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Chapter 12: Q37 E (page 706)

What is the half-life for the first-order decay of phosphorus-32?\(_{{\bf{15}}}^{{\bf{32}}}{\bf{P}} \to _{{\bf{16}}}^{{\bf{32}}}{\bf{S + }}{{\bf{e}}^{\bf{ - }}}\)The rate constant for the decay is\({\bf{4}}{\bf{.85 \times 1}}{{\bf{0}}^{{\bf{ - 2}}}}{\bf{da}}{{\bf{y}}^{{\bf{ - 1}}}}\).

Short Answer

Expert verified

The half-life is 14.3 days.

Step by step solution

01

Definition

The half-life of a sample is the time required for half of the sample to undergo the process.

02

First-order reaction half-life

The rate constant has an inverse relationship with the half-life.

A first-order reaction's half-life is specified as

\({t_{1/2}} = \frac{{0.693}}{k}\)

k is the rate constant of the given reaction, and this is the derived formula of half-life.

03

Calculation of Half-life

Replacing the values,

\(\begin{align}{t_{1/2}} &= \frac{{0.693}}{{4.85 \times {{10}^{ - 2}}}}\\{t_{1/2}} &= 14.3days\end{align}\)

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Most popular questions from this chapter

A study of the rate of dimerization of \({{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}\) gave the data shown in:

\({\bf{2}}{{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}} \to {{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\)

  1. Determine the average rate of dimerization between 0 s and 1600 s, and between 1600 s and 3200 s.
  2. Estimate the instantaneous rate of dimerization at 3200 s from a graph of time versus (\({{\bf{C}}_{\bf{4}}}{{\bf{H}}_{\bf{6}}}\)). What are the units of this rate?

(c) Determine the average rate of formation of \({{\bf{C}}_{\bf{8}}}{{\bf{H}}_{{\bf{12}}}}\) at 1600 s and the instantaneous rate of formation at 3200 s from the rates found in parts (a) and (b).

Fluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is (\({_{{\bf{518}}}^{\bf{9}}}{\bf{F}}\)⟶\({_{{\bf{18}}}^{\bf{8}}}{\bf{O + e - }}\).) Physicians use\(^{{\bf{18}}}{\bf{F}}\)to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment.

(a) What is the rate constant for the decomposition of fluorine-18?

(b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h?

(c) How long does it take for 99.99% of the\(^{{\bf{18}}}{\bf{F}}\)to decay?

Does the following data fit a second-order rate law?

Trial

Time(s)

(A) (M)

1

5

0.952

2

10

0.625

3

15

0.465

4

20

0.370

5

25

0.308

6

35

0.230

The decomposition of acetaldehyde is a second-order reaction with a rate constant of \({\bf{4}}{\bf{.71 \times 1}}{{\bf{0}}^{{\bf{ - 8 }}}}{\bf{L mo}}{{\bf{l}}^{{\bf{ - 1}}}}{\bf{ s}}{{\bf{ }}^{{\bf{ - 1}}}}\). What is the instantaneous rate of decomposition of acetaldehyde in a solution with a concentration of \({\bf{5}}{\bf{.55 \times 1}}{{\bf{0}}^{{\bf{ - 4}}}}{\bf{M}}\)?

Usethe data provided to graphically determine the order and rate constant of the following reaction: \({\bf{S}}{{\bf{O}}_{\bf{2}}}{\bf{C}}{{\bf{l}}_{\bf{2}}} \to {\bf{S}}{{\bf{O}}_{\bf{2}}}{\bf{ + C}}{{\bf{l}}_{\bf{2}}}\)

Time(hr)

0

5.00*\({\bf{1}}{{\bf{0}}^{\bf{3}}}\)

1.00*\({\bf{1}}{{\bf{0}}^{\bf{4}}}\)

1.50*\({\bf{1}}{{\bf{0}}^{\bf{4}}}\)

2.50*\({\bf{1}}{{\bf{0}}^{\bf{4}}}\)

3.00*104

4.00*104

\({\bf{(S}}{{\bf{O}}_{\bf{2}}}{\bf{C}}{{\bf{l}}_{\bf{2}}}{\bf{)}}\)(M)

0.100

0.0896

0.0802

0.0719

0.0577

0.0517

0.0415
See all solutions

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