Question

For the reaction \(2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{g})\) \(K_{\mathrm{c}}=2.8 \times 10^{2}\) at \(1000 \mathrm{K}\) (a) What is \(\Delta G^{\circ}\) at \(1000 \mathrm{K} ?\left[\text { Hint: What is } \mathrm{K}_{\mathrm{p}} ?\right]\) (b) If \(0.40 \mathrm{mol} \mathrm{SO}_{2}, 0.18 \mathrm{mol} \mathrm{O}_{2},\) and \(0.72 \mathrm{mol} \mathrm{SO}_{3}\) are mixed in a 2.50 L flask at \(1000 \mathrm{K}\), in what direction will a net reaction occur?

Short answer

The values of \(\Delta G^{\circ}\) and \(Q_c\) need to be calculated. The actual value of \(\Delta G^{\circ}\) is obtained from the solved Step 2 and the direction of reaction can be obtained from comparing \(Q_c\) and \(K_c\) which is determined in Step 4.

Step-by-step solution

Calculate \(\Delta G^{\circ}\)

Use the formula \(\Delta G^{\circ} = -RT \ln K_c\) to calculate the standard Gibbs free energy change. Here, \(R\) is the gas constant (\(8.314 \, \text{J mol}^{-1} \text{K}^{-1}\)), \(T\) is the temperature in Kelvin (1000 K), and \(K_c\) is the equilibrium constant. Substituting these values into the equation, we get \(\Delta G^{\circ} = -(8.314 \, \text{J mol}^{-1} \text{K}^{-1}) \cdot 1000 \, \text{K} \cdot \ln(2.8 \times 10^{2})\).

Solve for \(\Delta G^{\circ}\)

Solving the above calculation will give the value of \(\Delta G^{\circ}\) in Joules per mole. To convert it to kilojoules per mole, divide the result by 1000.

Calculation of reaction quotient Qc

The reaction quotient \(Q_c\) is given by \([SO_3]^2 / ([SO_2]^2[O_2])\). We can find the molar concentrations of \(SO_3\), \(SO_2\), and \(O_2\) by dividing the given moles by the volume of the container (2.50 L). Substituting these concentrations, we get \(Q_c = (0.72/2.50)^2 / [(0.40/2.50)^2 \cdot (0.18/2.50)]\).

Determine the reaction direction

If \(Q_c > K_c\), the reaction will move in the reverse direction to reach equilibrium. If \(Q_c < K_c\), the reaction will proceed forward. If \(Q_c = K_c\), the reaction is at equilibrium.

Key concepts

Gibbs Free Energy

Gibbs free energy, denoted as \( \Delta G \) and measured in joules per mole (J/mol), is a measurement of the available energy within a system that can be used to do work at constant temperature and pressure. It's a key concept to understand chemical thermodynamics and spontaneity.

When considering a chemical reaction at constant temperature and pressure, the Gibbs free energy change, \( \Delta G \), can tell us if a reaction will occur spontaneously. A negative value of \( \Delta G \) indicates a spontaneous process, while a positive value suggests that the reaction is non-spontaneous and will not occur on its own.

The relationship between Gibbs free energy and the equilibrium constant \( K_{c} \) is given by the equation:
\[ \Delta G^{\circ} = -RT \ln K_{c} \]
Here, \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. This equation plays a pivotal role in determining the spontaneity of a reaction. In the given exercise, the calculation of \( \Delta G^{\circ} \) at \(1000 K\) utilizes this equation to reveal the energy change associated with the reaction reaching equilibrium.

Equilibrium Constant

The equilibrium constant, represented as \( K \) either \( K_p \) for partial pressures or \( K_c \) for concentrations, is a fundamental aspect of chemical equilibrium. It is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their respective coefficients in the balanced equation.

For the reaction in the exercise, \(2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{g})\), the equilibrium constant \( K_c \) is given to be \(2.8 \times 10^{2}\) at \(1000 K\). This implies that at equilibrium, the ratio of the concentration of sulfur trioxide squared to the product of the squared concentration of sulfur dioxide and the concentration of oxygen is \(280\), which tells us that the reaction heavily favors the production of sulfur trioxide at this temperature.

Knowing the equilibrium constant is crucial for predicting the direction in which a reaction must shift to reach equilibrium, and it also helps in calculating the changes in Gibbs free energy associated with the reaction.

Reaction Quotient

The reaction quotient, \( Q_c \), is the ratio of the concentrations of the products and reactants, raised to their stoichiometric coefficients at any point in time, not necessarily at equilibrium. It is used to determine the direction in which a reaction must proceed to achieve equilibrium.

In the context of the exercise provided, we calculate \( Q_c \) by the formula:
\[ Q_c = \frac{[SO_3]^2}{[SO_2]^2[O_2]} \]
By plugging in the concentrations obtained from the given moles and volume of the container, we can compute the numerical value of \( Q_c \). Comparing \( Q_c \) with the equilibrium constant \( K_c \):

• If \( Q_c > K_c \) - the reaction will proceed in the reverse direction, indicating that there are too many products and the system will shift towards the reactants to reach equilibrium.


• If \( Q_c < K_c \) - the reaction will go forward, denoting a deficiency of products and the system will shift towards producing more products until equilibrium is attained.


• If \( Q_c = K_c \) - the reaction mixture is already at equilibrium and no further net change will occur.

In practice, calculating the reaction quotient allows chemists to predict how a reaction mixture will change over time and is essential for understanding the dynamics of chemical reactions outside of equilibrium conditions.