Question

The equilibrium constant \(K_{p}\) for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g) $$ \(4.0\) at \(1660^{\circ} \mathrm{C}\). Initially \(0.80\) mole \(\mathrm{H}_{2}\) and \(0.80\) mole \(\mathrm{CO}_{2}\) are injected into a \(5.0\) litre flask. What is the equilibrium concentration of \(\mathrm{CO}_{2}(g) ?\) (a) \(0.533 \mathrm{M}\) (b) \(0.0534 \mathrm{M}\) (c) \(0.535 \mathrm{M}\) (d) None of these

Short answer

The equilibrium concentration of \(\mathrm{CO}_{2}(g)\) is approximately \(0.2667\) M, which is not an option given, so the correct answer is (d) None of these.

Step-by-step solution

- Write the expression for the equilibrium constant

The equilibrium constant expression for the given reaction \(\mathrm{H}_{2}(g) + \mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2}\mathrm{O}(g) + \mathrm{CO}(g)\) is \[K_{p} = \frac{[\mathrm{H}_{2}\mathrm{O}][\mathrm{CO}]}{[\mathrm{H}_{2}][\mathrm{CO}_{2}]}\].

- Set up an ICE table

Create an Initial, Change, Equilibrium (ICE) table to keep track of the concentration changes of reactants and products. Initial concentrations (in moles per liter, M) of \(\mathrm{H}_{2}\) and \(\mathrm{CO}_{2}\) are both 0.80 M, the change for each will be \(-x\) when forming \(x\) M of \(\mathrm{H}_{2}\mathrm{O}\) and \(\mathrm{CO}\). The equilibrium concentrations will be 0.80 M \(-x\) for both \(\mathrm{H}_{2}\) and \(\mathrm{CO}_{2}\), and \(x\) M each for \(\mathrm{H}_{2}\mathrm{O}\) and \(\mathrm{CO}\).

- Express the equilibrium constant in terms of the variable x

Using the ICE table, substitute the equilibrium concentrations into the equilibrium expression to get: \[K_{p} = \frac{x^2}{(0.80 - x)^2}\]. Given that \(K_{p} = 4.0\), the equation to solve is \[4.0 = \frac{x^2}{(0.80 - x)^2}\].

- Simplify and solve for x

To simplify the equation, take the square root of both sides to get: \[2 = \frac{x}{0.80 - x}\]. Then, cross-multiply to obtain \[2 \times (0.80 - x) = x\], which simplifies to \[1.60 - 2x = x\]. After rearranging terms, we get \[1.60 = 3x\], and solving for \(x\) yields \[x = \frac{1.60}{3} \approx 0.5333\] M.

- Calculate the equilibrium concentration of CO2(g)

The equilibrium concentration of \(\mathrm{CO}_{2}(g)\) is found by subtracting \(x\) from the initial concentration: \[0.80 - x = 0.80 - 0.5333\] which gives approximately \[0.2667\] M.

Key concepts

ICE Table

An ICE table (Initial, Change, Equilibrium) is a valuable tool used to evaluate the concentrations of reactants and products over the course of a chemical reaction that has reached equilibrium. It helps systematically track the initial concentrations, the changes that occur as reactants convert to products, and the final equilibrium concentrations.

To construct an ICE table, begin by listing the reactants and products at the top. Below, in the corresponding columns, write the initial concentrations of the substances. During the reaction, the concentrations change, usually by an unknown quantity we label as 'x'. These changes are noted in the second row. The third row represents the concentrations at equilibrium, which are determined by combining the initial concentrations and the changes.

By applying the ICE table to the given reaction, students can visualize how the reactants and products shift as the reaction proceeds towards equilibrium. This methodical approach provides a clear path to calculating the unknown concentrations, ultimately leading to a better understanding of the reaction dynamics.

Le Chatelier's Principle

Le Chatelier's Principle is a guiding concept in chemistry, providing insight into how a chemical system at equilibrium reacts to external changes. When a system in equilibrium is subjected to a change in concentration, temperature, or pressure, it responds in a way that counteracts this disturbance and re-establishes equilibrium.

For instance, if you increase the concentration of a reactant, the system will shift towards producing more products to reduce the increased concentration of the reactant. Similarly, a temperature change can affect the equilibrium position depending on whether the reaction is exothermic or endothermic. Remembering that the system will always move to oppose the change introduced helps in predicting the direction of the shift and understanding the behavior of the reaction under various environmental alterations.

Chemical Equilibrium

Chemical equilibrium is a state in a reversible reaction where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products. It's important to note that equilibrium does not mean the reactants and products are in equal concentrations, but rather that their ratios remain constant over time.

At equilibrium, a reaction has reached a balance between the forward and reverse processes. This dynamic state is maintained as long as the external conditions remain constant. Understanding equilibrium allows chemists to determine how different conditions, such as changes in concentration or temperature, will affect reaction yields. It's crucial for optimizing reactions in industrial processes and for understanding natural chemical processes.

Reaction Quotient

The reaction quotient (Q) is a measure that indicates how far a reaction has progressed towards equilibrium. It is determined by plugging the current concentrations of the reactants and products into the same formula as the equilibrium constant (K).

If Q is equal to K, the reaction is at equilibrium. A Q less than K means the reaction will proceed in the forward direction to reach equilibrium, consuming reactants to form more products. Conversely, if Q is greater than K, the reaction will proceed in the reverse direction to re-establish equilibrium, forming more reactants from products.

In solving our example, you don't calculate Q directly, but you use the idea behind it. By setting up the equilibrium constant expression in terms of the changes in concentration (x), we mathematically 'forced' the reaction quotient to equal the known equilibrium constant (K), hence finding the change necessary to reach equilibrium.