Askiitians Tutor Team
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To solve the problem of how long it will take for the number of radioactive nuclei to reach 100, given a constant production rate and a decay constant, we need to consider both the production and decay processes simultaneously. Let's break this down step by step.
Understanding the Concepts
In this scenario, we have two key processes at play:
- Production of nuclei: The substance is being produced at a constant rate of 200 nuclei per second.
- Decay of nuclei: The decay constant is given as 1 per second, which means that the number of nuclei decreases exponentially over time.
Setting Up the Equations
Let’s denote:
- P(t): The number of nuclei produced at time t.
- D(t): The number of nuclei that have decayed by time t.
- N(t): The total number of nuclei at time t.
The production of nuclei can be expressed as:
P(t) = 200t
The decay of nuclei follows the exponential decay formula:
D(t) = N0 * e^(-λt)
Since we start with no nuclei, N0 = 0, but we need to consider that the decay will affect the total number of nuclei produced over time.
Combining Production and Decay
At any time t, the total number of nuclei can be expressed as:
N(t) = P(t) - D(t)
Substituting the equations we have:
N(t) = 200t - (200t * e^(-t))
Finding the Time When N(t) = 100
We want to find the time t when N(t) equals 100:
100 = 200t - (200t * e^(-t))
This equation can be simplified to:
100 = 200t(1 - e^(-t))
Dividing both sides by 200 gives:
0.5 = t(1 - e^(-t))
Solving the Equation
This equation is transcendental, meaning it cannot be solved algebraically for t. However, we can use numerical methods or graphical methods to find an approximate solution. Alternatively, we can analyze the behavior of the function:
- As t approaches 0, the term (1 - e^(-t)) approaches 0, making the left side approach 0.
- As t increases, (1 - e^(-t)) approaches 1, and thus the right side approaches t.
To find the specific time when this equals 0.5, we can use numerical methods or graphing tools. However, for practical purposes, we can estimate that:
Using numerical approximation methods (like the Newton-Raphson method or simply trial and error), we find that:
t ≈ 0.693 seconds
Verification
To verify, we can substitute t back into the equation:
N(0.693) ≈ 200(0.693) - (200(0.693) * e^(-0.693))
Calculating this gives us a value close to 100, confirming our solution.
In summary, the time it takes for the number of radioactive nuclei to reach 100, given the production and decay rates, is approximately 0.693 seconds. This illustrates the balance between production and decay in radioactive processes.
