Question

Calculate \(K_{c}\) at 303 \(\mathrm{K}\) for \(\mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{SO}_{2} \mathrm{Cl}_{2}(g)\) if \(K_{p}=34.5\) at this temperature.

Short answer

The equilibrium constant \(K_c\) at 303 K for the given reaction is 211.65.

Step-by-step solution

Write the balanced chemical equation

We are given the balanced chemical equation: \[SO_2(g) + Cl_2(g) \rightleftharpoons SO_2Cl_2(g)\]

Calculate the change in moles of gases, \(\Delta n\)

The change in moles of gases is the difference between the moles of gaseous products and the moles of gaseous reactants. From the balanced equation, we have: \(\Delta n = \text{moles of products} - \text{moles of reactants} = (1) - (1+1) = -1\)

Calculate the equilibrium constant \(K_c\)

Now, applying the relationship between \(K_c\) and \(K_p\), and given \(K_p = 34.5\), \(R = 0.0821\,\text{L}\,\text{atm}\,\text{K}^{-1}\,\text{mol}^{-1}\) and \(T = 303\,\text{K}\), we have: \[K_c = K_p(RT)^{(\Delta n)}\] Plugging in the values: \[K_c = 34.5 \times (0.0821 \times 303)^{-1}\] Calculating \(K_c\): \[K_c = 34.5 \times (0.0821 \times 303)^{-1} = 211.65\] So, the equilibrium constant \(K_c\) at 303 K for the given reaction is 211.65.

Key concepts

Chemical Equilibrium

Chemical equilibrium is an essential concept in chemical reactions, describing a state where the concentrations of reactants and products remain constant over time. This occurs when the rate of the forward reaction equals the rate of the backward reaction, making the dynamic state appear static to an outside observer. Equilibrium is reached in both reversible and dynamic chemical reactions.
Some key aspects of chemical equilibrium include:

• The equilibrium position is determined by the initial concentrations of reactants and products, and any changes in conditions like temperature, pressure, or concentration can shift the equilibrium.
• Le Chatelier's principle explains how equilibrium shifts in response to changes, such as adding more reactants or products.
• At equilibrium, despite the constant concentrations, molecules continuously react and recombine, showcasing the dynamic nature of these reactions.

Understanding these elements is fundamental in predicting how a reaction proceeds and how to manipulate conditions to favor the formation of desired products.

Kc and Kp Relationship

In chemical equilibria involving gases, we often encounter two forms of equilibrium constants: the concentration-based constant (\(K_c\)) and the pressure-based constant (\(K_p\)). The relationship between these two constants is central to predicting the behaviors of gaseous systems at equilibrium.
The relationship is given by the equation:\[K_p = K_c(RT)^{\Delta n}\]where:

• \(R\) is the universal gas constant (\(0.0821\, \text{L}\,\text{atm}\,\text{K}^{-1}\,\text{mol}^{-1}\)), representing the relationship between energy and temperature/volume.
• \(T\) is the temperature in Kelvin.
• \(\Delta n\) is the change in moles of gas as the reaction proceeds from reactants to products.

This equation highlights the impact of \(\Delta n\), the difference between the moles of gaseous products and reactants, on equilibrium constants. If \(\Delta n\) is zero, \(K_c\) is equal to \(K_p\), emphasizing the importance of reaction stoichiometry in calculating and comparing different equilibrium constants.

Thermodynamics

Thermodynamics is the study of energy and its transformations, playing a vital role in understanding chemical reactions and equilibria. It helps in determining the feasibility and directionality of reactions as well as potential for spontaneous change.
In the context of equilibrium:

• Gibbs free energy (\( \Delta G\)) determines whether a reaction is spontaneous. A negative \( \Delta G\) signifies a spontaneous process.
• At equilibrium, \( \Delta G\) equals zero, indicating no net change in energy as the rates of the forward and reverse reactions are balanced.
• The relationship between \( \Delta G\) and the equilibrium constant (\(K\)) is given by the equation: \[ \Delta G = -RT \ln K \] This shows that the greater the value of \(K\), the more likely the reaction will proceed in the forward direction.

These thermodynamic principles are integral to manipulating reaction conditions, enabling chemists to design processes that maximize output while minimizing energy consumption.