For the reaction $$\mathrm{A}+\mathrm{B} \longrightarrow \mathrm{C}$$ the rate expression is rate \(=k[\mathrm{~A}][\mathrm{B}]\) (a) Given three test tubes, with different concentrations of \(\mathrm{A}\) and \(\mathrm{B}\), which test tube has the smallest rate? (1) \(0.10 M \mathrm{~A}_{\mathrm{i}} 0.10 \mathrm{M} \mathrm{B}\) (2) \(0.15 \mathrm{MA}_{;} 0.15 \mathrm{M} \mathrm{B}\) (3) \(0.06 M \mathrm{~A} ; 1.0 M \mathrm{~B}\) (b) If the temperature is increased, describe (using the words increases, decreases, or remains the same) what happens to the rate, the value of \(k\), and \(E_{x}\)

Understanding the rate expression is crucial when analyzing how fast a chemical reaction proceeds. The rate expression for a reaction, also known as the rate law, represents the relationship between the reaction rate and the concentration of reactants.

In the provided exercise, the rate expression given is \( rate = k[A][B] \), where \( k \) is the rate constant, and \( [A] \) and \( [B] \) are the concentrations of the reactants A and B, respectively. The brackets denote concentration in molarity (M), and the rate is directly proportional to the product of these concentrations. This denotes that the reaction is first-order with respect to both A and B, implying that the rate of reaction will double if the concentration of either reactant is doubled, keeping everything else constant.

The simplicity of the rate expression allows us to predict how the reaction rate will change with varying concentrations of A and B without knowing the exact value of the rate constant, \( k \).

The impact of reactant concentration on the rate of a chemical reaction is straightforward yet profound. The rate expression encapsulates this relationship and informs us that the rate of reaction is contingent on the molar concentrations of the reactants involved.

Let's consider the example from the exercise. Comparing the given test tubes, we observe that even with different concentrations of A and B, it's the product of the concentrations that governs the rate. By comparing these products, we determined that Test tube 1, with lower concentrations of both reactants, has the smallest rate. This illustrates a general principle: as the concentration of reactants decreases, the rate of the reaction tends to diminish because there are fewer reactant molecules available to collide and react with each other.

Temperature is a pivotal factor in the rate of a chemical reaction. Increasing the temperature typically leads to a higher reaction rate. This is because molecules move more vigorously at higher temperatures, leading to more frequent and more energetic collisions among reactant molecules.

As per the provided exercise, when the temperature is increased, the rate of the reaction will increase as well. This is because an increase in temperature usually means that more molecules have enough kinetic energy to overcome the activation energy barrier of the reaction. However, it's crucial to note that the activation energy, \( E_x \), itself does not change with temperature; it is an inherent characteristic of the chemical reaction.

The Arrhenius equation is a formula that provides a quantitative basis for the effect of temperature on reaction rates. It is expressed as \( k = A e^{-\frac{E_x}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor (a constant), \( E_x \) represents the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

The equation clearly shows that the rate constant \( k \) increases exponentially with an increase in temperature. This exponential relationship underscores why even a slight rise in temperature can lead to a significant increase in the rate constant, and consequently, the reaction rate. It connects the macroscopic observation of increased reaction rate with temperature to the microscopic behavior of molecules, providing students with a deeper understanding of the underlying phenomena at play.