Question

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

Short answer

The value of \(K_p\) for the reaction \(\mathrm{PCl}_3(g)+\mathrm{Cl}_2(g) \rightleftharpoons \mathrm{PCl}_5(g)\) at \(500 \mathrm{~K}\) is approximately 0.00103.

Step-by-step solution

Determine the change in the number of moles of gas, \(\Delta n\)

To find \(\Delta n\), we need to determine the difference in the number of moles of gaseous products and the number of moles of gaseous reactants: \[\Delta n = n_{\text{products}} - n_{\text{reactants}}\] For the given reaction: \[\Delta n = (\text{moles of PCl}_5) - (\text{moles of PCl}_3 + \text{moles of Cl}_2)\]

Calculate the value of \(\Delta n\)

From the balanced chemical reaction, we can see that there is one mole of \(\mathrm{PCl}_5\) formed and one mole each of \(\mathrm{PCl}_3\) and \(\mathrm{Cl}_2\) consumed. Thus: \[\Delta n = 1 - (1 + 1) = -1\]

Calculate the value of \(K_p\) using the given formula

Now we have all the required values to plug into the relationship between \(K_c\) and \(K_p\): \[K_p = K_c(RT)^{\Delta n}\] We know: - \(K_c = 0.042\) - \(R = 0.0821 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}}\) (the ideal gas constant) - \(T = 500 \mathrm{~K}\) (the temperature in Kelvin) - \(\Delta n = -1\) Let's plug in these values: \[K_p = 0.042 \left(0.0821 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}} \cdot 500 \mathrm{~K}\right)^{-1}\]

Solve for \(K_p\)

Perform the required calculations to find the value of \(K_p\): \[K_p = 0.042 \frac{1}{(0.0821)(500)}\] \[K_p \approx 0.00103\] #Solution# The value of \(K_p\) for the reaction \(\mathrm{PCl}_3(g)+\mathrm{Cl}_2(g) \rightleftharpoons \mathrm{PCl}_5(g)\) at \(500 \mathrm{~K}\) is approximately 0.00103.

Key concepts

Kc and Kp relationship

In chemical equilibria involving gases, there are two different constants that students need to be familiar with: \(K_c\) and \(K_p\). These constants describe the ratio of concentrations or pressures of the reactants and products at equilibrium. Understanding their relationship is crucial for predicting the direction of a chemical reaction under different conditions.
1. **Definition:**
- **\(K_c\):** This is the equilibrium constant expressed in terms of the concentration of reactants and products. It is calculated using molarity units, where concentrations are in moles per liter.
- **\(K_p\):** This is the equilibrium constant expressed in terms of the partial pressures of gases. It is used when dealing with reactions involving gases.
2. **Relationship between \(K_c\) and \(K_p\):**
The conversion from \(K_c\) to \(K_p\) requires understanding the change in the number of moles of gas throughout the reaction, denoted as \(\Delta n\). The relationship is defined by the equation:
\[ K_p = K_c (RT)^{\Delta n} \] Where:
- \(R\) is the ideal gas constant.
- \(T\) is the temperature in Kelvin.
- \(\Delta n\) is the difference between the moles of gaseous products and reactants.
Through this relationship, you can interconvert \(K_c\) and \(K_p\), depending on whether you're given concentrations or pressures in a gas-phase equilibrium.

Ideal gas constant

The ideal gas constant, denoted by \(R\), is an essential factor in the calculations of both \(K_c\) and \(K_p\) when converting between them. This constant has a fixed value and acts as a bridge in thermodynamic equations involving gases.
1. **Value of \(R\):**
In calculations involving equilibrium constants, \(R\) is typically expressed as \(0.0821 \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}\). This specific unit format is chosen because it harmonizes with the units used in pressure calculations when expressed in atmospheres, volume in liters, and temperature in Kelvin.
2. **Role of \(R\):**
The ideal gas constant \(R\) introduces the influence of temperature and pressure into chemical equilibria calculations, especially when deducing \(K_p\) from \(K_c\). It underscores the connection between temperature, volume, and pressure for ideal gases.
For consistency in problem-solving, always ensure the units of \(R\) match the units used in other components of your equations. Pivoting correctly on \(R\) helps retain accuracy in your calculated values.

Equilibrium constant calculations

Calculating equilibrium constants like \(K_c\) and \(K_p\) is an essential skill in chemistry, allowing prediction of the composition of a reaction mixture at equilibrium. Let's explore the steps typically involved in these calculations.
1. **Balancing the Reaction:**
Begin with ensuring that the chemical equation is balanced. This step is crucial since the mole coefficients in the balanced equation are directly used in the expression of both \(K_c\) and \(K_p\).
2. **Writing the Equilibrium Expression:**
- **For \(K_c\):** The concentrations (in mol/L) of the products are raised to the power of their coefficients and divided by the concentrations of the reactants, similarly raised to their respective coefficients.
- **For \(K_p\):** Instead of concentrations, use partial pressures.
3. **Calculating \(\Delta n\):**
For reactions involving gases, calculate \(\Delta n\), which is the change in moles from reactants to products. This value is integral when converting between \(K_c\) and \(K_p\) using the formula \[ K_p = K_c (RT)^{\Delta n} \].
4. **Performing the Calculation:**
Plug the known quantities into the equilibrium expression. This includes substituting values for concentrations or partial pressures and constants for the temperature and \(R\), the ideal gas constant.
Overall, equilibrium constant calculations hinge upon a clear understanding of chemical reactions, balancing techniques, and the precise role of physical constants.