Question

Determine \(K_{c}\) for the reaction \(\mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})+\) \(\mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{g}),\) given the following data at \(298 \mathrm{K}\) $$\frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{NO}_{2}(\mathrm{g}) \quad K_{\mathrm{p}}=1.0 \times 10^{-9}$$ $$\operatorname{NOCl}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{NO}_{2} \mathrm{Cl}(\mathrm{g}) \quad K_{\mathrm{p}}=1.1 \times 10^{2}$$ $$\mathrm{NO}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{NO}_{2} \mathrm{Cl}(\mathrm{g}) \quad K_{\mathrm{p}}=0.3$$

Short answer

The equilibrium constant \(K_{c}\) for the given reaction is 6 \times 10^6

Step-by-step solution

Analyze the given reactions and required combined reaction

Observe the reactions, the goal is to combine them in a way that the resulting reaction matches the original reaction given. Mostly this process involves reversing some reactions and/or multiplying them by some factor such that the final combined reaction gets the same as the initial one.

Combine reactions

By reversing the first reaction and keeping the second one as it is, and adding half of the third reaction, the result is the original reaction:\n \(-1/2(1/2 \mathrm{N}_{2} + \mathrm{O}_{2} \rightarrow \mathrm{NO}_{2})\),\n \(+1(\mathrm{NOCl} + 1/2 \mathrm{O}_{2} \rightarrow \mathrm{NO}_{2} \mathrm{Cl})\),\n \(+1/2(\mathrm{NO}_{2} + 1/2 \mathrm{Cl}_{2} \rightarrow \mathrm{NO}_{2}\mathrm{Cl}),\),\n This forms our desired equation: \(\mathrm{N}_{2} + \mathrm{O}_{2} + \mathrm{Cl}_{2} \rightarrow 2\mathrm{NOCl}\)

Calculate new equilibrium constants

The equilibrium constant for the reverse of a reaction is the inverse of the original reaction, and when a reaction is multiplied by a factor n, its equilibrium constant is raised to the nth power. Thus, the new equilibrium constants are: \n \(-1/2(\mathrm{K}_{\mathrm{p1}'}= \frac{1}{(K_{p1}^{1/2})}) = \frac{1}{(1.0 \times 10^{-9})^{1/2}} = 10^9^{1/2} = 10^{4.5}\) \n \(+1(K_{p2}= 1.1 \times 10^{2}) = 1.1 \times 10^{2}\), \n \(+1/2(K_{p3'} = (0.3)^{1/2}) = (0.3)^{1/2} = 0.548\)

Multiply the equilibrium constants

When reactions are added together, their equilibrium constants multiply. Thus, the equilibrium constant for the original reaction is \(K_{c} = K_{\mathrm{p1}'} \times K_{p2} \times K_{p3'} = 10^{4.5} \times 1.1 \times 10^{2} \times 0.548 = 6 \times 10^{6}\)

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fascinating state where a chemical reaction and its reverse occur at the same rate. Imagine a seesaw perfectly balanced. In reactions, the concentrations of reactants and products remain constant over time when equilibrium is reached.
Think of it as a balance between forward and backward steps — neither side wins, they just keep moving at the same pace. This doesn’t mean the reactants and products are equal in concentration; instead, they are stable and unchanging as long as the system remains undisturbed.

• Equilibrium can be manipulated by changing conditions such as concentration, pressure, and temperature.
• The equilibrium constant, denoted as \( K \), provides insight into whether products or reactants dominate when equilibrium is achieved.
• A reaction with a large \( K \) tends to form more products, whereas a small \( K \) suggests more reactants.

Studying chemical equilibrium helps in understanding how reactions can be controlled and predicted, making it an essential topic in chemistry.

Reaction Mechanisms

Reaction mechanisms describe the step-by-step sequence of events that occur during a chemical reaction. Consider them like detailed instructions needed to bake a cake — each step builds upon the last until the cake, or end product, is complete.
Understanding these mechanisms allows chemists to understand how reactions occur on a molecular level. They can deduce the intermediate products, which aren’t always apparent in the overall reaction.
Just like following a recipe, each reaction step has its own rate and specific conditions under which it proceeds, often involving different transition states and intermediates that are crucial for the reaction’s progression.

• Reaction mechanisms provide insight into the pathways and intermediates appearing during a reaction.
• Each step in a mechanism often follows its specific kinetics laws.
• Complex reactions may involve multiple steps with varying speeds and sequences.

Exploring reaction mechanisms aid in predicting the outcomes and rates of reactions, essential for innovations in areas like pharmaceuticals and materials science.

Thermodynamics

Thermodynamics is like the study of energy in action. It explores how energy is transferred and transformed in chemical processes. When applied to chemical reactions, thermodynamics helps us understand why reactions occur and whether they are feasible under given conditions.
Three main laws underpin this field, dealing primarily with energy conservation, entropy, and the direction of energy change. In terms of chemical reactions, it tells us about energy exchanges and the favorability of reactions:

• The First Law of Thermodynamics states energy cannot be created or destroyed — only transformed.
• The Second Law introduces the concept of entropy, the idea that systems naturally progress from order to disorder.
• The Third Law relates to the lowest possible entropy (absolute zero) and its impact on materials.
• Gibbs free energy combines enthalpy and entropy to predict the direction of reactions.

When assessing thermodynamics in reactions, key considerations include \--\( \Delta H \), the enthalpy change, and \( \Delta G \), the Gibbs free energy change, \to understand reaction spontaneity and equilibrium positions.

Reaction Kinetics

Reaction kinetics is the study of rates at which chemical reactions proceed. Kinetics tells us how fast reactions occur, crucial for practical applications like synthesizing new materials or developing medicines.
It examines factors affecting reaction rates, such as concentration, temperature, and catalysts.
By understanding kinetics, scientists can speed up desired reactions or slow down undesired ones. Think of kinetics as the accelerator pedal in your reaction car — determining how fast you reach your chemical destination.

• The rate law expresses the relationship between reactant concentration and reaction rate.
• Activation energy, the initial energy required to start a reaction, is a key kinetic factor.
• Temperature generally speeds up reactions by providing more energy to reactant particles.
• Catalysts lower activation energy, thus increasing reaction rates without being consumed.

Understanding reaction kinetics is crucial for controlling industrial processes and developing commercial products. It helps design efficient and safe chemical procedures.