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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 the first-order decomposition of acetone at \(500^{\circ} \mathrm{C}\), $$\mathrm{CH}_{3}-\mathrm{CO}-\mathrm{CH}_{3}(g) \longrightarrow \text { products }$$ it is found that the concentration is \(0.0300 \mathrm{M}\) after \(200 \mathrm{~min}\) and \(0.0200 \mathrm{M}\) after 400 min. Calculate the following. (a) the rate constant (b) the half-life (c) the initial concentration

For the decomposition of HI, the activation energy is \(182 \mathrm{~kJ} / \mathrm{mol}\). The rate constant at \(850^{\circ} \mathrm{C}\) is \(0.0174 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{h}\). (a) What is the rate constant at \(700^{\circ} \mathrm{C} ?\) (b) At what temperature will the rate constant be a fourth of what it is at \(850^{\circ} \mathrm{C} ?\)

For the reaction $$2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g)$$ the rate constant is \(0.066 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(565^{\circ} \mathrm{C}\) and \(22.8 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min}\) at \(728^{\circ} \mathrm{C} .\) (a) What is the activation energy of the reaction? (b) What is \(k\) at \(485^{\circ} \mathrm{C}\) ? (c) At what temperature is \(k\), the rate constant, equal to \(11.6 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{min} ?\)

The following gas-phase reaction is second-order. $$2 \mathrm{C}_{2} \mathrm{H}_{4}(g) \longrightarrow \mathrm{C}_{4} \mathrm{H}_{8}(g)$$ Its half-life is \(1.51\) min when \(\left[\mathrm{C}_{2} \mathrm{H}_{4}\right]\) is \(0.250 \mathrm{M}\) (a) What is \(k\) for the reaction? (b) How long will it take to go from \(0.187 M\) to \(0.0915 \mathrm{M}\) ? (c) What is the rate of the reaction when \(\left[\mathrm{C}_{2} \mathrm{H}_{4}\right]\) is \(0.335 \mathrm{M} ?\)

The decomposition of dimethyl ether \(\left(\mathrm{CH}_{3} \mathrm{OCH}_{3}\right)\) to methane, carbon monoxide, and hydrogen gases is found to be first-order. At \(500^{\circ} \mathrm{C}\), a \(150.0\) -mg 35 sample of dimethyl ether is reduced to \(43.2 \mathrm{mg}\) after three quarters of an hour. Calculate (a) the rate constant. (b) the half-life at \(500^{\circ} \mathrm{C}\). (c) how long it will take to decompose \(95 \%\) of the dimethyl ether.
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