91Ó°ÊÓ

In chemical reaction kinetics, determining the order of a reaction is crucial for understanding how the rate of reaction relates to the concentration of reactants. A second-order reaction specifically means that the rate is proportional to the square of the concentration of one reactant. For the decomposition of NOâ‚‚, the units of the rate constant were given as \[4.7 \, / \,\mathrm{M \cdot s}\]This unit is characteristic of a second-order reaction because it matches the general format: \[1 / (M \cdot s)\]In practice, this implies that as the concentration of NOâ‚‚ doubles, the rate of the reaction increases by a factor of four. Understanding this proportional relationship helps us predict the behavior of the reaction under different conditions.

The rate law is a mathematical expression that describes the rate of a chemical reaction in terms of the concentration of reactants. For a second-order reaction involving NOâ‚‚, the rate law can be written as\[\text{rate} = k[\text{NO}_2]^2\]where \( k \) is the rate constant, and \([\text{NO}_2]\) is the concentration of the NOâ‚‚ reactant. For the given problem, this translates into\[\text{rate} = 4.7 \times (0.0182)^2\]This calculation gives the initial rate of reaction. Such equations let us determine how quickly a product forms, or a reactant is consumed, based on known concentrations. The rate law thus provides a direct way to relate concentration changes to the reaction rate, helping us to predict the amounts of substance over time.

The ideal gas law is pivotal for determining concentrations of gaseous reactants, as it relates several key properties: pressure, volume, temperature, and number of moles. The ideal gas law can be expressed as\[PV = nRT\]In this equation:- \( P \) is the pressure of the gas,- \( V \) is the volume,- \( n \) represents the number of moles,- \( R \) is the ideal gas constant (0.0821 L·atm/(K·mol)), and- \( T \) is the temperature in Kelvin.For NO₂ in this exercise, converting 746 mmHg to atm and using the temperature of 656 K allowed us to find the initial concentration. This calculation is essential because knowing the concentration directly impacts the rate law and helps predict reaction progress.

The rate constant \( k \) is a fundamental part of the rate law equation and is unique to each chemical reaction at a given temperature. It embodies the speed of the reaction: higher \( k \) values indicate faster reactions. In our exercise, the rate constant \( k \) for the decomposition of NOâ‚‚ was provided as\[4.7 \, / \, \mathrm{M \cdot s}\]It's essential to note that the rate constant depends on factors such as:

• The temperature of the reaction environment.
• The presence of a catalyst.

As seen in this problem, knowing \( k \) allows calculating both the rate of reaction and predicting concentrations of products like Oâ‚‚ over time. It provides a bridge between concentration changes and the observable reaction rate, emphasizing its central role in reaction kinetics.