Question

a) Calculate the reaction Gibbs free energy of \(\mathrm{H}_{2}(\mathrm{~g})+\) \(\mathrm{I}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{HI}(\mathrm{g})\) at \(700 . \mathrm{K}\) when the partial pressures of \(\mathrm{H}_{2}, \mathrm{I}_{2}\), and HI are \(0.35\) bar, \(0.18\) bar, and \(2.85\) bar, respectively. For this reaction, \(K=54\) at 700 . K. (b) Indicate whether this reaction mixture is likely to form reactants, is likely to form products, or is at equilibrium.

Short answer

First, calculate the reaction quotient Q, then use this to determine the Gibbs free energy change \( \Delta G \). The sign of \( \Delta G \) will indicate whether the reaction favors the formation of reactants, products, or is at equilibrium.

Step-by-step solution

Write the expression for the reaction quotient Q

The reaction quotient Q is calculated using the partial pressures of the reactants and products in the gaseous state. It is given by the expression: \[ Q = \frac{P_{\text{HI}}^2}{P_{\text{H}_2} \cdot P_{\text{I}_2}} \]where \( P_{\text{HI}} \), \( P_{\text{H}_2} \), and \( P_{\text{I}_2} \) are the partial pressures of HI, H2, and I2, respectively.

Calculate the reaction quotient Q

Substitute the given partial pressures into the Q expression to obtain the reaction quotient: \[ Q = \frac{(2.85 \, \text{bar})^2}{(0.35 \, \text{bar}) \cdot (0.18 \, \text{bar})} \]

Determine the Gibbs free energy change using the relationship

The change in Gibbs free energy (\( \Delta G \)) for the reaction under non-standard conditions can be found using the equation: \[ \Delta G = \Delta G^\circ + RT \ln(Q) \]However, since we are given the equilibrium constant (K) and not the standard Gibbs free energy change (\( \Delta G^\circ \)), we can relate Q to K by the equation: \[ \Delta G = RT \ln\left(\frac{Q}{K}\right) \]where R is the gas constant (8.314 J/(mol\cdot K)), T is the temperature in Kelvin, and ln is the natural logarithm.

Insert values and calculate \( \Delta G \)

Plug in the values of Q, K, R, and T to calculate \( \Delta G \): \[ \Delta G = (8.314 \, \text{J/(mol}\cdot\text{K)})(700 \, \text{K}) \ln\left(\frac{Q}{54}\right) \]

Evaluate the natural logarithm and \( \Delta G \)

After calculating the value of \( \ln\left(\frac{Q}{K}\right) \), plug it into the equation for \( \Delta G \) to find the Gibbs free energy change for the reaction.

Interpret the sign of \( \Delta G \)

If \( \Delta G \) is negative, then the reaction will proceed forward (form products). If \( \Delta G \) is positive, then the reaction will move in the reverse direction (form reactants). If \( \Delta G \) is close to zero, then the reaction is at equilibrium.

Key concepts

Reaction Quotient (Q)

In the context of chemical reactions, particularly those involving gases, the reaction quotient (Q) plays a pivotal role in determining the direction in which a reaction will proceed. It is a ratio that represents the current state of a reaction, calculated using the partial pressures of the reactants and products involved. The formula for Q is based on the balanced chemical equation, with products in the numerator and reactants in the denominator.

For example, if we're looking at the reaction of hydrogen gas with iodine to form hydrogen iodide, the reaction quotient is given by \[ Q = \frac{{P_{\text{HI}}^2}}{{P_{\text{H}_2} \cdot P_{\text{I}_2}}} \],where each pressure is raised to the power of its stoichiometric coefficient in the balanced equation. This value can then be used to determine how the reaction will shift to reach equilibrium by comparing it with the equilibrium constant (K).

Equilibrium Constant (K)

The equilibrium constant (K) tells us the ratio of concentrations or partial pressures of products to reactants at equilibrium for a given temperature. A high K value indicates that the reaction will favor the formation of products under equilibrium conditions. On the other hand, a low K value suggests that the reactants will be favored.

When comparing Q and K, if Q < K, the reaction will proceed forward to create more products and reach equilibrium. When Q > K, the reaction will shift in the reverse direction. If Q = K, the reaction is at equilibrium, indicating that the rate of the forward and reverse reactions are equal, and the concentrations of reactants and products will remain constant over time.

Partial Pressure

Partial pressure is a term that specifically relates to the pressure exerted by a single gas within a mixture of gases. It is an important factor when dealing with reactions involving gases, as it allows for the calculation of the reaction quotient (Q).

The partial pressure of a gas in a mixture can be found using Dalton’s Law of Partial Pressures, which states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. This concept lays the foundation for understanding the behavior of gases in chemical reactions and is crucial for calculating Q and assessing the Gibbs free energy changes in reactions.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is a mathematical function that is the inverse of the exponential function with base e, where e is an irrational and transcendental number approximately equal to 2.71828. In chemical thermodynamics, ln is used to convert the ratio of the reaction quotient (Q) to the equilibrium constant (K) into a value that can be used to calculate the Gibbs free energy change \( \Delta G \).

When the quotient \( \frac{Q}{K} \)is plugged into the natural logarithm within the Gibbs free energy equation, it yields a dimensionless value that, when multiplied by RT, gives the change in Gibbs free energy for the reaction in joules per mole.

Chemical Thermodynamics

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It applies principles such as enthalpy, entropy, and Gibbs free energy to predict whether a process will occur spontaneously at constant temperature and pressure.

One of the central equations in chemical thermodynamics is the Gibbs free energy equation, \( \Delta G = \Delta G^\circ + RT \ln(Q) \),which allows chemists to calculate the energy change associated with chemical reactions. This energy change will dictate whether a reaction is thermodynamically favorable, at equilibrium, or unfavorable. Understanding Gibbs free energy and its relationship with the reaction quotient and equilibrium constant is fundamental to the study of chemical kinetics and the direction of chemical reactions.