Question

If the equilibrium constant for the reaction \(\mathrm{CO}+\) \(\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\) is \(K,\) the equilibrium constant for the reaction \(\mathrm{CO}_{2}+3 \mathrm{N}_{2} \rightleftharpoons \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}+3 \mathrm{N}_{2}\) at the same temperature is \((a) 1 / K\) \((b) 1 /(K+3)\) \((c) 4 K\) \((d) K\) \((e) 1 / K^{2}\)

Short answer

Answer: (a) \(\frac{1}{K}\)

Step-by-step solution

Write the given reaction and the equilibrium constant

Reaction 1: \(\mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\), Equilibrium constant: \(K\)

Write the new reaction for which we want to find the equilibrium constant

Reaction 2: \(\mathrm{CO}_{2} + 3 \mathrm{N}_{2} \rightleftharpoons \mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} + 3 \mathrm{N}_{2}\)

Identify the relationship between Reaction 1 and Reaction 2

Reaction 2 is essentially the reverse of Reaction 1 (with the addition of 3 moles of \(\mathrm{N}_{2}\) on both sides, which does not affect the equilibrium constant as they are inert). So, Reaction 2 can be seen as the inverse of Reaction 1.

Calculate the new equilibrium constant for Reaction 2

For the inverse of a reaction, the new equilibrium constant is the reciprocal of the original equilibrium constant. Therefore, the equilibrium constant for Reaction 2 (\(K_{new}\)) is \(K_{new} = \frac{1}{K}\)

Match the result to the given options

The correct answer is \((a) \frac{1}{K}\)

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a reversible chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction and the concentrations of reactants and products remain constant over time. This concept is pivotal to understanding how reactions proceed and how they can be shifted or maintained. When we describe a reaction such as \(\mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\), the symbol \(\rightleftharpoons\) denotes that the reaction is reversible, suggesting that chemical equilibrium may be achieved under certain conditions.

At equilibrium, the ratio of the concentration of the products raised to their stoichiometric coefficients to the concentration of the reactants raised to their stoichiometric coefficients is a constant. This constant is known as the equilibrium constant (\(K\)). It provides insights into the position of equilibrium—whether the reactants or products are favored under given conditions. For instance, if \(K\) is much greater than 1, the formation of products is favored, and the equilibrium lies to the right. Conversely, if \(K\) is much less than 1, the reactants are favored and the equilibrium lies to the left.

Reaction Inversion

Reaction inversion refers to the process of reversing a chemical equation and its corresponding impact on the equilibrium constant. When we invert a chemical equation, the products of the original reaction become the reactants, and vice versa. An essential consequence of reaction inversion is that the equilibrium constant for the inverted reaction is the reciprocal of the equilibrium constant of the original reaction.

For example, if the equilibrium constant for \(\mathrm{CO} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}\) is \(K\), when this reaction is reversed to form \(\mathrm{CO}_{2} \rightleftharpoons \mathrm{CO} + \frac{1}{2} \mathrm{O}_{2}\), the new equilibrium constant (\(K_{new}\)) will be \(1/K\). Understanding reaction inversion is crucial for predicting how a change in conditions will affect the direction of a reaction and for calculating the equilibrium constant for the reverse reaction.

Thermodynamics

Thermodynamics is the branch of physical chemistry that deals with the relationships between heat and other forms of energy. It plays a vital role in predicting the spontaneity of chemical reactions and understanding the equilibrium state. Central to thermodynamics is the concept of Gibbs free energy (\(G\)), which determines whether a reaction can occur spontaneously. A negative \(\Delta G\) indicates a spontaneous forward reaction, while a positive \(\Delta G\) indicates a non-spontaneous forward reaction.

At equilibrium, \(\Delta G = 0\), and the system is at its lowest free energy under the given conditions. The equilibrium constant is directly related to the change in Gibbs free energy for the reaction by the equation \(\Delta G = -RT \ln(K)\), where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin. As reaction conditions such as temperature change, the value of \(K\) changes, reflecting a new equilibrium position, as predicted by the principles of thermodynamics.