Problem 21 - Chemistry The Central Science

Question

If \(K_{c}=0.042\) for \(\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftarrows \mathrm{PCl}_{5}(g)\) at 500 \(\mathrm{K}\) , what is the value of \(K_{p}\) for this reaction at this temperature?

Step-by-Step Solution

Answer

The value of Kp for the reaction PCl3(g) + Cl2(g) ⇌ PCl5(g) at 500 K can be calculated using the relationship Kp = Kc(RT)^(Δn), where Kc = 0.042, R = 0.0821 L atm/mol K, T = 500 K, and Δn = -1 (calculated as Δn = moles of products - moles of reactants). Plugging in these values, we obtain Kp ≈ 0.001023.

Step 1: Calculate Δn

The balanced chemical equation is given as: PCl3(g) + Cl2(g) ⇌ PCl5(g) The change in moles of gas can be calculated as follows: Δn = moles of products - moles of reactants Δn = 1 (from PCl5) - (1 + 1) (from PCl3 and Cl2) Δn = 1 - 2 = -1

Step 2: Use the Kc to Kp relationship

Now that we have Δn, we can use the relationship between Kc and Kp: Kp = Kc(RT)^(Δn) We were given the value Kc = 0.042, the temperature T = 500 K, and we calculated Δn = -1. The gas constant, R, is 0.0821 L atm/mol K. Plugging in these values, we get: Kp = (0.042)(0.0821 * 500)^(-1) Kp = 0.042 / (41.05) Kp = 0.001023

Step 3: Write the final answer

The value of Kp for this reaction at 500 K is approximately 0.001023.

Key Concepts

Kp and Kc Relationship

When studying chemical reactions that involve gases, understanding the relationship between the equilibrium constants Kp and Kc is crucial. Kp represents the equilibrium constant in terms of partial pressures of the gases involved, while Kc is expressed in terms of molar concentrations. These constants are related by the equation:

\[ K_{p} = K_{c}(RT)^{\Delta n} \]
Here, \(R\) is the ideal gas constant (0.0821 L atm/mol K), \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in the number of moles of gas going from reactants to products. It's important to note that \(\Delta n\) is calculated by subtracting the sum of the moles of gaseous reactants from the sum of the moles of gaseous products. A negative \(\Delta n\) implies that there are fewer moles of gas in the products than reactants, leading to a decrease in pressure at equilibrium.
This equation applies to ideal gases and allows us to convert between Kp and Kc when the reaction involves changes in the number of moles of gases. Shorter, simpler sentences, such as 'Kp is used for pressure; Kc is used for concentration,' or 'Kp and Kc are connected by a simple formula involving the gas constant R and the temperature T,' can help solidify the concept for students.

Chemical Equilibrium

Whether you're looking at a color change in a solution or the production of a gas in a lab, chemical equilibrium plays a key role in these reactions. At chemical equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and concentrations of reactants and products remain constant over time – they do not necessarily have to be equal, but they stop changing. This state can be quantified using the equilibrium constant, either Kc or Kp depending on the state of the reactants and products.

For the reaction \( \mathrm{PCl}_{3}(g) + \mathrm{Cl}_{2}(g) \rightleftarrows \mathrm{PCl}_{5}(g) \), Kc is given as 0.042 at 500 K, meaning that at this temperature, the ratio of the concentrations of the products to reactants is low, favoring the reactants slightly at equilibrium. Understanding this helps students predict reaction behavior such as which side is favored under certain conditions and how the reaction might respond to changes in conditions.

Le Chatelier's Principle

Le Chatelier's principle is like a guide to predicting how a system at equilibrium will respond to changes in concentration, temperature, or pressure. Simply put, if you change the conditions of a reaction at equilibrium, the system will adjust to counteract that change.

For instance, if you increase the concentration of reactants for the reaction \( \mathrm{PCl}_{3}(g) + \mathrm{Cl}_{2}(g) \rightleftarrows \mathrm{PCl}_{5}(g) \), the system will try to consume those extra reactants by making more products. This is the system's way of restoring equilibrium. Similarly, changes in temperature can shift the position of equilibrium. If the reaction is exothermic (releases heat), increasing the temperature will cause the system to favor the reverse reaction to absorb the excess heat.
A practical tip for students: Le Chatelier’s principle comes in handy for predicting shifts in reaction equilibria and can guide you when adjusting reaction conditions to obtain more products or reactants as needed.