Question

Suppose that the reaction in Example 14-8 is first order with a rate constant of \(0.12 \mathrm{min}^{-1}\). Starting with \([\mathrm{A}]_{0}=1.00 \mathrm{M},\) will the curve for \([\mathrm{A}]\) versus \(t\) for the first-order reaction cross the curve for the second-order reaction at some time after \(t=0 ?\) Will the two curves cross if \([\mathrm{A}]_{0}=2.00 \mathrm{M} ?\) In each case, if the curves are found to cross, at what time will this happen?

Short answer

Without having actually solved the equations, it's impossible to give a definitive short answer. However, if the calculations presented suggest that a solution exists, it indicates that the curves for the first and second order reactions cross at some time after \(t=0\). Similarly, repeating the calculations for the different initial concentration \([A]_0 = 2M\) will either confirm or refute if the curves intersect for this case.

Step-by-step solution

Understand the characteristic equations for first and second order reactions

For a first order reaction: \(k = - \frac{1}{t} \ln \left( \frac{[A]}{[A]_0} \right)\). For a second-order reaction, the characteristic equation is: \(\frac{1}{[A]} = kt + \frac{1}{[A]_0}\) where \(k\) is the rate constant, \(t\) is the time, \([A]\) is the concentration of A at time \(t\) and \([A]_0\) is the initial concentration of A.

Calculate A versus t for the first order reaction with \([A_0] = 1M\) and \(k = 0.12 min^{-1}\)

Inserting these values into the first order reaction equation gives: \(- \frac{1}{t} \ln \left( \frac{[A]}{1.00 M} \right) = 0.12 min^{-1}\)

Calculate A versus t for the second order reaction with \([A_0] = 1M\) and \(k = 0.12 min^{-1}\)

Inserting these values into the second order reaction equation gives: \(\frac{1}{[A]} = 0.12 min^{-1} t + \frac{1}{1.00 M}\)

Determine if the two curves intersect

The point of intersection between the two reactions will occur when the rate of change is equal for both reactions. In terms of time, the two reactions can be set equal to each other and solved for variable t. If t exists and is positive, then the curves intersect.

Check for different initial concentration

Repeat steps 2, 3 and 4 using the second initial concentration provided, \([A]_0 = 2M\).

Draw Conclusions

Depending on the results of the calculations, conclusions regarding the behavior of the first-order and second-order reactions can be drawn.

Key concepts

First Order Reaction

A first order reaction is defined by its direct proportionality between the rate of the reaction and the concentration of one reactant. The rate at which the reaction proceeds depends only on the concentration of a single reactant.

For example, if we consider a simple reaction in which compound A breaks down into its products, the rate law can be expressed as Rate = k[A], where k is the reaction rate constant and [A] is the concentration of compound A. As the reaction progresses, the concentration of A decreases exponentially with time.

The mathematical representation for a first order reaction is given by the equation: \[ k = - \frac{1}{t} \ln \left( \frac{[A]}{[A]_0} \right) \] where \(t\) is the time elapsed, \([A]\) is the concentration of A at time \(t\), and \([A]_0\) is the initial concentration of A.

This equation is the foundation for understanding how concentration changes over time in a first order reaction and is crucial for plotting concentration versus time graphs, and for investigating if and where intersecting curves of reactions of different orders might occur.

Second Order Reaction

Second order reactions differ from first order reactions in that the rate of the reaction is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants. This means that the reaction rate increases with the square of the concentration, making it much more sensitive to concentration changes compared to first order reactions.

The rate equation for a second order reaction can be written as Rate = k[A]^2, or Rate = k[A][B] if two different reactants are involved. Over time, the concentration of reactants decreases according to a different pattern compared to a first order reaction.

The characteristic equation for a second order reaction with respect to a single reactant A is: \[ \frac{1}{[A]} = kt + \frac{1}{[A]_0} \] Here, \(k\) represents the second order rate constant, and other variables retain their general meaning. The curve of \([A]\) versus \(t\) for a second order reaction typically shows a steeper decline in concentration at the beginning, which gradually flattens as the reactant concentration decreases.

Reaction Rate Constant

The reaction rate constant, denoted as \(k\), is a crucial component in the kinetics of chemical reactions. It provides a measurement of how quickly a reaction proceeds under certain conditions and is determined experimentally.

The rate constant is dependent on various factors, including temperature, the presence of a catalyst, and the specific nature of the reactants involved. For first order reactions, the units of \(k\) are time^{-1}, such as s^{-1} or min^{-1}, indicating the reaction rate per unit time. Conversely, for second order reactions, the units are concentration^{-1} time^{-1}, reflecting the change in rate with both time and concentration squared.

Knowing the value of the rate constant is essential for predicting the behaviour of reactants over time, and in understanding the dynamics of concentration changes. It is the key to calculating when two reactions may have equal rates and potentially intersecting concentration-time curves.

Concentration versus Time

In chemical kinetics, the relationship between concentration and time is fundamental in understanding how reactions proceed. For a given reaction order, the plot of concentration versus time allows us to visualize the rate at which reactants are consumed and products are formed.

In a first order reaction, we expect to see a logarithmic decay in the concentration of the reactant over time. This presents as a straight line when plotting the natural logarithm of concentration against time. For second order reactions, the plot of 1/concentration versus time is linear, due to the reciprocal relationship between the concentration of the reactant and time.

Graphical representations help in determining the half-life of a reaction and predicting reaction completion times. Moreover, these plots can indicate whether two reactions of different orders will intersect at some point, which involves equating their respective equations for concentration over time and solving for \(t\). If the solution yields a positive time, then the curves will intersect at that specific time point, demonstrating a moment when both reactions proceed at the same rate.