Problem 21 - Chemistry: Principles and Reactions

Question

When carbon monoxide reacts with hydrogen gas, methane and steam are formed. $$\mathrm{CO}(\mathrm{g})+3 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{CH}_{4}(g)+\mathrm{H}_{2} \mathrm{O}(g)$$ At \(1127^{\circ} \mathrm{C}\), analysis at equilibrium shows that \(P_{\mathrm{CO}}=0.921 \mathrm{~atm}\), \(P_{\mathrm{H}_{2}}=1.21 \mathrm{~atm}, P_{\mathrm{CH}_{4}}=0.0391 \mathrm{~atm}\), and \(P_{\mathrm{H}_{2} \mathrm{O}}=0.0124 \mathrm{~atm} .\) What is the equilibrium constant, \(K\), for the reaction at \(1127^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

Question: Calculate the equilibrium constant (K) for the given reaction at 1127°C, given the partial pressures at equilibrium as P_CO = 0.921 atm, P_H2 = 1.21 atm, P_CH4 = 0.0391 atm, and P_H2O = 0.0124 atm: CO(g) + 3H2(g) ⇌ CH4(g) + H2O(g)

Step 1: Write down the equilibrium constant expression for the given reaction

The equilibrium constant (K) for the reaction is given by the expression, K = (P_{CH4} * P_{H2O}) / (P_{CO} * (P_{H2})^3) This expression is derived from the balanced chemical equation given, using the Law of Mass Action.

Step 2: Substitute the given values

We are given that at equilibrium: P_{CO}=0.921 atm, P_{H2}=1.21 atm, P_{CH4}=0.0391 atm, and P_{H2O}=0.0124 atm. Substitute these values into the equilibrium constant expression. K = (0.0391 * 0.0124) / (0.921 * (1.21)^3)

Step 3: Calculate the equilibrium constant, K

Now it's a matter of performing the calculation. Don't forget to raise the partial pressure of hydrogen gas to the power of 3 as per its stoichiometric coefficient in the balanced chemical equation K = 0.00048504 / (0.921 * 1.770651) And when you calculate it further, K = 0.000247 This is the equilibrium constant for the reaction at 1127°C.

Key Concepts

Chemical Equilibrium

Chemical equilibrium refers to the state in a chemical reaction when the concentrations of reactants and products remain constant over time. This happens because the forward and reverse reactions occur at the same rate. Consider it like a seesaw that's perfectly balanced, with neither side going up or down.
Once a reaction reaches equilibrium, no net change is observed in the concentrations of substances involved, even though at the molecular level, both reactions are still proceeding.
In our example of carbon monoxide reacting with hydrogen to form methane and water, equilibrium is reached when the rate at which CO and H₂ react equals the rate at which CH₄ and H₂O break down back into CO and H₂.

• The equilibrium constant, K, is a numerical value that can provide insights into the concentrations of products and reactants at equilibrium.
• A large value of K indicates that, at equilibrium, products are favored, whereas a small value of K suggests that reactants dominate.

Reaction Kinetics

Reaction kinetics deals with the rate of chemical reactions and what factors influence these rates. It's like understanding how fast or slow a reaction goes from start to finish.
Various factors affect the rate of reactions, such as temperature, pressure, concentration, and the presence of catalysts.
The reaction rate provides insight into how quickly reactants transform into products, which is critical for processes in both industrial and biological systems.
For the reaction of CO and H₂ to CH₄ and H₂O, gaining insight into reaction kinetics helps us understand at what conditions the reaction happens fastest and most efficiently.
The temperature at which this specific reaction, at 1127°C, influences how rapidly equilibrium is achieved. Reaction kinetics provides tools to manipulate such reactions to one's advantage in industrial applications.

Law of Mass Action

The Law of Mass Action is a fundamental principle used to derive the equilibrium constant expression from a balanced chemical equation. It states that the rate of a chemical reaction is proportional to the product of the concentrations of the reactants, each raised to a power equal to its coefficient in the balanced equation.
This law forms the basis for calculating the equilibrium constant, K. By applying it, we can predict how the concentration of substances will shift as the reaction moves toward equilibrium.
In our case of CO reacting with H₂, the law tells us to set up the expression for K as \(K = \frac{P_{CH₄} \times P_{H₂O}}{P_{CO} \times (P_{H₂})^3}\).

• This equation uses partial pressures because the reaction occurs in a gaseous state, and each substance's pressure is proportional to its concentration.
• The exponents in the denominator indicate the reaction order for hydrogen, crucial for predicting how changes in pressure will affect equilibrium.

Understanding the Law of Mass Action empowers us to analyze how equilibrium will respond to changes in pressure, temperature, or concentration, thus making it an essential tool in chemical reactions.