Question

At a given temperature, why is the ratio \(k_{\mathrm{f}} / k_{\mathrm{r}}\) constant for a given reaction?

Short answer

The ratio \(\frac{k_f}{k_r}\) is constant for a given reaction at a given temperature because it is equal to the equilibrium constant, \(K_{eq}\), which is determined by the Gibbs free energy change, \(\Delta G^\circ\), and the temperature. Since \(\Delta G^\circ\) is constant for a given reaction and temperature, the equilibrium constant will also be constant, resulting in a constant ratio of forward and reverse rate constants: \(\frac{k_f}{k_r} = K_{eq} = constant\).

Step-by-step solution

Understand the Arrhenius Equation

The Arrhenius equation describes the temperature dependence of reaction rate constants. For a given reaction, the equation is: \(k = Ae^{\frac{-Ea}{RT}}\) where \(k\) is the rate constant, \(A\) is the pre-exponential factor (also called the frequency factor), \(Ea\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature. The forward and reverse reactions have their own rate constants, activation energies, and pre-exponential factors.

Relate Rate Constants to the Equilibrium Constant

At equilibrium, the rate of the forward reaction is equal to the rate of the reverse reaction. The equilibrium constant, \(K_{eq}\), is defined as the ratio of the forward and reverse rate constants at a given temperature: \(K_{eq} = \frac{k_f}{k_r}\)

Apply the Van 't Hoff Equation

The Van 't Hoff equation relates the equilibrium constant to the Gibbs free energy change, \(\Delta G^\circ\), for the reaction: \(\Delta G^\circ = -RT \ln K_{eq}\) From this equation, we can see that the equilibrium constant is determined by the Gibbs free energy change and the temperature.

Combine the Concepts

Since the Gibbs free energy change and temperature determine the equilibrium constant, the ratio of the forward and reverse rate constants, \(\frac{k_f}{k_r}\), is solely dependent on these factors as well: \(\frac{k_f}{k_r} = K_{eq} \) Since we are considering a given reaction at a given temperature, the Gibbs free energy change is constant, and accordingly, the equilibrium constant will also be constant: \(\frac{k_{f}}{k_{r}} = constant\) Thus, the ratio of the forward and reverse rate constants, \(\frac{k_f}{k_r}\), is constant for a given reaction at a given temperature.

Key concepts

Equilibrium Constant

Understanding the equilibrium constant is vital in the study of chemical reactions. It is a dimensionless number that represents the ratio of the concentration of the products to the reactants at equilibrium, each raised to the power of their respective stoichiometric coefficients in the balanced equation. The formula for the equilibrium constant, usually denoted as \( K_{eq} \), for a generic reaction \( aA + bB \rightleftharpoons cC + dD \) is expressed as:\
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\[ K_{eq} = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
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The equilibrium constant is a reflection of the extent to which a reaction occurs. A larger \( K_{eq} \) indicates that, at equilibrium, the concentration of products is higher than that of reactants, suggesting the reaction favors the formation of products. Conversely, a smaller \( K_{eq} \) suggests the reaction favors reactants. Temperature is a crucial factor affecting the equilibrium constant; a change in temperature will result in a new equilibrium constant. However, within the context of a given temperature, the equilibrium constant remains constant.

Van 't Hoff Equation

The Van 't Hoff equation is an important tool in thermodynamics that links the equilibrium constant \( K_{eq} \) to the temperature (T) and the standard Gibbs free energy change \( (\Delta G^\circ) \). The equation is:\
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\[ \Delta G^\circ = -RT \ln K_{eq} \]
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Where \( R \) is the ideal gas constant and \( T \) is the temperature in Kelvin. This relationship indicates that if the Gibbs free energy change is known, the equilibrium constant can be calculated, and vice versa. According to the Van 't Hoff equation, a negative \( \Delta G^\circ \) corresponds to a favorable reaction that will proceed spontaneously under standard conditions, and this results in a \( K_{eq} \) greater than one. Understanding how to apply the Van 't Hoff equation is essential when analyzing reaction spontaneity and equilibrium position.

Gibbs Free Energy Change

Gibbs free energy change, symbolized as \( \Delta G \), is a key concept in thermodynamics, representing the maximum reversible work that can be performed by a system at constant temperature and pressure. It combines enthalpy (\( \Delta H \)), entropy (\( \Delta S \)), and temperature (T) in the equation:\
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\[ \Delta G = \Delta H - T\Delta S \]
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A negative \( \Delta G \) indicates a spontaneous process, while a positive value suggests non-spontaneity. In the context of reaction rate constants, the relationship between \( \Delta G \) and the equilibrium constant \( K_{eq} \) ensures that achievable reactions have a consistent ratio of forward to reverse rate constants at a given temperature, since the \( \Delta G \) value for the reaction defines the equilibrium state.

Reaction Rate Constants

The rate constants of a reaction, symbolized as \( k_f \) for the forward reaction and \( k_r \) for the reverse reaction, embody the speeds at which these respective processes occur. These constants are derived from the Arrhenius equation:\
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\[ k = Ae^{-\frac{Ea}{RT}} \]
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The temperature-dependent behavior captured by this equation is important for understanding why reaction speeds change with temperature. A key insight is that the ratio of the rate constants \( \frac{k_f}{k_r} \) is actually the equilibrium constant \( K_{eq} \) for the reaction at a given temperature, hence maintaining a constant relationship between the speed of the forward and reverse reactions. By comprehending how reaction rate constants manifest and relate to equilibrium, students can better grasp the dynamic yet stable nature of chemical systems at equilibrium.