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Chapter 11: Problem 50

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

Short Answer

Expert verified
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 by step solution

01

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.
02

Calculate decomposed mass

First, we calculate the decomposed mass after 35 minutes. \(Decomposed \, mass = \% \, Decomposed \times Initial \, Mass \) \(Decomposed \, mass = \frac{37}{100} \times 282\) \(Decomposed \, mass = 104.34 \, mg\)
03

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.
04

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.
05

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.
06

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.

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Most popular questions from this chapter

In a first-order reaction, suppose that a quantity \(X\) of a reactant is added at regular intervals of time, \(\Delta t\). At first the amount of reactant in the system builds up; eventually, however, it levels off at a saturation value given by the expression $$\text { saturation value }=\frac{X}{1-10^{-a}} \quad \text { where } a=0.30 \frac{\Delta t}{t_{1 / 2}}$$ This analysis applies to prescription drugs, of which you take a certain amount each day. Suppose that you take \(0.100 \mathrm{~g}\) of a drug three times a day and that the half-life for elimination is \(2.0\) days. Using this equation, calculate the mass of the drug in the body at saturation. Suppose further that side effects show up when \(0.500 \mathrm{~g}\) of the drug accumulates in the body. As a pharmacist, what is the maximum dosage you could assign to a patient for an 8 -h period without causing side effects?

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If the activation energy of a reaction is \(4.86 \mathrm{~kJ}\), then what is the percent increase in the rate constant of a reaction when the temperature is increased from \(45^{\circ}\) to \(75^{\circ} \mathrm{C}\) ?

Nitrosyl chloride (NOCl) decomposes to nitrogen oxide and chlorine gases. (a) Write a balanced equation using smallest whole-number coefficients for the decomposition. (b) Write an expression for the reaction rate in terms of \(\Delta[\mathrm{NOCl}] .\) (c) The concentration of NOCl drops from \(0.580 M\) to \(0.238 M\) in \(8.00 \mathrm{~min}\). Calculate the average rate of reaction over this time interval.

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