Question

The water-gas shift reaction plays a central role in the chemical methods for obtaining cleaner fuels from coal: $$ \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) $$ At a given temperature, \(K_{\mathrm{p}}=2.7 .\) If \(0.13 \mathrm{~mol}\) of \(\mathrm{CO}, 0.56 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}, 0.62 \mathrm{~mol}\) of \(\mathrm{CO}_{2},\) and \(0.43 \mathrm{~mol}\) of \(\mathrm{H}_{2}\) are put in a \(2.0-\mathrm{L}\) flask, in which direction does the reaction proceed?

Short answer

The reaction will proceed in the reverse direction.

Step-by-step solution

Write the Reaction and Given Data

The balanced chemical equation for the water-gas shift reaction is \(\mathrm{CO}(g) + \mathrm{H}_{2}\mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g) + \mathrm{H}_{2}(g)\). The equilibrium constant, \(K_{\mathrm{p}}\), at a given temperature is 2.7. The initial moles of each component are as follows:\[\text{CO} = 0.13 \; \text{mol}\]\[\text{H}_2\text{O} = 0.56 \; \text{mol}\]\[\text{CO}_2 = 0.62 \; \text{mol}\]\[\text{H}_2 = 0.43 \; \text{mol}\]\

Calculate Initial Concentrations

Convert the moles of each gas to concentrations using the volume of the flask, \(2.0 \; \text{L}\).\[\left[ \text{CO} \right] = \frac{0.13 \; \text{mol}}{2.0 \; \text{L}} = 0.065 \; \text{M}\]\[\left[ \text{H}_2\text{O} \right] = \frac{0.56 \; \text{mol}}{2.0 \; \text{L}} = 0.28 \; \text{M}\]\[\left[ \text{CO}_2 \right] = \frac{0.62 \; \text{mol}}{2.0 \; \text{L}} = 0.31 \; \text{M}\]\[\left[ \text{H}_2 \right] = \frac{0.43 \; \text{mol}}{2.0 \; \text{L}} = 0.215 \; \text{M}\]\

Write the Reaction Quotient (Qp) Expression

The reaction quotient, \(Q_{\mathrm{p}}\), is given by:\[ Q_{\mathrm{p}} = \frac{ \left[ \mathrm{CO}_{2} \right] \left[ \mathrm{H}_{2} \right] }{ \left[ \mathrm{CO} \right] \left[ \mathrm{H}_2\mathrm{O} \right] }\]\ Substitute the initial concentrations into the expression:\[ Q_{\mathrm{p}} = \frac{ (0.31)(0.215) }{ (0.065)(0.28) } = \frac{0.06665}{0.0182} \approx 3.66\]

Compare Qp to Kp

Compare the calculated \(Q_{\mathrm{p}}\) to the given \(K_{\mathrm{p}} = 2.7\). Since \(Q_{\mathrm{p}} = 3.66\) is greater than \(K_{\mathrm{p}} = 2.7\), the reaction will shift to the left to reach equilibrium.

Key concepts

equilibrium constant

The equilibrium constant, denoted as \( K \), is a crucial concept in chemical reactions. It gives us an idea of the ratio of concentrations of products to reactants at equilibrium. When we have a reaction:
\(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant \(K\) for this reaction can be written as:
\[ K = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]The square brackets indicate the concentration of each species, and the exponents are the stoichiometric coefficients from the balanced equation.
Higher values of \( K \) mean more products than reactants at equilibrium. Conversely, lower values of \( K \) mean more reactants than products.Understanding \( K \) helps us predict the position of equilibrium and the extent of the reaction.
In the water-gas shift reaction, we are given \( K_p = 2.7 \). This value helps us in determining if the system is at equilibrium or if it will shift to reactants or products to reach equilibrium.

reaction quotient

The reaction quotient, \(Q\), provides a snapshot of a reaction's progress. While \(K\) uses equilibrium concentrations, \(Q\) uses the current, known concentrations at any point in time.
For a general reaction \( aA + bB \rightleftharpoons cC + dD \), the reaction quotient \(Q\) can be calculated similarly:\[ Q = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
By comparing \(Q\) with the equilibrium constant \(K\), we can predict the direction the reaction needs to proceed to reach equilibrium:

• If \(Q < K\), the reaction will shift to the right, producing more products.
• If \(Q > K\), the reaction will shift to the left, forming more reactants.
• If \(Q = K\), the reaction is at equilibrium.

In our example with the water-gas shift reaction: \[ Q_p = \frac{(0.31)(0.215)}{(0.065)(0.28)} \approx 3.66 \], which we then compare with \( K_p = 2.7 \).

direction of reaction

The direction in which a reaction proceeds depends on the relationship between the reaction quotient, \(Q\), and the equilibrium constant, \(K\).
If \(Q < K\), the system has too many reactants compared to products, so the reaction will move forward (to the right) to produce more products. Conversely, if \(Q > K\), there are too many products, and the reaction will go in reverse (to the left) to produce more reactants.
In our specific example, \( Q_p = 3.66 \), and \( K_p = 2.7 \). Since \( Q_p \) is greater than \( K_p \), the reaction favors the reactants. Thus, the system will shift to the left.
Understanding the direction of a reaction based on \( Q \) and \( K \) helps chemists control and optimize reactions, essential in various industrial and laboratory processes.