Problem 94 - Chemistry The Molecular Science

Question

The Gibbs free energy of formation, \(\Delta_{f} G^{\circ}\), of HI is \(+1.70 \mathrm{~kJ} / \mathrm{mol}\) at \(25^{\circ} \mathrm{C}\). Calculate the equilibrium constant for the reaction \(\mathrm{HI}(\mathrm{g}) \Longrightarrow \frac{1}{2} \mathrm{H}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{~J}_{2}(\mathrm{~g})\).

Step-by-Step Solution

Answer

The equilibrium constant (K) is approximately 0.456.

Step 1: Understanding the Gibb's Formula

The relationship between Gibbs free energy change (\(\Delta G^{\circ}\)) and the equilibrium constant (\(K\)) is given by the equation: \[\Delta G^{\circ} = -RT \ln K\] where \(R\) is the universal gas constant (8.314 \, \text{J/mol}\,\text{K}) and \(T\) is the temperature in Kelvin. Our goal is to calculate \(K\) using these values.

Step 2: Converting Energy Units

Convert the given Gibbs free energy from kilojoules to joules to match the units of \(R\).Given: \(\Delta_f G^{\circ} = +1.70 \, \text{kJ/mol}\).Since \(1 \, \text{kJ} = 1000 \, \text{J}\),\[\Delta_f G^{\circ} = 1.70\times 1000 = 1700 \, \text{J/mol}\].

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted by the symbol \(K\), is a critical concept in chemistry that provides insight into the extent of a reaction. It helps us understand whether a reaction favors the products or the reactants at equilibrium.

• If \(K > 1\), the reaction tends to form more products.
• If \(K < 1\), the reaction is inclined to retain more reactants.
• If \(K = 1\), the concentrations of products and reactants are equal at equilibrium.

The equilibrium constant is derived from the concentrations of reactants and products raised to the power of their stoichiometric coefficients at equilibrium. For instance, in the dissociation of HI to form \(\frac{1}{2} \text{H}_2\) and \(\frac{1}{2} \text{I}_2\), the equilibrium expression would be formulated on the basis of the balanced chemical equation, using activities or pressures when appropriate for gases.

Gibbs Free Energy Change

Gibbs free energy change, denoted as \(\Delta G^{\circ}\), is the energy associated with a chemical reaction that can do work at constant temperature and pressure. It's a fundamental concept in thermodynamics, giving insight into the spontaneity of a reaction.

• A negative \(\Delta G^{\circ}\) implies a spontaneous reaction under standard conditions.
• A positive \(\Delta G^{\circ}\) means the reaction is non-spontaneous as written but may become spontaneous if conditions change.
• When \(\Delta G^{\circ} = 0\), the system is in equilibrium.

The relationship between Gibbs free energy change and the equilibrium constant is given by the formula: \(\Delta G^{\circ} = -RT \ln K\). This equation indicates that the Gibbs energy change is directly related to the natural logarithm of the equilibrium constant. Thus, even small changes in \(\Delta G^{\circ}\) can significantly impact \(K\).

Thermodynamics

Thermodynamics is the branch of physical science that deals with the relations between heat and other forms of energy. In the context of Gibbs free energy and equilibrium, thermodynamics helps us understand how energy changes drive chemical reactions and determine their direction. The three main laws of thermodynamics provide the framework:

• The First Law states that energy can neither be created nor destroyed, only transformed.
• The Second Law asserts that total entropy, or disorder, of an isolated system can never decrease over time.
• The Third Law states that as temperature approaches absolute zero, the entropy of a system approaches a constant minimum.

These principles apply when considering Gibbs free energy, as they help in predicting whether reactions will occur spontaneously. Remember, a reaction proceeds spontaneously if it leads to a decrease in the system's free energy.

Energy Unit Conversion

Energy unit conversion is an essential step in many chemistry problems, especially when handling Gibbs free energy calculations. Here, the conversion ensures that units are consistent, particularly when using the universal gas constant \(R\), which is typically given in Joules per mole kelvin (\(\text{J/mol K}\)).A common conversion in thermodynamics involves converting energy values from kilojoules (\(\text{kJ}\)) to joules (\(\text{J}\)). Since 1 kJ equals 1000 J, any value in kJ needs to be multiplied by 1000 to convert it to Joules. For example, given that \(\Delta_f G^{\circ}\) for HI is \(+1.70 \text{kJ/mol}\), converting this involves the following calculation: \(\Delta_f G^{\circ} = 1.70 \times 1000 = 1700 \text{J/mol}\).Ensuring these conversions are done correctly is crucial because incorrect units can lead to errors in the calculation of the equilibrium constant \(K\) and other important thermodynamic properties.