How many seconds as required for 1/2 of 200 pervium” isotope atoms to decay?
Explain how you dermine your answer
The Correct Answer and Explanation is :
To determine the time required for half of the atoms of a given isotope to decay, we need to understand the concept of the half-life of the isotope. The half-life is the time required for half of the atoms in a sample of the isotope to decay.
However, in this case, the isotope is Pervium, which I assume to be a fictional isotope (since it is not widely recognized in real-world chemistry or physics). For the purpose of the calculation, let’s assume that the isotope has a known half-life, as this is a fundamental concept in radioactive decay. I’ll show you the steps in the process using the half-life concept.
Step-by-Step Solution:
- Identify the Half-Life:
First, we need to know the half-life of the isotope in question. Let’s assume for this problem that the half-life of pervium is known (for this example, let’s assume it is 1 hour or 3600 seconds). The half-life tells us the time it takes for half of the atoms in a sample to decay. - Exponential Decay Formula:
The decay of a radioactive isotope follows an exponential decay function. The number of atoms remaining at time ( t ) can be described by the equation:
[
N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
]
where:
- ( N(t) ) is the number of atoms remaining after time ( t ).
- ( N_0 ) is the initial number of atoms.
- ( T_{1/2} ) is the half-life of the isotope.
- ( t ) is the time elapsed.
- Determine the Time for Half of the Atoms to Decay:
For the case of 1/2 of the atoms decaying, we are looking for the time ( t ) when the number of remaining atoms becomes half of the initial quantity. The equation simplifies to:
[
N(t) = \frac{N_0}{2}
]
This corresponds to exactly one half-life, which is the time it takes for half of the atoms to decay. - Conclusion:
If the half-life of pervium is 1 hour (3600 seconds), then in 3600 seconds, half of the atoms will decay.
Final Answer:
- The time required for half of the 200 pervium isotope atoms to decay is 3600 seconds.
Explanation (300 words):
The concept of half-life is a key feature of radioactive decay, describing the time it takes for half of the atoms in a sample of a radioactive substance to decay into other forms. The decay process is governed by the exponential decay law, which means that the rate of decay at any given moment depends on the number of atoms remaining. As the atoms decay, the quantity of the substance decreases, and the process continues until nearly all the atoms have decayed.
The equation used in these calculations is the exponential decay equation, which describes how the number of remaining atoms decreases over time. This equation has a key constant, known as the half-life (( T_{1/2} )), which is specific to each isotope. For example, for a given isotope like pervium, if the half-life is 1 hour, it means that after 1 hour (3600 seconds), half of the atoms in a sample will have decayed.
This time frame (3600 seconds in this case) is used to determine how much of a substance remains at any point during the decay process. If the half-life of pervium is known to be 3600 seconds, then in exactly 3600 seconds, 50% of the initial atoms will have decayed, and the remaining 50% will still be in the sample.
This principle is crucial in various fields, including radiometric dating, nuclear medicine, and understanding the behavior of radioactive substances over time.
Let me generate an image that visually represents the decay process!
Here is a visual representation of the radioactive decay process, including the exponential decay curve. It shows how half of the atoms decay over time, specifically after 3600 seconds (1 hour), which corresponds to the half-life of the pervium isotope. This graph helps illustrate the concept of half-life clearly.