14.2.1 First-Order Reactions

Question 1

First-Order Reactions

Answer 1

• The rate law for a first-order reaction has the general form rate = k[A]. A more useful equation for first-order reactions is ln[A] t = –kt + ln[A] 0 .
• The rate of a first-order reaction can also be expressed in terms of half-life (t 1/2 ). t 1/2 = 0.693/k.

Question 2

note

Answer 2

• The rate law for a first-order reaction has the general form rate = k[A]. However, the plot of [A] versus t looks the same for a first-order reaction or a second-order reaction.
• Applying principles of calculus, a more useful equation can be derived: ln[A] t = –kt + ln[A] 0
• This equation is in the form of a straight line (y = mx + b), where y = ln[A] t , m = –k, x = t, and b = ln[A] 0 .
  A plot of ln[A] t versus t for a first-order reaction can be easily identified as a straight line.
• The rate of a first-order reaction can be expressed in terms of half-life (t 1/2 ).
• Half-life is the time required for a substance to decrease to half of its initial concentration.
• For a first-order reaction, t 1/2 = 0.693/k.
• The half-life for a first-order reaction is independent of the starting concentration. This allows for the simple calculation of the amount of substance remaining after a given time. This also allows for the simple calculation of the amount of time required for a given amount of material to react.

Question 3

Strontium-90 undergoes first-order radioactive decay. Its half-life is 28.8 years. What percent of an 8.0-gram sample of 90Sr will remain after a period of 75 years?

Answer 3

16%

Question 4

Cis−2-butene (cis-C4H8 ) forms trans−2-butene (trans-C4H8 ) according to the following equation.

This reaction is first order overall, and the half-life of cis−2-butene is 192 s. If the initial concentration of cis−2-butene in a sample is 0.36 M, what will be the concentration of cis−2-butene after 10.0 minutes (600. s)?

Answer 4

0.041 M

Question 5

Radon-222 is a radioactive isotope that has a first-order half-life of 3.82 days. Suppose you have a sample of gas in which the concentration of 222Rn is 3.4 × 10−4 M. What was the concentration of 222Rn in the sample three weeks earlier?

Answer 5

1.5 × 10^−2 M

Question 6

Consider the following first-order reaction.

A → B + C

The half-life of this reaction is 6.40 hours. Suppose you define the third-life (t1/3 ) of the reaction as the time at which one-third of the original amount of A remains. What is t1/3 for this reaction?

Answer 6

10 hours

Question 7

The following data are gathered for the radioactive decay of flourine-20
Time(s) 0.0 2.0 4.0 6.0
Concentration(M) 0.24 0.21 0.18 0.16
If the decay of 20F is a first-order reaction, which of the following is the best estimate of its half-life

Answer 7

10 s

Question 8

The following data are gathered for the radioactive decay of sodium-24.

If the decay of 24Na is a first-order reaction, which of the following is the best estimate of the rate constant of this reaction?

Answer 8

4.6 × 10^−2 hr−1

Question 9

Carbon-14 undergoes first-order radioactive decay and has a half-life of 5730 years. A sample of wood taken from a living tree has 3.6 times as much carbon-14 per gram as a sample taken from a wooden bowl. Approximately how old is the wood used to make the bowl? (Assume that the initial concentration of 14C in the wood of the bowl was equal to the concentration of 14C in living wood.)

Answer 9

10,600 years

Question 10

Suppose the reaction X → Y is first order in [X]. Which of the following graphs correctly depicts ln [X] versus time?

Answer 10

-

Question 11

In which of the following circumstances would the rate of a first order reaction increase?

Answer 11

when its rate constant increases and its half-life decreases

Question 12

The graph below shows ln [A] versus time for the reaction 2A → B. Which of the following conclusions can you draw about the reaction based on this graph?

Answer 12

The initial concentration of A is 2.0 M.