Relationship between Kc and Kp for homogeneous and heterogeneous equilibria

📚 Understanding $K_c$ and $K_p$

In chemical kinetics, $K_c$ and $K_p$ are equilibrium constants that describe the ratio of products to reactants at equilibrium. However, they differ in how they express concentrations: $K_c$ uses molar concentrations, while $K_p$ uses partial pressures, making $K_p$ suitable for reactions involving gases.

📜 History and Background

The concept of chemical equilibrium was first introduced by Claude Louis Berthollet in the early 19th century. However, the formal definitions of equilibrium constants like $K_c$ and $K_p$ were developed later, primarily through the work of Guldberg and Waage, who formulated the law of mass action.

• 👨‍🔬 Claude Louis Berthollet: 19th-century pioneer of chemical equilibrium concepts.
• ⚖️ Guldberg and Waage: Developed the Law of Mass Action, providing a foundation for equilibrium constants.

🔑 Key Principles

The relationship between $K_c$ and $K_p$ is given by the following equation:

$K_p = K_c(RT)^{\Delta n}$

Where:

• 🌡️ $R$ is the ideal gas constant (0.0821 L atm / (mol K)).
• ☀️ $T$ is the absolute temperature in Kelvin.
• ▵ $ \Delta n $ is the change in the number of moles of gas (moles of gaseous products - moles of gaseous reactants).

Homogeneous Equilibria

Homogeneous equilibria involve reactants and products in the same phase. If all reactants and products are gases, then the relationship $K_p = K_c(RT)^{\Delta n}$ applies directly. If $\Delta n = 0$, then $K_p = K_c$.

Heterogeneous Equilibria

Heterogeneous equilibria involve reactants and products in different phases. Solids and liquids do not appear in the $K_p$ expression because their 'concentration' (activity) is essentially constant. When calculating $\Delta n$, only gaseous species are considered.

• 🧱 Solids: Activity is considered to be 1 and is not included in equilibrium constant expressions.
• 💧 Liquids: Similar to solids, pure liquids are not included in the equilibrium constant expression.
• 💨 Gases: Gases are included in both $K_c$ and $K_p$ expressions.

⚗️ Example Calculations

Example 1: Homogeneous Equilibrium

Consider the reaction:

$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$

If $K_c = 0.5$ at 500K, calculate $K_p$.

$\Delta n = 2 - (1 + 3) = -2$

$K_p = 0.5 * (0.0821 * 500)^{-2} = 0.5 / (41.05)^2 = 0.000296$

Example 2: Heterogeneous Equilibrium

Consider the reaction:

$CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$

If $K_c = 0.04$ at 800K, calculate $K_p$. Note that $K_c$ here refers to the concentration of $CO_2$ gas.

$\Delta n = 1 - 0 = 1$ (only $CO_2$ is a gas)

$K_p = 0.04 * (0.0821 * 800)^{1} = 0.04 * 65.68 = 2.6272$

🧪 Real-world Examples

• 🏭 Haber-Bosch Process: The industrial synthesis of ammonia, a crucial process for fertilizer production. The equilibrium between nitrogen, hydrogen, and ammonia gases relies on $K_p$ calculations to optimize yield.
• 🌋 Limestone Decomposition: The decomposition of calcium carbonate ($CaCO_3$) into calcium oxide ($CaO$) and carbon dioxide ($CO_2$) is vital in cement production. Understanding $K_p$ helps control the reaction conditions.

✍️ Conclusion

Understanding the relationship between $K_c$ and $K_p$ is crucial for predicting and controlling chemical reactions, particularly those involving gases. The equation $K_p = K_c(RT)^{\Delta n}$ allows for conversions between these equilibrium constants, providing a more complete picture of the equilibrium state. Remember to only consider gaseous species when calculating $\Delta n$ for heterogeneous equilibria!

❓ Practice Quiz

1. ❓ What is the relationship between $K_p$ and $K_c$?
2. 🧪 How does $\Delta n$ affect the relationship between $K_p$ and $K_c$?
3. 💨 For the reaction $2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$, if $K_c = 2.8 \times 10^2$ at 1000K, calculate $K_p$.
4. 🧱 For the reaction $C(s) + CO_2(g) \rightleftharpoons 2CO(g)$, how is $\Delta n$ calculated?
5. 💡 What is the significance of knowing $K_p$ and $K_c$ in industrial processes?
6. 📚 Explain the difference between homogeneous and heterogeneous equilibria with respect to $K_p$ and $K_c$.
7. 📈 If $K_p = K_c$, what does this imply about the reaction?