 kJ/mol
• ΔGf° [H₂O(g)] = -228.6 kJ/mol
• ΔGf° [TiO₂(s)] = -889.4 kJ/mol
• ΔGf° [HCl(g)] = -95.3 kJ/mol

These values represent the Gibbs free energy change when one mole of each compound is formed from its elements in their standard states. We will use these values in the next section to calculate the standard reaction free energy (ΔG°).

Importance of Accurate Data

It is crucial to use accurate and reliable thermodynamic data for these calculations. Minor variations in the values can lead to significant differences in the final result for ΔG°. Therefore, always ensure that the data source is reputable and that the values are appropriate for the temperature and conditions under consideration. Using the ALEKS Data tab, as specified, helps ensure consistency and accuracy in the calculations.

Calculating ΔG° Using the Standard Gibbs Free Energies of Formation

Now that we have gathered the necessary standard Gibbs free energies of formation (ΔGf°) for each reactant and product, we can proceed with calculating the standard reaction free energy (ΔG°). The standard reaction free energy is the change in Gibbs free energy that occurs when a reaction is carried out under standard conditions. It can be calculated using the following equation:

$$ΔG°_{rxn} = ΣnΔGf°(products) - ΣnΔGf°(reactants)$$

Where:

• ΔG°rxn is the standard reaction free energy.
• ΣnΔGf°(products) is the sum of the standard Gibbs free energies of formation of the products, each multiplied by its stoichiometric coefficient.
• ΣnΔGf°(reactants) is the sum of the standard Gibbs free energies of formation of the reactants, each multiplied by its stoichiometric coefficient.
• n represents the stoichiometric coefficient for each substance in the balanced chemical equation.

Applying the Equation to Our Reaction

For the reaction:

$$TiCl_4(g) + 2 H_2 O(g) \rightarrow TiO_2(s) + 4 HCl(g)$$

We can apply the equation as follows:

$$ΔG°_{rxn} = [1 × ΔGf°(TiO_2(s)) + 4 × ΔGf°(HCl(g))] - [1 × ΔGf°(TiCl_4(g)) + 2 × ΔGf°(H_2O(g))]$$

Plugging in the Values

Using the example data from the previous section:

• ΔGf° [TiCl₄(g)] = -737.2 kJ/mol
• ΔGf° [H₂O(g)] = -228.6 kJ/mol
• ΔGf° [TiO₂(s)] = -889.4 kJ/mol
• ΔGf° [HCl(g)] = -95.3 kJ/mol

We substitute these values into the equation:

$$ΔG°_{rxn} = [1 × (-889.4 \text{ kJ/mol}) + 4 × (-95.3 \text{ kJ/mol})] - [1 × (-737.2 \text{ kJ/mol}) + 2 × (-228.6 \text{ kJ/mol})]$$

Performing the Calculation

First, we calculate the terms within the brackets:

$$ΔG°_{rxn} = [-889.4 \text{ kJ/mol} - 381.2 \text{ kJ/mol}] - [-737.2 \text{ kJ/mol} - 457.2 \text{ kJ/mol}]$$

$$ΔG°_{rxn} = [-1270.6 \text{ kJ/mol}] - [-1194.4 \text{ kJ/mol}]$$

Now, we subtract the second term from the first:

$$ΔG°_{rxn} = -1270.6 \text{ kJ/mol} + 1194.4 \text{ kJ/mol}$$

$$ΔG°_{rxn} = -76.2 \text{ kJ/mol}$$

Rounding the Answer

The problem statement asks us to round the answer to zero decimal places. Therefore:

$$ΔG°_{rxn} ≈ -76 \text{ kJ/mol}$$

Interpretation of the Result

The calculated standard reaction free energy (ΔG°) is -76 kJ/mol. This negative value indicates that the reaction is spontaneous under standard conditions. In other words, the reaction will proceed in the forward direction without the need for external energy input. The magnitude of ΔG° also suggests that the reaction is reasonably favorable, as a larger negative value typically corresponds to a greater driving force for the reaction.

Conclusion

In this article, we have demonstrated how to calculate the standard reaction free energy (ΔG°) for the chemical reaction:

$$TiCl_4(g) + 2 H_2 O(g) \rightarrow TiO_2(s) + 4 HCl(g)$$

We began by understanding the significance of ΔG° as an indicator of reaction spontaneity and its relationship to the equilibrium constant. We then discussed the importance of collecting accurate standard thermodynamic data, such as the Gibbs free energies of formation (ΔGf°), from reliable sources like the ALEKS Data tab. Using these values, we applied the equation:

$$ΔG°_{rxn} = ΣnΔGf°(products) - ΣnΔGf°(reactants)$$

to calculate ΔG° for the reaction. We found that the standard reaction free energy (ΔG°) is approximately -76 kJ/mol, indicating that the reaction is spontaneous under standard conditions.

Key Takeaways

• The standard reaction free energy (ΔG°) is a critical parameter for determining the spontaneity of a chemical reaction.
• A negative ΔG° indicates a spontaneous reaction, while a positive ΔG° indicates a non-spontaneous reaction.
• Accurate thermodynamic data is essential for calculating ΔG°.
• The equation ΔG°rxn = ΣnΔGf°(products) - ΣnΔGf°(reactants) is used to calculate ΔG° from standard Gibbs free energies of formation.
• The magnitude of ΔG° provides insights into the favorability of the reaction.

By mastering the calculation of ΔG°, we gain a powerful tool for predicting and understanding chemical reactions, which is invaluable in various fields, including chemistry, chemical engineering, and materials science. This understanding allows us to design and optimize chemical processes, predict reaction outcomes, and develop new technologies with greater efficiency and effectiveness.

Practice Problems

To solidify your understanding of calculating the standard reaction free energy (ΔG°), here are a few practice problems. Use the principles and methods discussed in this article, along with thermodynamic data from a reliable source (such as the ALEKS Data tab or a chemistry textbook), to solve these problems.

1. Calculate the standard reaction free energy (ΔG°) for the following reaction:

   $$N_2(g) + 3 H_2(g) \rightarrow 2 NH_3(g)$$

2. Determine the spontaneity of the following reaction under standard conditions:

   $$2 SO_2(g) + O_2(g) \rightarrow 2 SO_3(g)$$

3. Find the standard free energy change for the following reaction:

   $$CH_4(g) + 2 O_2(g) \rightarrow CO_2(g) + 2 H_2O(g)$$

4. What is the standard Gibbs free energy change for the reaction below?

   $$C(s) + O_2(g) \rightarrow CO_2(g)$$

5. Calculate the Gibbs free energy change at standard conditions for this reaction:

   $$H_2(g) + I_2(g) \rightarrow 2 HI(g)$$

For each problem, follow these steps:

1. Identify the reactants and products in the balanced chemical equation.

2. Gather the standard Gibbs free energies of formation (ΔGf°) for each reactant and product from a reliable thermodynamic data source.

3. Apply the equation:

   $$ΔG°_{rxn} = ΣnΔGf°(products) - ΣnΔGf°(reactants)$$

   where n represents the stoichiometric coefficient for each substance.

4. Calculate ΔG°rxn and round your answer to the appropriate number of decimal places.

5. Interpret the result: Is the reaction spontaneous (ΔG° < 0), non-spontaneous (ΔG° > 0), or at equilibrium (ΔG° = 0) under standard conditions?

Working through these practice problems will help you develop confidence in your ability to calculate and interpret standard reaction free energies, a crucial skill in the study of chemical thermodynamics. Remember to always double-check your data and calculations to ensure accuracy.

By understanding and applying the concepts discussed in this article, you will be well-equipped to tackle a wide range of thermodynamic problems and gain a deeper appreciation for the spontaneity and feasibility of chemical reactions.