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The rearrangement of methyl isonitrile \(\left(\mathrm{CH}_{3} \mathrm{NC}\right)\) to acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) is a first-order reaction and has a rate constant of \(5.11 \times 10^{-5} \mathrm{~s}^{-1}\) at \(472 \mathrm{~K}\) \(\mathrm{CH}_{3}-\mathrm{N} \equiv \mathrm{C} \rightarrow \mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{N}\) If the initial concentration of \(\mathrm{CH}_{3} \mathrm{NC}\) is \(0.0340 \mathrm{M}\) : (a) What is the molarity of \(\mathrm{CH}_{3} \mathrm{NC}\) after \(2.00 \mathrm{~h} ?\) (b) How many minutes does it take for the \(\mathrm{CH}_{3} \mathrm{NC}\) concentration to drop to \(0.0300 \mathrm{M} ?\) (c) How many minutes does it take for \(20 \%\) of the \(\mathrm{CH}_{3} \mathrm{NC}\) to react?

Step by step solution

01

Understand the First-order Reaction

For a first-order reaction, the rate of the reaction is proportional to the concentration of a single reactant. The equation for a first-order reaction is given by: \[ \ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \] where \([A]_t\) is the concentration of the reactant at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time.

02

Calculate Concentration After 2 Hours (Part a)

We need to find the concentration of the reactant \([\text{CH}_3\text{NC}]\) after 2 hours. Convert 2 hours into seconds: \(2 \text{ hours} = 2 \times 3600 \text{ s} = 7200 \text{ s}\). Use the formula: \[ \ln \left( \frac{[A]_t}{0.0340} \right) = -5.11 \times 10^{-5} \times 7200 \] Calculate the right side and solve for \([A]_t\).

03

Solve for [A]_t after 2 hours

From the equation in the previous step: \[ \ln \left( \frac{[A]_t}{0.0340} \right) = -0.36792 \] \[ [A]_t = 0.0340 \times e^{-0.36792} \] Calculate \([A]_t\) to get the concentration after 2 hours: \( [A]_t \approx 0.0266 \text{ M} \).

04

Calculate Time for Concentration to Drop to 0.0300 M (Part b)

We are asked how long it takes for the concentration to drop to \(0.0300\, \text{M}\). Use the formula: \[ \ln \left( \frac{0.0300}{0.0340} \right) = -5.11 \times 10^{-5} \times t \] Solve the equation for \(t\) to find the time in seconds.

05

Solve for Time t and Convert to Minutes (Part b)

\[ \ln \left( \frac{0.0300}{0.0340} \right) = -0.124938 \] Setting: \[ -0.124938 = -5.11 \times 10^{-5} \times t \] \[ t = \frac{0.124938}{5.11 \times 10^{-5}} \] Solve for \(t\) and convert to minutes: \( t \approx 2447 \text{ s} \) or approximately \(40.8 \text{ minutes}\).

06

Calculate Time for 20% Reaction (Part c)

If 20% reacts, 80% remains, thus \([A]_t = 0.80 \times 0.0340 = 0.0272 \text{ M}\). Use the formula: \[ \ln \left( \frac{0.0272}{0.0340} \right) = -5.11 \times 10^{-5} \times t \] Solve the equation for \(t\).

07

Solve for Time t and Convert to Minutes (Part c)

\[ \ln \left( \frac{0.0272}{0.0340} \right) = -0.22314 \] Setting: \[ -0.22314 = -5.11 \times 10^{-5} \times t \] \[ t = \frac{0.22314}{5.11 \times 10^{-5}} \] Solve for \(t\) and convert to minutes: \( t \approx 4368 \text{ s} \) or approximately \(72.8 \text{ minutes}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Constant

The rate constant is a crucial element in understanding reaction kinetics. In a first-order reaction, the rate constant, denoted as \( k \), defines the speed at which a reaction proceeds. For the rearrangement of methyl isonitrile to acetonitrile, given as \( 5.11 \times 10^{-5} \mathrm{~s}^{-1} \) at \( 472 \mathrm{~K} \), it tells us how fast the reaction occurs at this temperature. The rate constant is specific to each reaction and is influenced by factors such as temperature and the nature of the reactants.

• Higher rate constants indicate faster reactions.
• A first-order reaction rate depends linearly on the concentration of one reactant.

Understanding and calculating the rate constant is essential for predicting how long it takes for a reaction to occur under given conditions.

Reaction Kinetics

Reaction kinetics involves the study of how chemical reactions occur and how factors affect their rates. In first-order reactions, like the conversion of methyl isonitrile to acetonitrile, kinetics tells us that the rate of the reaction is directly proportional to the concentration of the methyl isonitrile. This means if the concentration of methyl isonitrile decreases, the reaction rate decreases as well.

• The rate equation is: \( \,\ln \left( \frac{[A]_t}{[A]_0} \right) = -kt \).
• Kinetics can help us understand both the speed and the mechanisms of reactions.

Studying kinetics allows chemists to optimize reactions, like ensuring that enough time is provided for a desired conversion or product yield.

Concentration Change

Concentration change is a key aspect of first-order reactions, which involves the alteration in quantity of reactants or products over time. In the exercise, concentration change is calculated to determine how the amount of methyl isonitrile decreases over set periods. The exponential decay in concentration is characteristic of first-order kinetics.

• The formula \( [A]_t = [A]_0 \, e^{-kt} \) is used.
• Calculations involve solving for \( [A]_t \) after known intervals.

By analyzing concentration changes, you can predict how long a reactant will last or how much product will be formed in a specific time period.

Methyl Isonitrile

Methyl isonitrile (\( \mathrm{CH}_3\mathrm{NC} \)) is the reactant in this rearrangement reaction. It is a type of molecule known as an isonitrile, which has a carbon atom bonded to a nitrogen atom. In the context of the reaction, methyl isonitrile undergoes a structural rearrangement to form acetonitrile. It’s important to understand its properties and behavior during the reaction, which follows first-order kinetics.

• Its conversion is dictated solely by its concentration.
• Understanding its rearrangement helps in predicting reaction outcomes.

This rearrangement is a typical example of how molecular transformations can occur spontaneously under specific conditions like temperature.

Acetonitrile

Acetonitrile (\( \mathrm{CH}_3\mathrm{CN} \)) is the product of the methyl isonitrile rearrangement. This compound is widely used in industrial and laboratory settings, especially as a solvent and a synthetic intermediate. The transition from methyl isonitrile to acetonitrile is notable because it represents the final step in its transformation and involves a simple shift in bonding from the nitrogen to the carbon atom. This reaction and its product have significant practical applications:

• Acetonitrile's formation is a direct result of the rearrangement.
• It is less reactive compared to its precursor, making it stable for various uses.

Examining such reactions offers insights into chemical stability and synthesis possibilities.