Problem 16 - Chemistry and Chemical Reactivity

Question

The decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) in \(\mathrm{CCl}_{4}\) is a first-order reaction. If \(2.56 \mathrm{mg}\) of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is present initially and \(2.50 \mathrm{mg}\) is present after 4.26 minutes at \(55^{\circ} \mathrm{C}\) what is the value of the rate constant, \(k\) ?

Step-by-Step Solution

Answer

The rate constant \(k\) is approximately \(0.00541 \, \text{min}^{-1}\).

Step 1: Understanding the problem

We are dealing with a first-order reaction and need to find the rate constant, \(k\). We know the initial and remaining amounts of \(\mathrm{N}_2\mathrm{O}_5\), and the reaction time.

Step 2: Apply first-order kinetic formula

For a first-order reaction, the relationship between concentration and time is given by \[\ln \left( \frac{[A]_0}{[A]} \right) = kt\] where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant.

Step 3: Convert mg to concentration proportion

The amounts given are in milligrams. Since the units are consistent, we do not need to convert to molarity for the proportion:- \([A]_0 = 2.56 \text{ mg}\)- \([A] = 2.50 \text{ mg}\)

Step 4: Use the equation to solve for k

Substitute the values into the equation \[\ln \left( \frac{2.56}{2.50} \right) = k \times 4.26 \]This simplifies to \[\ln(1.024) = 4.26k\] Evaluate \(\ln(1.024)\) and solve for \(k\).

Step 5: Calculate k

Evaluate the left side: \[\ln(1.024) = 0.02303\]Solve for \(k\):\[k = \frac{0.02303}{4.26} \approx 0.00541 \, \text{min}^{-1}\]

Key Concepts

Rate Constant Calculation

The rate constant, denoted by the symbol \( k \), is a crucial component in understanding the speed of a chemical reaction. In a first-order reaction, the rate constant helps determine how quickly the reactant concentration falls over time.
For a first-order reaction, the calculation involves measuring the initial concentration and the concentration at a given time. Using the formula \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \), you can isolate \( k \), the rate constant, to quantify the reaction's speed. This formula allows us to see how the reaction proceeds over a specific period.
Knowing the rate constant lets us predict the behavior of the reaction under similar conditions, which is valuable for controlling reaction conditions in practical applications.

Kinetic Equations

Kinetic equations are mathematical expressions that describe the relation between the concentrations of reactants or products and time. For a first-order reaction, the kinetic equation is \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \), where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \( t \), and \( k \) is the rate constant.
These equations offer a clear and predictable pattern of chemical changes over time. They reveal how fast a reactant consumes and a product forms, based on the reaction order, especially helpful in analyzing reaction processes.

• The equation connects the natural logarithm of the concentration ratio to a linear relationship with time.
• It simplifies to calculate how different factors like temperature impact the rate.

Kinetic equations are useful tools for chemical engineers and chemists alike, making predictions about how different concentrations will behave over time.

Reaction Order

The order of a reaction indicates how the concentration of reactants affects the rate of the reaction. In chemical kinetics, the reaction order is determined experimentally and can be zero, first, second, or higher.
A **first-order reaction** means the rate is directly proportional to the concentration of one reactant. In our case, the decomposition of \( \mathrm{N}_{2} \mathrm{O}_{5} \) is solely dependent on its concentration. This is represented by the equation \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \).
The practical importance of knowing the reaction order is pivotal as it aids:

• In predicting how changing a reactant's concentration might affect the rate.
• In deciding optimal conditions for industrial reactions.

Understanding reaction orders helps in mechanistic studies of complex reactions, offering insights into the step-by-step sequence of events happening during a reaction.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate of a chemical reaction and the factors affecting it. It's pivotal in both scientific studies and industrial applications.
Understanding chemical kinetics involves:

• Identifying the rate of reaction which is the change of concentration over time.
• Determining the rate laws and reaction orders to establish mathematical models for reaction rates.
• Investigating how different conditions like temperature, concentration, and catalysts impact reaction speeds.

Kinetics is crucial for understanding how to control reactions precisely, which is important in fields like pharmaceuticals, where reaction rates must be controlled to be productive in drug formulation. Furthermore, it helps chemists develop methods for energy-efficient processes by minimizing time and resource wastage.