Problem 33 - Chemistry The Molecular Nature of Matter

Question

Write the equilibrium law for each of the following reactions in terms of molar concentrations: (a) \(\mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g)\) (b) \(\frac{1}{2} \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{HCl}(g)\) How does \(K_{\mathrm{c}}\) for reaction (a) compare with \(K_{\mathrm{c}}\) for reaction (b)?

Step-by-Step Solution

Answer

For reaction (a), \(K_{\mathrm{c}} = \frac{[\mathrm{HCl}]^2}{[\mathrm{H}_{2}][\mathrm{Cl}_{2}]}\). For reaction (b), \(K_{\mathrm{c}} = \frac{[\mathrm{HCl}]}{[\mathrm{H}_{2}]^{\frac{1}{2}} [\mathrm{Cl}_{2}]^{\frac{1}{2}}}\). \(K_{\mathrm{c}}\) for reaction (b) is the square root of \(K_{\mathrm{c}}\) for reaction (a).

Step 1: Write the Equilibrium Constant Expression for Reaction (a)

The equilibrium expression is based on the balanced chemical equation. For the reaction \(\mathrm{H}_{2}(g) + \mathrm{Cl}_{2}(g) \rightleftharpoons 2\mathrm{HCl}(g)\), the equilibrium constant \(K_{\mathrm{c}}\) is given by the following formula: \[K_{\mathrm{c}} = \frac{[\mathrm{HCl}]^2}{[\mathrm{H}_{2}][\mathrm{Cl}_{2}]}\] where the concentrations are expressed in molarity (M).

Step 2: Write the Equilibrium Constant Expression for Reaction (b)

Reaction (b) is similar to reaction (a) but has fractional coefficients: \(\frac{1}{2} \mathrm{H}_{2}(g) + \frac{1}{2} \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{HCl}(g)\). The equilibrium constant \(K_{\mathrm{c}}\) is written as: \[K_{\mathrm{c}} = \frac{[\mathrm{HCl}]}{[\mathrm{H}_{2}]^{\frac{1}{2}} [\mathrm{Cl}_{2}]^{\frac{1}{2}}}\]

Step 3: Compare the Equilibrium Constants \(K_{\mathrm{c}}\)

To compare the two equilibrium constants, notice that reaction (b) is essentially one half of reaction (a). Thus, if we take the square root of the equilibrium expression for reaction (a), we can derive the expression for reaction (b). So, \(K_{\mathrm{c}}\) for reaction (b) will be the square root of \(K_{\mathrm{c}}\) for reaction (a). Hence, the relationship between the two constants is \[K_{\mathrm{c}, \text{reaction (b)}} = \sqrt{K_{\mathrm{c}, \text{reaction (a)}}}\].

Key Concepts

Equilibrium Law

Understanding the equilibrium law is crucial for anyone studying chemistry, especially when dissecting complex chemical reactions. The equilibrium law, also known as the law of mass action, provides a way to quantify the balance point in a reversible reaction. At this point, the rate of the forward reaction is equal to the rate of the reverse reaction, meaning that the concentrations of reactants and products remain constant over time.

At equilibrium, the rate of production of reactants is equal to the rate of production of products. For a general reaction, where reactants A and B convert to products C and D, the equilibrium law is represented mathematically as: \[K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}\]where the letters \(a, b, c,\) and \(d\) correspond to the stoichiometric coefficients in the balanced equation, and \([A], [B], [C],\) and \([D]\) are the molar concentrations of the reactants and products, respectively. The symbol \(K_c\) represents the equilibrium constant and indicates the extent of the reaction at equilibrium; a high value suggests a reaction favoring product formation, whereas a low value suggests a reaction favoring the reactants.

It's essential to note that the equilibrium constant is only influenced by changes in temperature; concentrations and pressures don't affect its value but instead shift the position of equilibrium to maintain the constant.

Molar Concentrations

When dealing with chemical reactions, understanding molar concentrations is key to grasping how reactants transform into products. Molar concentration, often simply called concentration, is a measure of the amount of a substance within a defined volume of solution. The standard unit for molar concentration is mole per liter (M or mol/L). In the context of chemical reactions and equilibrium, molar concentrations become central to the calculation of the equilibrium constant \(K_c\).

For example, in the reaction \(H_{2}(g) + Cl_{2}(g) \rightleftharpoons 2HCl(g)\), the molar concentrations of hydrogen, chlorine, and hydrochloric acid gases at equilibrium are pivotal in calculating the equilibrium constant using the equilibrium law. The part of the law \([HCl]^2\) indicates that the concentration of hydrochloric acid gas is squared due to its coefficient of 2 in the balanced reaction equation. It signifies that changes in the concentration of hydrochloric acid will have a more significant effect on the value of the equilibrium constant compared to equal changes in the concentrations of hydrogen or chlorine.

Reaction Kinetics

Reaction kinetics is the branch of chemistry that studies the rates of chemical processes and the factors affecting these rates. Kinetics delves into how quickly reactants turn into products and which factors like temperature, concentration, and catalysts can modify this speed. Crucially, reaction kinetics is different from equilibrium in that kinetics is concerned with the speed of a reaction, not the position of balance that the reaction reaches.

For instance, a fast reaction kinetics means that reactants are converted into products at a high rate. However, it is not necessarily an indication of whether the final mixture will contain more products or reactants at equilibrium; this is what the equilibrium law determines. Both kinetics and equilibrium provide a comprehensive understanding of chemical processes. Kinetics describes the journey from reactants to products, while the equilibrium law describes the final ratio of reactants to products when the system reaches stability.