Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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![### Radioactive Decay and Nuclear Reactions
#### A. Calculate the Amount of Radioactive Material Left
**Problem:**
Calculate the amount of radioactive material left of a 250g sample, after 30 days, whose half-life is 6 days.
**Solution Approach:**
To determine the remaining amount of a radioactive substance, we use the formula for exponential decay related to half-life:
\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^\frac{t}{T_{1/2}} \]
Where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \),
- \( N_0 \) is the initial quantity of the substance,
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the elapsed time.
Given:
- \( N_0 = 250 \) grams,
- \( T_{1/2} = 6 \) days,
- \( t = 30 \) days.
First, calculate the number of half-lives that have passed:
\[ \text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{30 \text{ days}}{6 \text{ days}} = 5 \]
Then, calculate the remaining amount:
\[ N(t) = 250 \times \left(\frac{1}{2}\right)^5 \]
\[ N(t) = 250 \times \frac{1}{32} \]
\[ N(t) \approx 7.8125 \text{ grams} \]
So, after 30 days, approximately 7.8 grams of the 250-gram sample of the radioactive material will remain.
#### B. Products of Alpha-Particle Decay
**Problem:**
Write out the products of α-particle decay of the Thorium nucleus below.
\[ ^{234}_{90} \text{Th} \rightarrow \]
**Solution Approach:**
When a nucleus undergoes alpha decay, it emits an alpha particle, which is a helium nucleus \( _2^4 \text{He} \). The new element formed will have its atomic number reduced by 2 (as it loses 2 protons) and its mass number reduced by 4 (as it loses 2 protons and 2 neutrons).
\[ ^{234}_{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e803b3c-73d2-4762-ae70-129489ba72a5%2F9b6bbfd3-bbad-4404-a45d-1b1e039df148%2Fc2v1ra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Radioactive Decay and Nuclear Reactions
#### A. Calculate the Amount of Radioactive Material Left
**Problem:**
Calculate the amount of radioactive material left of a 250g sample, after 30 days, whose half-life is 6 days.
**Solution Approach:**
To determine the remaining amount of a radioactive substance, we use the formula for exponential decay related to half-life:
\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^\frac{t}{T_{1/2}} \]
Where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \),
- \( N_0 \) is the initial quantity of the substance,
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the elapsed time.
Given:
- \( N_0 = 250 \) grams,
- \( T_{1/2} = 6 \) days,
- \( t = 30 \) days.
First, calculate the number of half-lives that have passed:
\[ \text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{30 \text{ days}}{6 \text{ days}} = 5 \]
Then, calculate the remaining amount:
\[ N(t) = 250 \times \left(\frac{1}{2}\right)^5 \]
\[ N(t) = 250 \times \frac{1}{32} \]
\[ N(t) \approx 7.8125 \text{ grams} \]
So, after 30 days, approximately 7.8 grams of the 250-gram sample of the radioactive material will remain.
#### B. Products of Alpha-Particle Decay
**Problem:**
Write out the products of α-particle decay of the Thorium nucleus below.
\[ ^{234}_{90} \text{Th} \rightarrow \]
**Solution Approach:**
When a nucleus undergoes alpha decay, it emits an alpha particle, which is a helium nucleus \( _2^4 \text{He} \). The new element formed will have its atomic number reduced by 2 (as it loses 2 protons) and its mass number reduced by 4 (as it loses 2 protons and 2 neutrons).
\[ ^{234}_{
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