Problem 50 - Chemistry: Principles and Reactions

Question

The decomposition of \(\mathrm{A}\) at \(85^{\circ} \mathrm{C}\) is a zero-order reaction. It takes 35 minutes to decompose \(37 \%\) of an initial mass of \(282 \mathrm{mg}\). (a) What is \(k\) at \(85^{\circ} \mathrm{C}\) ? (b) What is the half-life of \(282 \mathrm{mg}\) at \(85^{\circ} \mathrm{C}\) ? (c) What is the rate of decomposition for \(282 \mathrm{mg}\) at \(85^{\circ} \mathrm{C} ?\) (d) If one starts with \(464 \mathrm{mg}\), what is the rate of its decomposition at \(85^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Answer: The rate constant (k) for the zero-order reaction at 85°C is 2.981 mg/min. The half-life of 282 mg is 47.27 minutes. The rate of decomposition for both 282 mg and 464 mg at 85°C is the same, which is 2.981 mg/min.

Step 1: Calculate the zero-order reaction formula

For a zero-order reaction, the rate of decomposition is expressed as: \(rate = k[A]^{0}\) Since \([A]^0 = 1\), the formula simplifies to: \(rate = k\) We are also given that the initial mass of A is 282 mg, and after 35 minutes, 37% of that has decomposed. We can use this information to proceed further with the calculations.

Step 2: Calculate decomposed mass

First, we calculate the decomposed mass after 35 minutes. \(Decomposed \, mass = \% \, Decomposed \times Initial \, Mass \) \(Decomposed \, mass = \frac{37}{100} \times 282\) \(Decomposed \, mass = 104.34 \, mg\)

Step 3: Calculate the rate constant k

Since it is a zero-order reaction, the rate of reaction equals the rate constant k. Hence from the data about the decomposition, we can find k: \(k = \frac{Decomposed \, mass}{Time}\) \(k = \frac{104.34 \, mg}{35 \, minutes}\) \(k = 2.981 \, mg/min\) (a) The rate constant k at 85°C is 2.981 mg/min.

Step 4: Calculate the half-life (t) of 282 mg

For a zero-order reaction, the half-life is given by the formula: \(t = \frac{Initial \,\, Mass}{2k}\) Inserting the calculated k value and initial mass: \(t = \frac{282}{2 \times 2.981}\) \(t = 47.27 \, minutes\) (b) The half-life of 282 mg at 85°C is 47.27 minutes.

Step 5: Find the rate of decomposition for 282 mg at 85°C

For a zero-order reaction, the rate of decomposition equals the rate constant k. Hence, (c) The rate of decomposition for 282 mg at 85°C is 2.981 mg/min.

Step 6: Find the rate of decomposition for 464 mg at 85°C

We already calculated the rate constant k and for a zero-order reaction, the rate of decomposition does not depend on the initial mass. Thus, the decomposition rate remains the same regardless of the initial mass. (d) The rate of decomposition for 464 mg at 85°C is 2.981 mg/min.

Key Concepts

reaction rate constant

In the context of chemical reactions, the reaction rate constant (k) is a crucial parameter that defines the speed at which reactants are converted into products. For zero-order reactions, the reaction rate is directly proportional to this constant, meaning it remains consistent over time.

For zero-order reactions, the equation simplifies to:

• \(rate = k[A]^0 = k\)

In this form, the concentration of the reactant does not affect the rate, as indicated by the exponent of zero. This situation often occurs in reactions where the active sites on a catalyst are fully saturated.

Given in the problem, decomposing substance A at 85°C demonstrates a zero-order reaction. Therefore, the reaction rate constant \(k\) can be calculated using the relationship:\[k = \frac{\text{Decomposed mass}}{\text{Time}}\]By understanding the mass decomposed and time taken, \(k\) helps in predicting how fast a reaction will proceed under the same conditions.

half-life calculation

Half-life is an important concept in chemistry and physics, signifying the time required for a substance to reduce to half its initial amount. However, in zero-order reactions, the half-life is a function of the initial concentration and the reaction rate constant.
For zero-order reactions, the half-life \(t_{1/2}\) is derived using the formula:

• \(t_{1/2} = \frac{\text{Initial Mass}}{2k}\)

This formula reveals that the half-life is inversely related to the rate constant \(k\). Thus, if \(k\) increases, the time needed for half of the substance to decompose decreases.

In the provided problem, the half-life of a 282 mg sample at 85°C calculated to be approximately 47.27 minutes, emphasizing how the reaction rate constant significantly affects the decomposition timeline for zero-order reactions.

rate of decomposition

The rate of decomposition represents the speed at which a substance breaks down into simpler products. In zero-order reactions, this rate is constant and strictly dictated by the reaction rate constant \(k\).

For the decomposition of A, both 282 mg and 464 mg samples at 85°C will decompose at the same rate, given by \(k\). This is due to the distinctive characteristic of zero-order reactions, where the rate is independent of the concentration of the reactant.

• \(rate = k\)

This constancy in rate indicates that regardless of the initial quantity, the breakdown process will proceed uniformly over time, and the same quantity will decompose per unit time.

In practical terms, understanding the rate of decomposition is vital for predicting how long a reaction will take to reach completion, which is particularly useful in industrial and laboratory settings where time plays a crucial role in planning and efficiency.