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?

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.

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 \( K_p = 2.7 \).

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.