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 inital 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 \(85^{\circ} \mathrm{C} ?\)

Short answer

Answer (a): The rate constant k at 85°C is 2.98 mg/min. Answer (b): The half-life of 282 mg at 85°C is 47.32 minutes. Answer (c): The rate of decomposition for 282 mg at 85°C is 2.98 mg/min. Answer (d): The rate of decomposition for 464 mg at 85°C is 4.90 mg/min.

Step-by-step solution

Calculate the amount decomposed at time t = 35 minutes

We are given that 37% of the initial mass 282 mg is decomposed after 35 minutes. To find the amount of A decomposed, multiply the percentage by the initial mass: \(Amount\;Decomposed = 0.37 * 282\,mg = 104.34\, mg\)

Determine the concentration at time t

We are given that the initial mass is 282 mg. After 35 minutes, 104.34 mg has decomposed. Calculate the concentration at time t by subtracting the amount decomposed from the initial mass: \([A]_{t} = [A]_{0} - Amount\;Decomposed = 282\,mg - 104.34\,mg = 177.66\,mg\)

Calculate the rate constant (k) using the zero-order reaction equation

Now, we can use the zero-order equation: \([A]_{t}=[A]_{0}-kt\) Plug in the values we have found: \(177.66\, mg = 282\, mg - k(35\, min)\) Solve for k: \(k = \frac{282\,mg - 177.66\, mg}{35\, min} = 2.98\, \frac{mg}{min}\) Answer (a): The rate constant k at 85°C is 2.98 mg/min.

Calculate the half-life for 282 mg at 85°C using the zero-order reaction equation

We need to find the time taken (t) when 50% of the initial mass (282 mg) has decomposed (141 mg). Use the zero-order equation again: \([A]_{t}=[A]_{0}-kt\) Plug in the values we know: \(141\, mg = 282\, mg - 2.98\, \frac{mg}{min}(t)\) Solve for t: \(t = \frac{282\,mg - 141\, mg}{2.98\, \frac{mg}{min}} = 47.32\, min\) Answer (b): The half-life of 282 mg at 85°C is 47.32 minutes.

Calculate the rate of decomposition for 282 mg at 85°C

We've already found the reaction rate constant (k) for this zero-order reaction, which is the rate of decomposition: Rate of decomposition = k = 2.98 mg/min Answer (c): The rate of decomposition for 282 mg at 85°C is 2.98 mg/min.

Calculate the rate of decomposition for 464 mg at 85°C using the rate constant

For a zero-order reaction, the rate of decomposition is proportional to the initial mass. We can use the proportionality and the rate constant to find the rate of decomposition for 464 mg at 85°C: \(\frac{Rate\;of\;Decomposition\;for\;464\,mg}{Rate\;of\;Decomposition\;for\;282\,mg} = \frac{464\,mg}{282\,mg}\) Solve for the rate of decomposition for 464 mg: \(Rate\;of\;Decomposition\;for\;464\,mg = 2.98\, \frac{mg}{min} * \frac{464\,mg}{282\,mg} = 4.90\, \frac{mg}{min}\) Answer (d): The rate of decomposition for 464 mg at 85°C is 4.90 mg/min.

Key concepts

Chemical Kinetics

Understanding the pace at which chemical reactions occur is essential in various fields, from industrial processes to biological systems. Chemical kinetics is the branch of chemistry that deals with measuring and studying the rates of chemical reactions. The principles of kinetics help explain not only how fast reactions proceed but also the steps involved, known as the reaction mechanism.

For a zero-order reaction, the reaction rate is constant and does not depend on the concentration of the reactant. This means that the reactants are consumed at a steady rate irrespective of their amount. Zero-order kinetics often occur in reactions where a catalyst is saturated with the reactant, or where the reaction occurs on the surface of a solid. In the context of the exercise, the decomposition of substance A at a specific temperature follows zero-order kinetics, and the rate constant determined from the data provided is a fixed value, indicative of the reaction's inherent nature under the given conditions.

Half-Life Calculation

The half-life of a reactant is the time required for its amount to be reduced by half of its initial value during a chemical reaction. It is a key concept when dealing with radioactive substances, medications in pharmacokinetics, and various types of chemical reactions.

In zero-order reactions, unlike first-order reactions, the half-life is not constant and varies depending on the initial concentration of the reactant. As we saw in the exercise, to calculate the half-life for a zero-order reaction, we use the rate constant and the initial mass. Because the reaction rate is independent of the concentration for zero-order reactions, the time it takes for the reactant's mass to reach half of its original value can be determined directly from the rate of reaction.

Reaction Rate Constant

At the heart of analyzing reaction kinetics is the reaction rate constant, commonly denoted by the symbol k. This constant is a quantitative measure of the speed of a chemical reaction, and its value can be influenced by various factors such as temperature, pressure, and the presence of a catalyst.

In the provided exercise, the reaction rate constant is derived using the zero-order kinetic equation. The rate constant k has units that reflect the reaction order; for zero-order reactions, it is expressed in terms of concentration over time (e.g., mg/min). Notably, in zero-order kinetics, the rate constant gives the slope of the line when plotting the reactant concentration against time, representing the reaction's constant rate of decrease over time. Even when different initial amounts of the reactant are present, such as in the 282 mg and 464 mg scenarios, the k value remains the same, demonstrating the invariant nature of this constant for a given reaction under constant conditions.