For a gas reaction at \(400 \mathrm{K}\) the rate is reported as $$-\frac{d p_{\mathrm{A}}}{d t}=3.66 p_{\mathrm{A}}^{2}, \quad \mathrm{atm} / \mathrm{hr}$$ (a) What are the units of the rate constant? (b) What is the value of the rate constant for this reaction if the rate equation is expressed as $$-r_{\mathrm{A}}=-\frac{1}{V} \frac{d N_{\mathrm{A}}}{d t}=k C_{\mathrm{A}}^{2}, \quad \operatorname{mol} / \mathrm{m}^{3} \cdot \mathrm{s}$$

The reaction rate constant, often symbolized as 'k', is a parameter that quantifies the speed of a chemical reaction. For a given reaction at a specified temperature, its value remains constant and is independent of the concentrations of the reactants. However, it does vary with temperature, and understanding its units is crucial for solving reaction rate problems.

Using the original exercise as an example, the rate constant can be derived from the given equation \( -\frac{d p_{\mathrm{A}}}{d t} = k p_{\mathrm{A}}^2 \). Here, we see that the rate at which the pressure of reactant 'A' decreases is proportional to the square of its partial pressure. This indicates a second-order reaction in respect to 'A'. To find the units of 'k', we isolate it and apply the units from the rate expression, resulting in the units \( \frac{1}{\mathrm{atm} \cdot \mathrm{hr}} \).

Considering this within the context of chemical reaction engineering ensures accurate progression from laboratory reactions to industrial-scale processes. A deep understanding of the reaction order and corresponding rate constant's units helps engineers to scale reactions appropriately and ensures the efficiency of processes.

The rate law of a chemical reaction is an equation that links the reaction rate with the concentrations or pressures of the reactants and the specific rate constant. It reflects the relationship between the reactants' concentration and the speed of the reaction, showing how the reaction rate changes as the concentration of reactants changes over time.

In the step-by-step solution, the rate law is expressed as \( -r_{\mathrm{A}} = k C_{\mathrm{A}}^2 \), signifying a second-order reaction with respect to the concentration of reactant 'A' (\(C_{\mathrm{A}}\)). It is crucial to know the reaction order because it directly affects how the concentration affects the rate: for a second-order reaction, the rate is proportional to the square of the concentration. Understanding the rate law is a fundamental aspect of chemical kinetics, as it helps predict the rate at which a chemical reaction proceeds under different concentrations, a key factor in the design and control of chemical reactors.

The ideal gas law is a fundamental equation in chemistry that describes the relationship between the pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and amount (in moles, \(n\)) of an ideal gas. Expressed as \( PV = nRT \), where 'R' is the universal gas constant, this law is crucial for converting back and forth between the units of pressure and concentration. This conversion process is highlighted in steps 4 and 6 of the solution.

By rearranging the ideal gas law to \( C = \frac{P}{RT} \), where 'C' is the concentration, it becomes possible to convert pressures (measured in atmospheres) to concentrations (moles per cubic meter), necessary for employing the rate law in terms of concentration rather than pressure. This conversion is especially useful in chemical reaction engineering applications where dealing with gas-phase reactions at various scales is common. Knowing how to apply the ideal gas law is essential for predicting the behavior of gases under different conditions, which is pivotal in both laboratory settings and industrial chemical processes.