Question

For the first order decomposition of azomethane at \(600^{\circ} \mathrm{K}\), it takes 30 minutes for the original concentration to decrease to half its value. After \(60.0\) minutes have elapsed, what percentage of the azomethane originally present remains?

Short answer

After 60 minutes, approximately \(25\%\) of the azomethane originally present remains, using the first-order reaction equation and the given half-life.

Step-by-step solution

Identify the given values and the question asked

The given values in the problem are: 1. The half-life of azomethane (t½) is 30 minutes. 2. The reaction is a first-order decomposition reaction. 3. We are asked to find the percentage of azomethane remaining after 60 minutes.

Determine the rate constant (k) from the half-life

For first-order reactions, we can use the formula: \[ t_{1/2} = \frac{0.693}{k} \] We are now going to solve for \(k\), the rate constant. Plug in the half-life of azomethane: \[ 30 \text{ minutes} = \frac{0.693}{k} \] \[ k = \frac{0.693}{30} \text{ minutes}^{-1} \]

Apply the first-order reaction equation

Now, we will use the first-order reaction equation to relate the concentration of azomethane initially (\(A_{0}\)) and at time t (\(A_{t}\)): \[ \ln{\frac{A_t}{A_0}} = -kt \] We are given that the elapsed time (t) is 60 minutes, so we can substitute the values for k and \(t\): \[ \ln{\frac{A_t}{A_0}} = -\frac{0.693}{30} (60) \]

Solve for the remaining azomethane concentration

Now we just need to solve for \( \frac{A_t}{A_0} \), the fraction of azomethane remaining at time t: \[ \ln{\frac{A_t}{A_0}} = -0.693(2) \] \[ e^{\ln{\frac{A_t}{A_0}}} = e^{-0.693(2)} \] \[ \frac{A_t}{A_0} = e^{-1.386} \] Now, we'll convert the fraction to a percentage by multiplying by 100: \[ \text{Percentage remaining} = \frac{A_t}{A_0} \times 100 = e^{-1.386} \times 100 \]

Calculate the final answer

Now we just need to evaluate the expression to obtain the percentage of azomethane remaining at 60 minutes: \[ \text{Percentage remaining} \approx e^{-1.386} \times 100 \approx 25 \% \] So, after 60 minutes, about 25% of the azomethane originally present remains.

Key concepts

Rate Constant

The rate constant, often denoted as 'k', is a crucial parameter in the realm of chemistry, particularly in reaction kinetics. It tells us how quickly a reaction proceeds under certain conditions. In a first-order reaction, the rate constant is inversely proportional to the half-life of the reactant, making it a significant figure for predicting how long it takes for the concentration of a reactant to decrease to half its initial value.

Understanding the rate constant is fundamental when studying chemical reactions, as it encapsulates the effect of factors like temperature and the presence of a catalyst on the reaction rate. The mathematical relationship highlighted in the solution illustrates how we can calculate the rate constant using the half-life formula:
\[\begin{equation}\[ k = \frac{0.693}{t_{1/2}} \]\end{equation}\]
where \(t_{1/2}\) represents the half-life. It's vital to note that the units of k are dependent on the order of reaction—it is \(min^{-1}\) for a first-order reaction in this case.

Half-Life

Half-life, symbolized as \(t_{1/2}\), is a term commonly used in both chemistry and physics. It defines the period required for a quantity to reduce to half of its initial value. In the context of first-order reactions, the half-life is a constant value, independent of the initial concentration of the reactant. This attribute of the half-life is particularly useful because it simplifies the prediction of the concentration of the reactant at any given time.

Moreover, the half-life gives a straightforward insight into the stability or persistence of a substance in a system. For the decay of azomethane at \(600^{\text{k}}\), the half-life of 30 minutes indicates that after every 30 minutes, only 50% of the azomethane will remain. This predictability can be especially valuable in industrial processes and pharmacokinetics where time-dependent concentration changes are pivotal.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical reactions occur and the factors that influence these rates. It's a cornerstone of chemical kinetics, which not only looks at the speed of reactions but also tries to understand the steps (reaction mechanism) by which a reaction proceeds. For a first-order reaction, the rate is directly proportional to the concentration of one reactant.

Key equations used in reaction kinetics for first-order reactions include the rate law, which for a first-order reaction takes the form:
\[\begin{equation}\[ Rate = k[A] \]\end{equation}\]
where \(k\) is the rate constant, and \([A]\) is the concentration of the reactant. Another crucial aspect of reaction kinetics is the integrated rate law, which for a first-order reaction, gives us the relationship between the concentrations of the reactant over time, allowing us to determine the remaining concentration after any given period.

Exponential Decay

Exponential decay describes the process in which a quantity decreases at a rate that is proportional to its current value. For a first-order chemical reaction, the concentration of a reactant decreases exponentially over time, which perfectly aligns with the concept of exponential decay. Such decay processes are pervasive in nature and can be observed in radioactive decay, population decline, and, as shown in the given problem, chemical decomposition.

Mathematically, the exponential decay can be represented as:
\[\begin{equation}\[ A_t = A_0e^{-kt} \]\end{equation}\]
where \(A_0\) is the initial concentration, \(A_t\) is the concentration at time \(t\), and \(k\) is the rate constant. In our example, by understanding exponential decay, we can predict how much azomethane remains after 60 minutes, which is a practical skill in scenarios such as predicting medication levels in blood over time.

Natural Logarithm

The natural logarithm is an operation that corresponds to the inverse of the exponential function. Represented as \(\ln\), it plays an integral role in solving equations involving exponential relationships—like those found in first-order reaction kinetics. In the context of our azomethane reaction, the natural logarithm helps us relate the rate constant and the concentrations of the reactant at different times.

The first-order reaction equation provided earlier,
\[\begin{equation}\[ \ln{\frac{A_t}{A_0}} = -kt \]\end{equation}\]
can be rearranged to isolate \(A_t/A_0\) through the use of natural logarithms. This equation demonstrates the transformation of an exponential equation into a linear one, where the natural logarithm of the concentration ratio gives a value that can be easily related to the time elapsed and the rate constant.