Question

Use data from Appendix \(\mathrm{C}\) to calculate the equilibrium constant, \(K\), at \(298 \mathrm{~K}\) for each of the following reactions: (a) \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)\) (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) (c) \(3 \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{6}(g)\)

Short answer

The equilibrium constants for the given reactions at \(298 \mathrm{~K}\) can be calculated using the relation between Gibbs free energy change, \(\Delta G\), and the equilibrium constant, \(K\). After calculating \(\Delta G\) for each reaction using standard Gibbs free energy of formation values from Appendix C, we can find the equilibrium constants using the equation \(K = e^{-\frac{\Delta G}{RT}}\).

Step-by-step solution

Write down the Gibbs free energy - equilibrium constant relation

Recall the relationship between the equilibrium constant and the Gibbs free energy change for a reaction: \[ \Delta G = -RT \ln K \] Where \(\Delta G\) is the Gibbs free energy change, \(R\) is the universal gas constant \((8.314 \mathrm{~J/(mol\cdot K)})\), \(T\) is the temperature in Kelvin, and \(K\) is the equilibrium constant.

Calculate Gibbs free energy change for each reaction

Use the standard Gibbs free energy of formation values (\(\Delta G^\circ_\mathrm{f}\)) for each species from Appendix C to determine the \(\Delta G\) of each reaction using the formula: \[ \Delta G = \sum n_{products} \Delta G^\circ_\mathrm{f(products)} - \sum n_{reactants} \Delta G^\circ_\mathrm{f(reactants)} \] Let's calculate the \(\Delta G\) for each reaction: (a) \(\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)\) \[ \Delta G = 2 \Delta G^\circ_\mathrm{f(HI)} - (\Delta G^\circ_\mathrm{f(H_{2})} + \Delta G^\circ_\mathrm{f(I_{2})}) \] (b) \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) \[ \Delta G = (\Delta G^\circ_\mathrm{f(C_{2}H_{4})} + \Delta G^\circ_\mathrm{f(H_{2}O)}) - \Delta G^\circ_\mathrm{f(C_{2}H_{5}OH)} \] (c) \(3 \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{6}(g)\) \[ \Delta G = \Delta G^\circ_\mathrm{f(C_{6}H_{6})} - 3 \Delta G^\circ_\mathrm{f(C_{2}H_{2})} \] Using the values of \(\Delta G^{\circ}_{\mathrm{f}}\) from Appendix C, we can plug them into these equations to calculate the corresponding \(\Delta G\) for each reaction.

Calculate the equilibrium constant for each reaction using the Gibbs free energy change

Now that we have the \(\Delta G\) for each reaction, we can use the relationship between \(\Delta G\) and \(K\) to find the equilibrium constant for each reaction: \[ K = e^{-\frac{\Delta G}{RT}} \] where \(e\) is the base of the natural logarithm. For each reaction, plug in the values of \(\Delta G\), \(R\), and \(T\) to calculate the equilibrium constant \(K\). By following these steps, we will be able to calculate the equilibrium constant for each reaction at \(298 \mathrm{~K}\).

Key concepts

Gibbs free energy

Gibbs free energy (\(\Delta G\)) is a critical term in thermodynamics that provides valuable insights into the spontaneity of a chemical reaction. In simpler terms, it indicates whether a reaction will occur on its own, without any external help. A negative value of \(\Delta G\) shows that the reaction is spontaneous, while a positive value suggests non-spontaneity.

To find \(\Delta G\), one often uses the equation \(\Delta G = \sum n_{products} \Delta G^\circ_\mathrm{f(products)} - \sum n_{reactants} \Delta G^\circ_\mathrm{f(reactants)}\). This formula helps calculate the change in free energy for a reaction using standard Gibbs free energy of formation (\(\Delta G^\circ_\mathrm{f}\)) values found in tables like Appendix C.

These values are crucial because they provide the basis for determining how systems evolve, and understanding this can be a potent tool in predicting reaction outcomes.

Chemical equilibrium

Chemical equilibrium occurs when the forward and reverse reactions in a chemical process occur at the same rate, resulting in no overall change in the concentrations of reactants and products. At this point, the system is said to be in balance, or equilibrium.

It's important to note that equilibrium does not mean that the reactants and products are equal in amount, but rather that their concentrations remain constant over time.

• The position of equilibrium is determined by the equilibrium constant (\(K\)), a value that quantifies the relationship between product and reactant concentrations at equilibrium.
• A large \(K\) value indicates that a reaction favors products, while a small \(K\) value suggests a preference for reactants.

Understanding chemical equilibrium is essential for harnessing control over reactions, allowing chemists to predict how systems respond to changes in conditions like temperature, pressure, and concentration.

Reaction quotient

The reaction quotient (\(Q\)) is a useful concept that reflects the ratio of the concentrations of products to reactants at any given point in time during a reaction. It is similar to the equilibrium constant (\(K\)), but it applies to conditions that are not necessarily at equilibrium.

To calculate \(Q\), the formula used is:\[ Q = \frac{[\text{products}]^{n}}{[\text{reactants}]^{m}} \]where \(n\) and \(m\) are the stoichiometric coefficients from the balanced chemical equation.

• If \(Q = K\), the system is at equilibrium.
• If \(Q < K\), the reaction will proceed in the forward direction to reach equilibrium.
• If \(Q > K\), the reaction will shift in the reverse direction until equilibrium is reached.

By comparing \(Q\) and \(K\), one can predict the direction in which a reaction will shift to achieve equilibrium, making it a valuable tool for forecasting reaction behavior.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy. In the context of chemical reactions, it provides a framework for understanding how energy changes during a process, which is vital for predicting reaction feasibility and direction.

The key concepts of thermodynamics relevant to chemical reactions include:

• First Law: Conservation of Energy - Energy cannot be created or destroyed, only transformed, meaning that the total energy of a system and its surroundings is constant.
• Second Law: Entropy - In any energy exchange, if no energy enters or leaves the system, the potential energy of the state will always be less than the initial state, often associated with increased disorder or entropy.
• Third Law: Absolute Entropy - As temperature approaches absolute zero, the entropy of a perfect crystal approaches zero.

By applying these laws, thermodynamics helps chemists understand and predict the spontaneity and equilibrium positions of chemical reactions, making it indispensable in scientific investigations and practical applications alike.