Question

What is the rate of decay from 1.00 mol of radioactive nuclides having the following half-lives: \(12,000\) years? 12 hours? 12 seconds?

Short answer

The decay rates for the given half-lives are: 1. For 12,000 years: \( \frac{ln(2)}{12000}\ mol\ year^{-1} \) 2. For 12 hours: \( 730.5 \times ln(2)\ mol\ year^{-1} \) 3. For 12 seconds: \( 31,622,400 \times ln(2)\ mol\ year^{-1} \)

Step-by-step solution

Convert half-lives to decay constants

We will convert the given half-lives to decay constants using the formula: \( k = \frac{ln(2)}{T} \), where T is the half-life and k is the decay constant. 1. For 12,000 years: \( k = \frac{ln(2)}{12000} \) 2. For 12 hours: Convert 12 hours to years: \( 12 \frac{hours}{1\ day} \times \frac{1\ day}{24\ hours} \times \frac{1\ year}{365.25\ days} = \frac{1}{730.5}\ years \) \( k = \frac{ln(2)}{\frac{1}{730.5}} \) 3. For 12 seconds: Convert 12 seconds to years: \( 12 \frac{seconds}{1\ minute} \times \frac{1\ minute}{60\ seconds} \times \frac{1\ hour}{60\ minutes} \times \frac{1\ day}{24\ hours} \times \frac{1\ year}{365.25\ days} = \frac{1}{31,622,400}\ years \) \( k = \frac{ln(2)}{\frac{1}{31,622,400}} \)

Calculate decay rates for each half-life

We will now calculate the decay rates for each half-life by multiplying the decay constant (k) by the initial number of moles (1 mol). 1. For 12,000 years: Decay Rate = \( k \times 1\ mol = \frac{ln(2)}{12000} \times 1\ mol \) 2. For 12 hours: Decay Rate = \( k \times 1\ mol = \frac{ln(2)}{\frac{1}{730.5}} \times 1\ mol \) 3. For 12 seconds: Decay Rate = \( k \times 1\ mol = \frac{ln(2)}{\frac{1}{31,622,400}} \times 1\ mol \)

Simplify the expressions for decay rates

Finally, we will simplify the expressions for the decay rates. 1. For 12,000 years: Decay Rate = \( \frac{ln(2)}{12000}\ mol\ year^{-1} \) 2. For 12 hours: Decay Rate = \( 730.5 \times ln(2)\ mol\ year^{-1} \) 3. For 12 seconds: Decay Rate = \( 31,622,400 \times ln(2)\ mol\ year^{-1} \) So, the decay rates for the given half-lives are: 1. For 12,000 years: \( \frac{ln(2)}{12000}\ mol\ year^{-1} \) 2. For 12 hours: \( 730.5 \times ln(2)\ mol\ year^{-1} \) 3. For 12 seconds: \( 31,622,400 \times ln(2)\ mol\ year^{-1} \)

Key concepts

half-life calculation

Understanding half-life is crucial in the study of radioactive decay. The half-life of a substance is the time required for half of the radioactive nuclides in a sample to decay. This concept helps in predicting how long a radioactive material will remain active. The formula to find the decay constant from the half-life is:

• \( k = \frac{\ln(2)}{T} \)

where \( k \) is the decay constant and \( T \) is the half-life. The natural logarithm of 2, denoted as \( \ln(2) \), is a constant that is approximately 0.693.
By plugging the half-life into the formula, one can derive the decay constant, which is useful for various calculations, including determining the rate of decay. The half-life provides insight into the stability of a radioactive element and helps in planning for safety and environmental considerations.

radioactive decay

Radioactive decay is a process by which unstable atomic nuclei lose energy by emitting radiation. During this process, a nuclide transforms into a different nuclide or less unstable configuration. The rate at which this transformation occurs is characterized by the decay constant.
There are several types of radioactive decay, including alpha decay, beta decay, and gamma decay, each involving the emission of different particles or radiation. Understanding this concept is vital in fields like geology, archeology, and medicine, where radioactive decay is used in techniques such as radiometric dating and cancer treatment.

• Stability of a nuclide is defined by its half-life and decay constant.
• The process follows an exponential decay pattern, which means the number of undecayed nuclei decreases at a rate proportional to its current value.

Knowing the principles of radioactive decay allows scientists to calculate the age of ancient objects, predict future decay scenarios, and handle radioactive materials safely.

rate of decay

The rate of decay is determined by the decay constant and the number of moles of radioactive nuclides present initially. The rate tells us how quickly a radioactive substance is decaying. This rate is essential in fields that require precision in timing and decay predictability.
The formula used to calculate the rate of decay is:

• \( \, \text{Decay Rate} = k \times N_0 \, \)

where \( k \) is the decay constant, and \( N_0 \) is the initial number of moles of the substance. To break it down further:

• The decay rate allows scientists to determine how much of a radioactive substance will be present after a given time.
• By understanding the rate of decay, proper safety protocols can be followed, and the material's lifecycle managed.

Practically, the rate of decay is crucial for endeavors such as carbon dating, radio-pharmaceuticals, and nuclear energy management.

mole concept

The mole is a fundamental concept in chemistry used to describe the amount of a substance. One mole is equivalent to Avogadro's number, approximately \( 6.022 \times 10^{23} \) entities, be it atoms, molecules, or ions.
In the context of radioactive decay, the mole concept helps quantify the number of nuclides in a sample. When discussing the rate of decay, knowing the number of moles allows us to apply the decay constant effectively.

• A mole provides a bridge between atomic scale measurements and macroscopic applications.
• It simplifies complex calculations by allowing chemists and physicists to use average atomic masses.

Being a basic unit in the International System of Units (SI), the concept is pivotal in understanding chemical reactions, stoichiometry, and the understanding of gases as well. Hence, it's a cornerstone in accurately measuring and predicting behaviors in the world of chemistry and physics.