Question

Consider the reaction: \begin{equation}\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}(g)\end{equation} An equilibrium mixture of this reaction at a certain temperature has \([\mathrm{CO}]=0.105 \mathrm{M},\left[\mathrm{H}_{2}\right]=0.114 \mathrm{M},\) and \(\left[\mathrm{CH}_{3} \mathrm{OH}\right]=0.185 \mathrm{M} .\) What is the value of the equilibrium constant \(\left(K_{\mathrm{c}}\right)\) at this temperature?

Short answer

The value of the equilibrium constant \(K_{c}\) is 135.53.

Step-by-step solution

Write the Expression for the Equilibrium Constant

The equilibrium constant expression for the given reaction \(\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}(g)\) is \[K_{c} = \frac{[\mathrm{CH}_{3} \mathrm{OH}]}{[\mathrm{CO}][\mathrm{H}_{2}]^{2}}\]

Plug in the Equilibrium Concentrations

Substitute the given concentrations into the equilibrium constant expression.\[K_{c} = \frac{0.185}{0.105 \times (0.114)^2}\]

Calculate the Value of the Equilibrium Constant

Perform the calculation to find the value of \(K_{c}\).\[K_{c} = \frac{0.185}{0.105 \times 0.013}\]\[K_{c} = \frac{0.185}{0.001365}\]\[K_{c} = 135.53\]

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is crucial when examining chemical reactions that can occur in both the forward and reverse direction. Unlike a 'completed' reaction, a reaction in equilibrium is not static; instead, there is a steady exchange where reactants and products are formed at the same rate. This dynamic process eventually leads to a state where the concentrations of reactants and products remain constant over time, although both the forward and reverse reactions are still occurring. It's important to note that chemical equilibrium does not mean the reactants and products are present in equal amounts, but rather that their ratios do not change with time.

For students aiming to grasp this concept, consider a visual analogy like a seesaw that balances despite the continuous motion; the reactants and products are constantly interchanging, yet their concentrations are steady, illustrating the balance that defines equilibrium.

Reaction Concentration

The concentration of a species in a reaction, commonly represented in molarity (M), which refers to moles per liter, dictates the behavior and direction of a chemical reaction. It is a snapshot of how much reactant or product is present in a given volume at any point in time. During a reaction, as reactants are converted to products, their concentrations will change until equilibrium is reached. At this point, as we've learned, the concentrations stabilize. When calculating the equilibrium constant, for instance, we use the equilibrium concentrations to determine the extent to which a reaction has proceeded. It's essential to get precise measurements of these concentrations to correctly compute the equilibrium constant.

Le Chatelier's Principle

Le Chatelier's principle provides a qualitative means of predicting the shift of a chemical equilibrium in response to changes in conditions such as concentration, pressure, or temperature. It essentially states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will shift to counteract the change. In simpler terms, the system adjusts to regain balance. For example, adding more reactants to a reaction mixture in equilibrium will shift the equilibrium to produce more products.

Applications to Le Chatelier's Principle

• If the concentration of a reactant is increased, the reaction will shift towards producing more products to lower the reactant's concentration back to equilibrium.
• Conversely, removing a reactant will cause the reaction to shift towards the reactants to increase their concentration again.
• Changes in temperature and pressure can also lead to shifts in equilibrium according to Le Chatelier's principle.

Students can think of this principle as a system's effort to maintain balance when faced with external changes, much like how a person might shift their weight to remain stable on a wobbly platform.

Equilibrium Constant Expression

The equilibrium constant expression, denoted as K or Kc when involving concentrations, mathematically represents the ratio of the concentrations of products to reactants at equilibrium, raised to the power of their respective coefficients in the balanced chemical equation. For the reaction in the exercise, the equilibrium constant expression was used to quantify how far the reaction proceeds towards the products at a given temperature. This expression is vital because it tells us the relative amounts of reactants and products at equilibrium and can help predict the direction of the reaction if the system is disturbed. Keep in mind that the equilibrium constant is just that—a constant for a given reaction at a specified temperature. Change the temperature, and the value of the equilibrium constant will also change, reflecting a new equilibrium position.