20.2.1 Rates of Disintegration Reactions

Question 1

Rates of Disintegration Reactions

Answer 1

• Just as for any first-order reaction, half-lives can be used for calculations involving nuclear disintegration reactions.
• Activity is the rate of nuclear decay.
• Measurements of activities and half-lives can be used to make calculations for nuclear disintegration reactions.

Question 2

note

Answer 2

• All nuclear disintegration reactions are first-order. Therefore, the quantity of parent nuclei determines the rate that daughter nuclei are produced.
• For example, strontium-90 undergoes beta decay, producing yttrium-90 and a beta particle. This reaction has a half-life of 29 years. If 10 g of 90 Sr are present to begin with, 5 g 90 Sr will be present after 29 years, 2.5 g 90 Sr after another 29 years, etc.
• Just as for any first-order reaction, half-lives can be used for calculations involving nuclear disintegration reactions.
• Many radioactive nuclei have extremely long half-lives. This is the heart of the nuclear waste problem—once nuclear waste is formed, it will remain radioactive for a large number of years.
• For a first-order reaction, the concentration of reactant at time t is related to the initial concentration of reactant by the equation [reactant]t = [reactant] 0 e –kt .
• For a nuclear disintegration reaction, the number of nuclei is used instead of the concentration of reactant. Therefore, N = N i e –kt .
• Activity (A) is the rate of nuclear decay (A = kN). Since A is related to N by a constant, A = A i e –kt .
• The SI unit for activity is the becquerel (Bq).
  1 Bq = 1 disintegration / s.
• Another common unit for activity is the curie (Ci). 1 Ci = 3.7 x 10 10 disintegrations / s. Since 1 Ci is so large, mCi (1/1000 of a curie) are often used.
• Activity is measured using a Geiger counter. Geiger counters contain an anode surrounded by argon gas.
  When ionizing radiation such as a beta particle strikes an argon atom, an argon ion (Ar + ) is created. This allows for the completion of the circuit, and a sound is produced.
• Problem: A sample of 245 Am with a half-life of 7.37 x 10 3 y has an activity of 1.00 Bq. How many 245 Am nuclei are present?
• Since A = kN and k = 0.693 / t 1/2 , N = A · t 1/2 / 0.693.
• The half-life must first be converted to seconds. Plugging this value into the equation N = A · t 1/2 / 0.693 yields the number of 245 Am nuclei (3.35 x 10 11 nuclei).

Question 3

note 2

Answer 3

• Problem: How many 245 Am nuclei remain after 100 years?
• The number of nuclei at a given time (N) is related to the initial number of nuclei (N i ) by the equation N = N i e –kt . The half-life is used to solve for k (9.4 x 10 –5 y –1 ). Plugging the known values into the equation yields the number of 245 Am nuclei after 100 years (3.32 x 10 11 nuclei). About 99% of the nuclei remain. This makes sense, since the half-life of 245 Am is over 7000 years—after only 100 years, many more than half of the initial 245 Am nuclei remain.

Question 4

Which of the following statements about nuclear reactions is not correct?

Answer 4

At = Ai e^k t

Question 5

Which of the following radioactive isotopes would be in greatest abundance after about 28 years?

Answer 5

100 g of 210Pb, whose t½ = 22.3 y.

Question 6

A sample of 245Am has a t½ of 7.37 × 103 y and an activity value, A, of 1.00 Bq. What is the value of k, and how many nuclei are there initially and after 500 years?

Answer 6

k = 9.4 × 10−5 / y;
Ni = 3.35 × 1011 nuclei;
N500 = 3.20 × 1011 nuclei.

Question 7

Which statement about the kinetics of nuclear reactions is not correct?

Answer 7

t½ = ln (2) k = 0.693k.

Question 8

Which statement about reactions is not correct?

Answer 8

The term half-life refers exclusively to first-order nuclear reactions.

Question 9

Which statement about the activity term, A = kN = Aie−kt is not correct?

Answer 9

A is expressed in units of disintegrations.

Question 10

Which statement about nuclear reactions is not correct?

Answer 10

Almost all nuclear reactions are second-order reactions.

Question 11

Nitrogen-13 decays by beta emission to carbon-13
13 7 N -> 13 6 C + 0 +1 e
Suppose that you start with 16 g of 13 N 7 and that its half-life is 10 minutes. Which statement about this decay reaction is not correct?

Answer 11

After six half-lives, there are about 0.125 g of 13 7 N left

Question 12

Suppose that you start with 40 g of 90Sr at the beginning of the reaction below
90Sr -> 90 Y + 0 -1 e
Which statement about this reaction is not correct?

Answer 12

90Y is definitely a stable isotope.

Question 13

These are the half-lives of some heavy isotopes.

238U = 4.5 × 10^9 y
235U = 7.1 × 10^8 y
239Pu = 2.4 × 10^4 y

Which statement about half-lives is not correct?

Answer 13

After about five or six half-lives, virtually all of a starting isotope will have decayed.