The rate constant for a first order reaction is \(60 \mathrm{~s}^{-1}\). How much time will it take to reduce the initial concentration of the reactant to its \(1 / 16^{\text {th }}\) value?

Understanding the rate constant of a reaction is crucial because it tells us how quickly a reaction proceeds. For a first-order reaction, the rate constant (\( k \)) has the unit of inverse seconds \( \mathrm{s}^{-1} \) and is used to calculate the reaction rate. Essentially, it is the proportionality constant in the rate law equation, which expresses the rate of a chemical reaction as a function of the concentration of the reactants.

For the given exercise where the rate constant is \( 60 \mathrm{s}^{-1} \) for a first-order reaction, we can anticipate that the reaction will progress rapidly because the rate constant value is high. This means that, in one second, a significant proportion of the reactant will be converted to the product. By understanding the rate constant, students can not only predict the time it will take for a certain amount of reactant to be consumed but also compare the speed of different reactions.

The concentration ratio is a way to express the relationship between the initial and final concentration of the reactant in a chemical reaction. In our exercise, the concentration of the reactant decreases to its \( 1/16^\text{th} \) value, which tells us that the final concentration is significantly lower than the initial one.

This ratio is critical in the equation for first-order kinetics because it represents the extent of the reaction. In general, determining how the concentration of a reactant changes over time is a fundamental aspect of studying reaction kinetics. With the help of this ratio, we utilize mathematical concepts, such as natural logs, to simplify the calculation process needed to determine various parameters, including reaction time. It is the simplification of complex reactions to manageable calculations that makes this approach so valuable for chemists and students alike.

In reaction kinetics, the natural logarithm (\( \ln \) ) is often used in the context of first-order reactions to simplify equations and solve for unknowns, such as time. It is helpful because it transforms multiplicative relationships into additive ones, making it easier to manipulate and isolate variables algebraically.

In the exercise, taking the natural log of the concentration ratio simplifies the calculation of the reaction time. We start from the principle that for a first-order reaction, the log of the concentration ratio is equal to the product of the rate constant and time. For example, in \( \ln(16) \) , the expression can be further simplified because 16 is a power of 2 (\( 2^4 \) ), which helps us solve for time (\( t \) ) when we already know the rate constant (\( k \) ). This use of the natural logarithm is an elegant example of how mathematical tools are applied in chemical kinetics to decode the timing and progression of reactions.