Question

At a temperature of 700 \(\mathrm{K}\) , the forward and reverse rate constants for the reaction \(2 \mathrm{HI}(g) \rightleftharpoons \mathrm{H}_{2}(g)+\mathrm{I}_{2}(g)\) are \(k_{f}=1.8 \times 10^{-3} \mathrm{M}^{-1} \mathrm{s}^{-1}\) and \(k_{r}=0.063 \mathrm{M}^{-1} \mathrm{s}^{-1}\) (a) What is the value of the equilibrium constant \(K_{c}\) at 700 \(\mathrm{K} ?\) (b) Is the forward reaction endothermic or exothermic if the rate constants for the same reaction have values of \(k_{f}=0.097 M^{-1} \mathrm{s}^{-1}\) and \(k_{r}=2.6 \mathrm{M}^{-1} \mathrm{s}^{-1}\) at 800 \(\mathrm{K} ?\)

Short answer

a) The equilibrium constant, Kc at 700K is 0.029. b) The forward reaction is endothermic.

Step-by-step solution

Part (a): Finding the equilibrium constant Kc

To find the value of the equilibrium constant, Kc, at 700K, we use the relationship between Kc and the forward and reverse rate constants given by: \[ K_c = \frac{k_f}{k_r} \] Given, kf = 1.8 x 10^-3 M^-1 s^-1 and kr = 0.063 M^-1 s^-1. Substitute these values into the formula: \[ K_c = \frac{1.8 \times 10^{-3}}{0.063} \]

Part (a): Calculating Kc

Now, divide 1.8 x 10^-3 by 0.063 to find the value of the equilibrium constant Kc: \[ K_c = 0.029 \] Thus, the equilibrium constant Kc at 700K is 0.029.

Part (b): Determining if the forward reaction is endothermic or exothermic

To determine if the forward reaction is endothermic or exothermic, we need to examine the rate constants at two different temperatures. We know that: 1. If the ratio of kf to kr increases as the temperature increases, the forward reaction is endothermic, i.e., it absorbs heat. 2. If the ratio of kf to kr decreases as the temperature increases, the forward reaction is exothermic, i.e., it releases heat. The given rate constants at 800K are kf = 0.097 M^-1 s^-1 and kr = 2.6 M^-1 s^-1. Calculate the ratio of kf to kr at 800K: \[ \frac{k_{f}(800K)}{k_{r}(800K)} = \frac{0.097}{2.6} \]

Part (b): Calculating the ratio of rate constants at 800K

Divide 0.097 by 2.6 to find the ratio of kf to kr at 800K: \[ \frac{k_{f}(800K)}{k_{r}(800K)} = 0.037 \] Now, compare this ratio to the Kc value we found at 700K (Kc = 0.029). Since the ratio of kf to kr at 800K is greater than the Kc value at 700K, the forward reaction is endothermic - it absorbs heat. In conclusion: a) The equilibrium constant Kc at 700K is 0.029. b) The forward reaction is endothermic.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fascinating concept in chemistry, where the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products.
This equilibrium can be represented by the equilibrium constant, denoted as \( K_c \).
In the context of the given reaction, \( 2 \mathrm{HI}(g) \rightleftharpoons \mathrm{H}_2(g) + \mathrm{I}_2(g) \), the equilibrium constant can be calculated using the formula:

• \( K_c = \frac{k_f}{k_r} \)

where \( k_f \) and \( k_r \) are the forward and reverse rate constants, respectively.

At 700 K, the forward rate constant is \( 1.8 \times 10^{-3} \ \mathrm{M}^{-1} \ \mathrm{s}^{-1} \) and the reverse rate constant is \( 0.063 \ \mathrm{M}^{-1} \ \mathrm{s}^{-1} \). By substituting these values into the formula, the equilibrium constant \( K_c \) is calculated as \( 0.029 \).
This value tells chemists the ratio of the concentration of products to reactants at equilibrium and helps predict how a system will respond to changes in concentrations, temperature, or pressure via Le Chatelier's Principle.
Understanding \( K_c \) is crucial for understanding how chemical reactions can be manipulated in processes like synthesis and extraction.

Endothermic and Exothermic Reactions

Reactions are often classified as either endothermic or exothermic, based on how they interact with heat.
Endothermic reactions absorb heat, while exothermic reactions release heat.

In the exercise, the determination of whether the forward reaction is endothermic or exothermic was carried out by comparing the rate constants at different temperatures.
The relationship between the rate constants at two different temperatures provides insight into the heat exchange of the reaction:

• If the ratio \( \frac{k_f}{k_r} \) increases with increasing temperature, the reaction is endothermic because the forward reaction speeds up more than the reverse, indicating heat is absorbed to drive the reaction forward.
• If the ratio \( \frac{k_f}{k_r} \) decreases, the reaction is exothermic as it does not require additional heat input to proceed more efficiently.

At 800 K, the calculated ratio \( \frac{k_f}{k_r} \) was \( 0.037 \), which was higher than the \( K_c \) value at 700 K, indicating an endothermic reaction.
This understanding helps in designing chemical processes where temperature control is crucial, such as in industrial synthesis or thermal regulation in catalysis.

Reaction Kinetics

Reaction kinetics focuses on understanding the speed of chemical reactions and the factors that influence this speed.
It is central to controlling chemical processes efficiently.
Kinetic analysis is used to determine how different variables such as concentration, temperature, and catalysts affect reaction rates.

The rate constants \( k_f \) and \( k_r \) signify the speed of the forward and reverse reactions, respectively.
These constants are influenced by temperature changes, as described by the Arrhenius equation:

• \( k = A \exp{\left(\frac{-E_a}{RT}\right)}\)

where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

In laboratory and industrial settings, understanding reaction kinetics allows chemists to optimize reactions by choosing suitable conditions that maximize product yield and minimize time and energy consumption.
For instance, by knowing that the forward reaction is endothermic, scientists can deliberate on how to supply appropriate heat to sustain and enhance the reaction's progress without causing unwanted side reactions or thermal degradation of sensitive components.