Question

The decomposition of gaseous dimethyl ether at ordinary pressures is first order. Its half-life is 25.0 minutes at \(500^{\circ} \mathrm{C}\) $$\mathrm{CH}_{3} \mathrm{OCH}_{3}(\mathrm{g}) \rightarrow \mathrm{CH}_{4}(\mathrm{g})+\mathrm{CO}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{g})$$ (a) Starting with \(8.00 \mathrm{g}\) of dimethyl ether, what mass remains (in grams) after 125 minutes and after 145 minutes? (b) Calculate the time in minutes required to decrease \(7.60 \mathrm{ng}\) (nanograms) to 2.25 ng. (c) What fraction of the original dimethyl ether remains after 150 minutes?

Short answer

Due to the length of the solution, focus on one part at a time and follow the detailed steps for accurate results.

Step-by-step solution

Understanding First-Order Kinetics

For a first-order reaction, the rate law can be expressed as \( rac{d[A]}{dt} = -k[A] \), where \([A]\) is the concentration of the reactant and \(k\) is the rate constant. The integrated form of this equation can be expressed as: \([A] = [A]_0 e^{-kt}\).

Calculating the Rate Constant (k)

The half-life for a first-order reaction is given by \( t_{1/2} = \frac{0.693}{k} \). Substituting the given half-life:\[ k = \frac{0.693}{25.0 \, ext{minutes}} = 0.02772 \, ext{minute}^{-1} \]

Using the First-Order Equation

The general equation for first-order kinetics allows us to calculate the concentration at any time \( t \) using:\[ [A] = [A]_0 e^{-kt} \]

Finding Mass After 125 Minutes

First, convert 8.00 g of dimethyl ether to moles using its molar mass (\(46.08 \, \text{g/mol}\)). Thus, \([A]_0 = \frac{8.00}{46.08} \, \text{mol}\).At 125 minutes:\[ [A] = [A]_0 e^{-0.02772 \times 125} \]

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions, or how quickly reactants turn into products. Understanding kinetics allows chemists to predict how long a reaction will take under certain conditions, which is essential in both industrial and laboratory settings.
A key part of kinetics is the reaction order, which helps determine how the concentration of reactants affects the rate. In our example, we deal with a first-order reaction, where the rate of the reaction is directly proportional to the concentration of a single reactant.

• First-order reactions are straightforward because they depend linearly on a single reactant's concentration.
• They are often involved in processes like radioactive decay or the decomposition of gases, such as dimethyl ether.

Mastering the fundamentals of chemical kinetics provides a solid foundation for understanding more complex reaction mechanisms.

Rate Constant

The rate constant, denoted by the symbol \(k\), is a crucial parameter in the study of chemical kinetics. It relates the speed of a reaction to the concentration of the reactants. For a first-order reaction, the rate constant has dimensions of time inverse, such as \( ext{min}^{-1}\).
Determining the rate constant involves understanding the reaction's half-life. Since the half-life is the time required for the concentration of a reactant to reach half its initial value, it gives a straightforward way to calculate \(k\) for first-order reactions. In our case, the dimethyl ether had a half-life of 25 minutes, which allowed us to calculate:
\[ k = \frac{0.693}{25.0 \, \text{minutes}} = 0.02772 \, \text{minute}^{-1} \]
This calculation is vital for predictive modeling of how long the reactant will take to decrease to a given amount. The rate constant gives insight into the reaction's speed under the specific conditions - in this instance, at \(500^{\circ} \text{C}\).

Half-life

The concept of half-life is instrumental in understanding first-order reactions. It is defined as the time required for a reactant's concentration to decrease to half its original amount. This measure is constant for first-order reactions irrespective of the initial concentration. Because of this fixed ratio, it simplifies calculations in kinetic studies.
The half-life of dimethyl ether at \(500^{\circ} \text{C}\) was given as 25 minutes. This allowed us to calculate the time it would take for any given quantity of the ether to reduce to a specified amount, like in the challenges of finding the mass remaining after a certain time, or calculating the needed time for a significant reduction in nanograms.

• For practical purposes, knowing the half-life helps in planning reactions and adjusting conditions to either speed up or slow down the reaction rate.
• It also lends itself to straightforward calculations of residual quantities and prediction.

Being familiar with half-life allows for a more intuitive grasp on how a system behaves over time.

Integrated Rate Law

The integrated rate law provides a mathematical description of the concentration of a reactant depending on time. For a first-order reaction, the integrated rate law is expressed as:
\[ [A] = [A]_0 e^{-kt} \]
Here, \([A]_0\) is the initial concentration, \([A]\) is the remaining concentration at time \(t\), and \(e\) is the base of the natural logarithm.
In practical terms, the integrated rate law allows us to calculate the concentration of a reactant at any time during the reaction, which was crucial for solving our exercise problems.

• This equation simplifies the process of determining how much reactant remains after a given time.
• It was used directly to solve the problem of finding how much dimethyl ether remains after 125 minutes and other time intervals.

The beauty of the integrated rate law is its simplicity and effectiveness in predicting concentrations over time, whether for solving practical problems or theoretical analyses.