Question

For the process $$\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g)$$ it is found that the equilibrium concentrations at a particular temperature are \(\left[\mathrm{H}_{2}\right]=1.4 \mathrm{M},\left[\mathrm{CO}_{2}\right]=1.3\) \(M,[\mathrm{CO}]=0.71 \mathrm{M},\) and \(\left[\mathrm{H}_{2} \mathrm{O}\right]=0.66 \mathrm{M} .\) Calculate the equilibrium constant \(K\) for the reaction under these conditions.

Short answer

Given the equilibrium concentrations, the equilibrium constant K for the reaction \(\mathrm{CO}(g) + \mathrm{H}_2\mathrm{O}(g) \rightleftharpoons \mathrm{CO}_2(g) + \mathrm{H}_2(g)\) can be calculated using the expression \(K_c = \frac{[\mathrm{CO}_2][\mathrm{H}_2]}{[\mathrm{CO}][\mathrm{H}_2\mathrm{O}]}\). With the given concentrations, this evaluates to \(K_c \approx \frac{(1.3)(1.4)}{(0.71)(0.66)} \approx 3.80\).

Step-by-step solution

First, let's ensure we have the balanced chemical equation: \(\mathrm{CO}(g) + \mathrm{H}_2\mathrm{O}(g) \rightleftharpoons \mathrm{CO}_2(g) + \mathrm{H}_2(g)\) #Step 2: Write the Expression for Kc#

Next, we need to write the equilibrium constant expression, which includes the concentrations of reactants and products. For this reaction, the expression for Kc is: \(K_c = \frac{[\mathrm{CO}_2][\mathrm{H}_2]}{[\mathrm{CO}][\mathrm{H}_2\mathrm{O}]}\) #Step 3: Substitute the Given Concentrations in Kc Expression#

Now, we will substitute the given equilibrium concentrations in the Kc expression: \(K_c = \frac{[1.3 \mathrm{M}][1.4 \mathrm{M}]}{[0.71 \mathrm{M}][0.66 \mathrm{M}]}\) #Step 4: Calculate Kc#

Finally, we will calculate the equilibrium constant, Kc: \(K_c = \frac{(1.3)(1.4)}{(0.71)(0.66)}\) \(K_c \approx 3.80\) So, the equilibrium constant Kc for this reaction under these conditions is approximately 3.80.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a crucial concept in chemistry, representing a state where both reactants and products coexist without any net change in their concentrations over time. In the given reaction between carbon monoxide and water vapor, a dynamic equilibrium is established. This means the forward and reverse reactions occur at the same rate. At equilibrium, the amounts of the reactant gases ( ext{CO} and ext{H}_2 ext{O} ) and the product gases ( ext{CO}_2 and ext{H}_2 ) stabilize.

• Dynamic Nature: Even at equilibrium, the molecules of reactants are constantly converting to products and vice versa.
• No Net Change: Though the individual molecules react, the observable concentrations remain fixed.

Understanding this "no net change" property is fundamental when calculating equilibrium constants, as it distinguishes chemical equilibrium from mere cessation of reactions.

Reaction Quotient

The reaction quotient, Q, provides a snapshot of a reaction's progression at any given instant. The key difference between Q and the equilibrium constant (K) lies in their timing. While K describes the reaction at equilibrium, Q can be calculated at any moment. For the given reaction, before equilibrium is achieved, you may calculate Q using the concentrations:

• Predicted Direction: Comparing Q with K tells us if the system will proceed forward or backward (towards reactants).
• Initial Conditions: It helps define whether the reaction quotient indicates more products or reactants relative to equilibrium.

If Q < K, the forwards reaction will proceed to make more products. When Q > K, the reverse reaction should proceed. If Q = K, then the reaction is at equilibrium.

Concentration

Concentration plays a vital role in determining the position of equilibrium in a chemical reaction. It directly affects the equilibrium constant when calculating using concentrations. In the exercise, the initial concentrations were used to determine K_c. Higher concentrations of products or reactants can shift the equilibrium position:

• Effect on Rate: Changes in concentrations impact the rate of reactions, influencing how fast equilibrium is reached.
• Shifts in Equilibrium: According to Le Châtelier’s Principle, changes in concentrations can cause shifts to either the left or the right.

In our exercise, the concentrations of ext{CO}, ext{H}_2 ext{O}, ext{CO}_2, and ext{H}_2 were necessary to compute K_c by positioning the equilibrium status as stable with the provided values.

Equilibrium Expression

The equilibrium expression is fundamental in calculating equilibrium constants, transforming complex chemical interactions into simple quotients. It involves ratios of the concentrations of products over reactants, expressed mathematically. For the reaction given, the equilibrium expression for K_c is provided as:\[ K_c = \frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]} \]This aids in linking observable concentrations to the intrinsic tendency of the reaction to reach equilibrium.

• Concentration Only: Calculations with K_c involve only the concentrations of aqueous or gaseous species.
• Standard Form: It's standardized based on the stoichiometry and states of matter.

The equilibrium expression captures the balance of reactants and products and aids in understanding the behavior of reactions under different conditions.