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 \(K_c\) for the given reaction at 303 K is approximately \(862.36\).

Step-by-step solution

Write down the balanced chemical equation

We are given the equilibrium reaction: \[ \mathrm{SO}_{2}(g) + \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{SO}_{2}\mathrm{Cl}_{2}(g) \]

Understand the relationship between Kp and Kc

The relationship between equilibrium constants Kp (in terms of partial pressure) and Kc (in terms of molar concentration) is given by the equation: \[K_p = K_c(RT)^{\Delta n}\] where \(K_p\), \(K_c\) - equilibrium constants, \(R\) - the ideal gas constant (\(R=0.0821 \frac{L \cdot atm}{mol \cdot K}\)), \(T\) - the absolute temperature (in Kelvin), \(\Delta n = n_{products} - n_{reactants}\) - the change in the number of moles of gaseous species.

Calculate the change in the number of moles Δn

For the given reaction, there is 1 mole of gaseous product (SO₂Cl₂) and 2 moles of gaseous reactants (SO₂ and Cl₂). Therefore the change in the number of moles Δn is: \[\Delta n = n_{products} - n_{reactants} = 1 - 2 = -1\]

Insert values into the Kp-Kc relationship and solve for Kc

Now we substitute the given values for Kp, R, T, and Δn into the relationship between Kp and Kc equation: \[K_p = K_c(RT)^{\Delta n}\] \[34.5 = K_c(0.0821 \cdot 303)^{-1}\] Now we solve for Kc: \[K_c = 34.5 \cdot (0.0821 \cdot 303)^1 =34.5 \cdot (24.99) = 862.36\] So, the value of the equilibrium constant Kc for the given reaction at 303 K is approximately \(862.36\).

Key concepts

Equilibrium Constants Relationship

Understanding the relationship between the equilibrium constants, like the ones denoted as Kp and Kc, is essential when dealing with chemical reactions taking place in a gaseous medium. Kp is the equilibrium constant based on the partial pressures of the gases involved, whereas Kc is defined based on the molar concentrations.

To connect these constants, we use the formula:
\[K_p = K_c(RT)^{\Delta n}\]
where \(R\) represents the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in moles of the gaseous substances from reactants to products. This relationship is particularly useful when you have the value for one constant (like Kp from experimental data) and need to find the other (Kc for instance). The formula accounts for the different units and states of the chemical species present in equilibrium.

Calculated Mole Changes

The change in moles, denoted as \(\Delta n\), in an equilibrium reaction is a key factor in the relationship between Kp and Kc. Here's how we calculate it:

• Count the number of moles of gaseous products in the balanced equation.
• Count the number of moles of gaseous reactants in the same balanced equation.
• Subtract the number of moles of reactants from the number of moles of products.

For the given reaction, \(\Delta n = n_{products} - n_{reactants} = 1 - 2 = -1\). This calculated change in moles will dictate the exponent for the term \((RT)^{\Delta n}\) in our Kp-Kc equation. A negative \(\Delta n\) will mean that a reaction produces fewer moles of gaseous products than it consumes reactants, leading to a decrease in pressure.

Ideal Gas Law Constants

The ideal gas law is represented by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. The ideal gas constant \(R\) has different values depending on the units used for pressure and volume. In the context of Kp and Kc relationship, we often use \(R = 0.0821\frac{L \cdot atm}{mol \cdot K}\).

This constant is crucial in bridging the gap between behaviors of ideal gases under different conditions, and it enables calculation of one state variable when the others are known. When using R in equations relating to equilibrium, it's important to maintain consistent units across all variables to ensure accuracy in your calculations.

Chemical Equilibrium

Chemical equilibrium occurs in a reversible reaction when the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. This state can be quantified using an equilibrium constant, such as Kc or Kp, dependent on whether concentration or pressure is being measured.

It's important to note that equilibrium doesn’t mean the amounts of reactants and products are equal, but that their ratios stay constant. Factors that affect chemical equilibrium include concentration, temperature, and pressure. Le Chatelier's Principle also explains how the equilibrium position will shift to counteract changes in these factors. Understanding equilibrium is essential for predicting how the conditions can alter the outcome of a chemical reaction.