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Answer :
Final answer:
The half-life of the radioisotope is calculated using the formula for half-life that involves the natural logarithm of 2 divided by the decay rate. The decay rate is calculated using the initial and final count rates and the time passed. After calculations, the half-life is found to be approximately 304 hours.
Explanation:
The problem involves the concept of half-life associated with radioactive decay.
To solve this problem, we first need to calculate the rate of decay with the formula: k = ln(2) / t₁₂, where t₁₂ is the half-life. However, we are finding the half-life, so we manipulate the equation to be: t₁₂ = ln(2) / k.
The rate k can be calculated with the equation k = ln(N₀ / N) / Δt, where N₀ is the initial count rate (3000 cpm), N is the final count rate (2736 cpm), and Δt is the time passed (48 hours or 2880 minutes).
Substituting the given values in, we get: k = ln(3000 / 2736) / 2880 ≈ 0.000038 min⁻¹.
Then, substituting k into the half-life formula, we get: t₁₂ = ln(2) / k ≈ 18240 minutes or about 304 hours.
So, the half-life of the radioisotope is 304 hours.
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The activity of a radioisotope is 3000 counts per minute at one time and 2736 counts per minute 48 hours later, then the half life time is 365 hours.
How do we calculate the half life time?
Half life time for the first order reaction will be calculated by uisng the below equation as:
k = 0.693/t, where
t = half life time
k is the rate constant and its value will be calculatedy using the below equation as:
ln(No/N) = kT, where
No = initial concentration of radioisotope = 3000
N = remaining contentration of radioisotope = 2736
T = time require for above change = 48 hours
On putting these values on above equation, we get
k = ln(3000/2736) / 48 = 0.0019
Now putting this value in the first equation, we get
t = 0.693/0.0019 = 364.7 = 365 hours
Hence half life time is 365 hours.
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