Question

If it takes 75.0 min for the concentration of a reactant to drop to \(25.0 \%\) of its initial value in a first-order reaction, what is the rate constant for the reaction in the units \(\min ^{-1} ?\)

Short answer

The rate constant for the reaction is approximately 0.0185 \(\min^{-1}\).

Step-by-step solution

Understand the first-order reaction

For a first-order reaction, the rate of the reaction is directly proportional to the concentration of the reactant. The rate constant, k, can be determined using the first-order reaction formula: \[ k = \frac{\ln(\frac{[A]_0}{[A]})}{t} \] where \([A]_0\) is the initial concentration, [A] is the concentration at time t, and t is the time.

Set up the equation

Given that the concentration of the reactant drops to 25% of its initial value in 75.0 minutes, we can substitute into the equation: \[ k = \frac{\ln(\frac{[A]_0}{0.25[A]_0})}{75.0 \min} = \frac{\ln(4)}{75.0 \min} \]

Calculate the rate constant

Now we calculate the rate constant using natural log values: \[ k = \frac{\ln(4)}{75.0 \min} \approx \frac{1.3863}{75.0 \min} \approx 0.0185 \min^{-1} \]

Key concepts

Understanding Chemical Kinetics

Chemical kinetics is the study of reaction rates, how reaction rates change under various conditions, and what molecular events occur during the overall reaction. It's a cornerstone of physical chemistry that helps us understand the speed at which a chemical reaction occurs and which factors will affect this speed.

In the context of a first-order reaction, the rate at which reactants are transformed into products is directly tied to the concentration of the reactant. As the concentration decreases, so does the reaction rate. This relationship is quantified by what we call the rate constant, symbolized as 'k'. In a first-order scenario, the time it takes for the concentration of a reactant to halve, known as the half-life, remains constant, regardless of the concentration. This is a unique property of first-order kinetics and is used extensively in processes such as radioactive decay and chemical shelf-life prediction.

Deciphering the Rate Law

The rate law is an equation that tells us the relationship between the reaction rate and the concentrations of reactants. For first-order reactions, the rate law has a very simple form:
\[ \text{rate} = k[A] \]
where 'rate' is the speed of the reaction, 'k' is the first-order rate constant, and '[A]' is the concentration of the reactant A. What is important to note is that the rate law must be determined experimentally; it is not something that can be gleaned simply from a reaction's balanced chemical equation.

The rate law provides the foundation for calculating how fast a product is formed or a reactant is consumed, and it provides the mechanistic insight on how different reactants come together to form products.

Navigating the Natural Logarithm

The natural logarithm, commonly abbreviated as 'ln', is a mathematical function that is the inverse of the exponential function where the base is the constant 'e' (approximately equal to 2.71828). In chemical kinetics, and specifically in first-order reaction rate equations, taking the natural logarithm of the ratio of initial reactant concentration to the remaining concentration allows us to linearize the relationship and isolate the rate constant 'k'.

For the reaction in question, the natural logarithm helps us solve for the rate constant k by using the formula:\[ k = \frac{\ln(\frac{[A]_0}{[A]})}{t} \]
This formula is derived from integrating the first-order rate law equation, and the natural logarithm allows us to deal with the exponential nature of the reaction's progress in a linear fashion.

Focusing on Reaction Concentration

Reaction concentration plays a pivotal role in determining the rate of a chemical reaction. For a first-order reaction, the rate at which the reaction proceeds is directly proportional to the concentration of the single reactant. In our exercise, we observe a change in the reactant concentration from its initial value to 25% of that value over a given time. To find the rate constant 'k', we apply the relationship for a first-order reaction and the fact that the reactant concentration decreases with time.

The concentration terms in our example are formatted as a fraction, representing the decrease over time from the initial concentration \( [A]_0 \) to \( [A] \) at a later time. By applying the correct mathematical operations—specifically, the natural logarithm—we effectively navigate the calculation to determine the rate constant, illustrating how the interplay between concentration and time dictates the progression of the reaction.