To understand the relationship between the rate constants \( k \), \( k_1 \), and \( k'_1 \) in the reaction you provided, we need to delve into the principles of chemical kinetics and how rate expressions are derived from the stoichiometry of a reaction. The chemical equation for your reaction is:
2NO(g) + 2H2(g) ⇌ N2(g) + 2H2O(g)
Rate Expressions Overview
In chemical kinetics, the rate of a reaction can be expressed in terms of the change in concentration of reactants and products over time. The rate expressions you provided represent the same reaction but from different perspectives, based on the consumption of reactants or the formation of products. Let's analyze each one.
Understanding the Rate Expressions
- Rate of Formation of N2: {d[N2]/dt} = k1[NO][H2]
- Rate of Formation of H2O: {d[H2O]/dt} = k[NO][H2]
- Rate of Consumption of NO: {-d[NO]/dt} = k'1[NO][H2]
- Rate of Consumption of H2: {-d[H2]/dt} = k1[NO][H2]
Linking the Rate Constants
When analyzing these rate expressions, we notice that they are all based on the concentration of the reactants, NO and H2. The constants \( k \), \( k_1 \), and \( k'_1 \) are linked through the stoichiometry of the reaction.
Stoichiometry and Rate Constants
Since the reaction involves the consumption of 2 moles of NO and 2 moles of H2 to produce 1 mole of N2 and 2 moles of H2O, we can derive relationships between these constants based on the stoichiometric coefficients:
- The rate of formation of products (N2 and H2O) is related to the rate of consumption of reactants (NO and H2).
- Due to the stoichiometric coefficients, we can infer that the rate of formation of a product is equal to half the rate of consumption of a reactant. Therefore:
From the consumption of NO: Since 2 moles of NO are consumed for every mole of N2 produced:
{-d[NO]/dt} = 2 * {d[N2]/dt} ⇒ k'1 = 2k1
From the formation of H2O: Similarly, for H2O, which is produced in a 2:2 ratio with respect to H2:
{d[H2O]/dt} = 2 * {-d[H2]/dt} ⇒ k = k1
Conclusion on Rate Constants
In summary, we can express the relationships among the rate constants as follows:
This means that the rate constant for the formation of N2 and H2O can be directly related back to the rate constants associated with the consumption of reactants, reflecting the stoichiometric ratios of the reaction. Understanding these relationships is crucial for analyzing kinetics in chemical reactions.
