Radioactive Decay Worksheet Answers: Clear & Concise Solutions
Table of Contents :
Radioactive decay is a fascinating topic in the realm of nuclear chemistry and physics. It revolves around the process by which unstable atomic nuclei lose energy by emitting radiation. This process leads to the transformation of the original element into another element or a different isotope of the same element. In this article, we will discuss radioactive decay, the types of decay processes, and provide clear and concise answers to common questions, especially in the context of worksheets that many students encounter.
Understanding Radioactive Decay
Radioactive decay can be defined as the spontaneous transformation of an unstable atomic nucleus into a more stable one. This process is accompanied by the release of energy in the form of radiation, which can be alpha particles, beta particles, or gamma rays.
Types of Radioactive Decay
There are three primary types of radioactive decay:
-
Alpha Decay (α-decay):
- In this process, an atomic nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons.
- This leads to a decrease in the atomic number by 2 and the mass number by 4.
Example: [ ^{238}{92}U \rightarrow ^{234}{90}Th + ^{4}_{2}He ]
-
Beta Decay (β-decay):
- Beta decay occurs when a neutron in an unstable nucleus transforms into a proton, emitting a beta particle (an electron).
- The atomic number increases by 1 while the mass number remains unchanged.
Example: [ ^{14}{6}C \rightarrow ^{14}{7}N + ^{0}_{-1}e ]
-
Gamma Decay (γ-decay):
- Gamma decay involves the emission of gamma rays, which are high-energy electromagnetic radiation.
- It usually occurs after other types of decay when the nucleus is in an excited state, returning to a more stable state without changing its number of protons or neutrons.
Example: [ ^{60}{27}Co^* \rightarrow ^{60}{27}Co + \gamma ]
The Importance of Decay Constants
In studying radioactive decay, understanding the decay constant (λ) is crucial. It represents the probability per unit time that a given nucleus will decay. This can also be represented by the half-life (T½), which is the time required for half of the radioactive nuclei in a sample to decay.
Half-life Formula
The relationship between the half-life and the decay constant is given by the formula:
[ T_{1/2} = \frac{0.693}{\lambda} ]
Solving Radioactive Decay Problems: Examples and Answers
Example Problem 1: Alpha Decay
Question: A sample contains 80 g of Uranium-238. How much will remain after 4 half-lives?
Solution:
-
Find the half-life of Uranium-238: Approximately 4.5 billion years.
-
After each half-life, the amount of the radioactive isotope halves:
- After 1 half-life: ( 80 g \div 2 = 40 g )
- After 2 half-lives: ( 40 g \div 2 = 20 g )
- After 3 half-lives: ( 20 g \div 2 = 10 g )
- After 4 half-lives: ( 10 g \div 2 = 5 g )
Therefore, 5 g of Uranium-238 will remain after 4 half-lives.
Example Problem 2: Beta Decay
Question: If a sample of Carbon-14 has an initial mass of 20 g, how much will remain after 2 half-lives? (Half-life of Carbon-14 is about 5730 years)
Solution:
- After 1 half-life: ( 20 g \div 2 = 10 g )
- After 2 half-lives: ( 10 g \div 2 = 5 g )
Thus, 5 g of Carbon-14 will remain after 2 half-lives.
Example Problem 3: Calculating the Decay Constant
Question: Given the half-life of a certain isotope is 10 years, find its decay constant.
Solution: Using the half-life formula:
[ \lambda = \frac{0.693}{T_{1/2}} = \frac{0.693}{10 \text{ years}} = 0.0693 \text{ year}^{-1} ]
So, the decay constant is 0.0693 year⁻¹.
<table> <tr> <th>Decay Type</th> <th>Particle Emitted</th> <th>Change in Atomic Number</th> <th>Change in Mass Number</th> </tr> <tr> <td>Alpha Decay</td> <td>2 protons, 2 neutrons</td> <td>-2</td> <td>-4</td> </tr> <tr> <td>Beta Decay</td> <td>Electron</td> <td>+1</td> <td>0</td> </tr> <tr> <td>Gamma Decay</td> <td>Gamma radiation</td> <td>0</td> <td>0</td> </tr> </table>
Important Notes
"Always remember that the decay process is random, and we can only predict the behavior of a large number of nuclei. For individual nuclei, the exact time of decay cannot be determined."
Common Mistakes in Radioactive Decay Calculations
- Not using the correct half-life: Ensure you have the right half-life for the isotope you are dealing with.
- Misunderstanding the types of decay: Each type of decay affects the atomic and mass numbers differently.
- Not accounting for units: When calculating decay constants, ensure you have consistent units throughout the calculations.
Conclusion
Understanding radioactive decay is crucial for students and professionals alike in fields such as nuclear physics, chemistry, and environmental science. By grasping the key concepts and practicing with various problems, learners can develop a strong foundational knowledge that will serve them well in future studies. Remember, mastery of radioactive decay processes enhances your ability to engage with the wonders of nuclear science effectively.