Question

Methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is produced commercially by the catalyzed reaction of carbon monoxide and hydrogen: \(\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}(g)\). An equilibrium mix- ture in a 2.00-L vessel is found to contain \(0.0406 \mathrm{~mol}\) \(\mathrm{CH}_{3} \mathrm{OH}, 0.170 \mathrm{~mol} \mathrm{CO}\), and \(0.302 \mathrm{~mol} \mathrm{H}_{2}\) at \(500 \mathrm{~K}\). Cal- culate \(K_{c}\) at this temperature.

Short answer

The equilibrium constant (\(K_c\)) at 500 K for the reaction between carbon monoxide and hydrogen to produce methanol is 1.25.

Step-by-step solution

Write down the reaction and the expression for \(K_c\)

The reaction is given by the following equation: \[CO(g) + 2 H_2(g) \rightleftharpoons CH_3OH(g)\] The expression for the equilibrium constant \(K_c\) is defined as follows: \[K_c = \frac{[CH_3OH]}{[CO] \times [H_2]^2}\] Where [CH3OH], [CO], and [H2] are the equilibrium concentrations of methanol, carbon monoxide, and hydrogen, respectively.

Convert moles to concentrations

We are given the moles of each substance present in a 2.00 L vessel at equilibrium. To find the concentrations, we can use the formula: \[[Substance] = \frac{mol \ of \ substance}{volume}\] For each substance, \[[CH_3OH] = \frac{0.0406 mol}{2.00 L} = 0.0203 M\] \[[CO] = \frac{0.170 mol}{2.00 L} = 0.0850 M\] \[[H_2] = \frac{0.302 mol}{2.00 L} = 0.151 M\]

Substitute the equilibrium concentrations into the \(K_c\) expression

Now, we can substitute the equilibrium concentrations we found in step 2 into the expression for \(K_c\): \[K_c = \frac{0.0203}{(0.0850) \times (0.151)^2}\]

Solve for \(K_c\)

Finally, we can calculate the value of \(K_c\): \[K_c = \frac{0.0203}{(0.0850) \times (0.151)^2} = 1.25\] So, the equilibrium constant \(K_c\) at 500 K for this reaction is 1.25.

Key concepts

Chemical Equilibrium

In a chemical reaction, equilibrium is achieved when the rate of the forward reaction equals the rate of the reverse reaction. This means the concentrations of the reactants and products remain constant over time. It's a dynamic balance, not a static one. At equilibrium, reactions are still occurring, but there's no net change in concentration of reactants and products.
In our example involving methanol production, once equilibrium is reached, the concentrations of carbon monoxide, hydrogen, and methanol remain consistent.
Understanding chemical equilibrium is crucial because it helps in predicting the concentrations of reactants and products under specific conditions. It also plays a pivotal role in optimizing reactions in industrial processes for maximum yield.

Reaction Quotient

The reaction quotient, Q, serves as a tool for determining the direction in which a reaction mixture will proceed to achieve equilibrium. It is calculated using the same expression as the equilibrium constant, but with the initial concentrations instead of the equilibrium concentrations.
Here's how to think about it:

• If Q < K, the reaction will proceed forward to form more products.
• If Q > K, the reaction will go in reverse to form more reactants.
• If Q = K, the reaction is at equilibrium.

For the methanol synthesis reaction, if we start with a different set of concentrations, comparing Q and K will tell us which direction the reaction will naturally shift to reach equilibrium.

Le Chatelier's Principle

Le Chatelier's Principle provides insight into how a system at equilibrium responds to stress. Stress can be in the form of changes in concentration, pressure, or temperature. According to this principle, if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change and re-establish equilibrium.
For example, in our methanol production reaction:

• If we increase the pressure, the system will shift towards the side with fewer moles of gas, which is the product side, thereby increasing the production of methanol.
• Increasing the concentration of hydrogen or carbon monoxide will also shift the equilibrium to produce more methanol, in order to counterbalance the added reactants.
• Conversely, removing methanol will have the same effect — the system will shift to produce more methanol to restore balance.

Using Le Chatelier's Principle helps in strategically maximizing desired product formation in chemical reactions.