key term - [A]_t = [A]_0 e^{-kt}

Definition

[A]_t = [A]_0 e^{-kt} is an equation that describes the concentration of a reactant (A) at a given time (t) in a first-order chemical reaction. It relates the initial concentration of the reactant ([A]_0) to its concentration at time t ([A]_t) and the rate constant (k) of the reaction.

5 Must Know Facts For Your Next Test

1. The equation [A]_t = [A]_0 e^{-kt} is derived from the rate law for a first-order reaction, which states that the rate of the reaction is proportional to the concentration of the reactant.
2. The rate constant (k) determines the speed at which the reaction occurs, with a higher value of k indicating a faster reaction.
3. The exponential term e^{-kt} represents the fraction of the initial concentration that remains at time t, as the reactant is consumed over time.
4. The equation can be used to predict the concentration of the reactant at any given time during the course of the reaction, provided that the initial concentration ([A]_0) and the rate constant (k) are known.
5. The half-life (t_1/2) of a first-order reaction, which is the time it takes for the concentration of the reactant to decrease by half, can be calculated using the equation t_1/2 = ln(2)/k.

Review Questions

• Explain the relationship between the rate constant (k) and the exponential term e^{-kt} in the equation [A]_t = [A]_0 e^{-kt}.
  • The rate constant (k) and the exponential term e^{-kt} are directly related in the equation [A]_t = [A]_0 e^{-kt}. The rate constant (k) determines the speed at which the reaction occurs, with a higher value of k indicating a faster reaction. The exponential term e^{-kt} represents the fraction of the initial concentration that remains at time t, as the reactant is consumed over time. As the value of k increases, the rate of the exponential decay also increases, leading to a faster decrease in the concentration of the reactant over time.
• Describe how the equation [A]_t = [A]_0 e^{-kt} can be used to determine the half-life of a first-order reaction.
  • The equation [A]_t = [A]_0 e^{-kt} can be used to determine the half-life of a first-order reaction. The half-life (t_1/2) is the time it takes for the concentration of the reactant to decrease by half. This can be calculated using the equation t_1/2 = ln(2)/k, where ln(2) is the natural logarithm of 2 and k is the rate constant. By rearranging the original equation and setting [A]_t = [A]_0/2, the half-life can be derived, which represents the time it takes for the concentration of the reactant to decrease to half of its initial value.
• Analyze how the initial concentration ([A]_0) and the rate constant (k) influence the concentration of the reactant ([A]_t) over time in the equation [A]_t = [A]_0 e^{-kt}.
  • The initial concentration ([A]_0) and the rate constant (k) both have a significant impact on the concentration of the reactant ([A]_t) over time in the equation [A]_t = [A]_0 e^{-kt}. A higher initial concentration ([A]_0) will result in a higher concentration of the reactant at any given time t, as the exponential term e^{-kt} will be multiplied by a larger value. Similarly, a higher rate constant (k) will lead to a faster decrease in the concentration of the reactant over time, as the exponential term e^{-kt} will decrease more rapidly. By understanding the relationships between these variables, one can predict and analyze the behavior of first-order reactions.

Related terms

First-Order Reaction:

A chemical reaction in which the rate of the reaction is proportional to the concentration of a single reactant.

Rate Constant (k): The proportionality constant that determines the rate of a chemical reaction, representing the probability of a reaction occurring per unit of time.

Exponential Decay:

The process by which a quantity decreases at a rate proportional to its current value, resulting in an exponential decrease over time.