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A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCi. After 4.00 hours, its activity is 8.00 mCi.

Find the half-life.

Answer :

The half-life of a radioactive isotope refers to the time it takes for half of the initial amount of the isotope to decay. In this case, we are given that a freshly prepared sample of the isotope has an initial activity of 10.0 mCi and after 4.00 hours, its activity decreases to 8.00 mCi.

To find the half-life, we can use the following formula:
N = N₀ * (1/2)^(t / T₁/₂)
Where: N is the final activity (8.00 mCi in this case), N₀ is the initial activity (10.0 mCi), t is the time elapsed (4.00 hours)
T₁/₂ is the half-life (what we need to find)
We can rearrange the formula to solve for T₁/₂:
(1/2)^(t / T₁/₂) = N / N₀
Taking the logarithm of both sides of the equation:
log[(1/2)^(t / T₁/₂)] = log(N / N₀)
Using the property of logarithms, we can bring the exponent down:
(t / T₁/₂) * log(1/2) = log(N / N₀)

Now we can solve for T₁/₂:
T₁/₂ = (t * log(1/2)) / log(N / N₀)
Plugging in the given values:
T₁/₂ = (4.00 * log(1/2)) / log(8.00 / 10.0)
Simplifying the expression:
T₁/₂ = (4.00 * log(1/2)) / log(0.8)
Using a logarithm calculator, we find that log(1/2) ≈ -0.3010 and log(0.8) ≈ -0.0969.
Substituting these values into the equation:
T₁/₂ = (4.00 * -0.3010) / -0.0969
Calculating this expression, we find:
T₁/₂ ≈ 12.4 hours
Therefore, the half-life of the radioactive isotope is approximately 12.4 hours.

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Rewritten by : Barada

To find the half-life of the radioactive isotope, we can use the formula:

N(t) = N₀ * (1/2)^(t/T)

Where:
N(t) = activity at time t
N₀ = initial activity
t = time
T = half-life

Given that the initial activity N₀ is 10.0 mCi and the activity after 4.00 hours is 8.00 mCi, we can plug these values into the formula:

8.00 = 10.0 * (1/2)^(4.00/T)

To solve for T, we can rearrange the equation:

(1/2)^(4.00/T) = 8.00/10.0

Simplifying the right side:

(1/2)^(4.00/T) = 0.80

Taking the logarithm of both sides:

log(1/2)^(4.00/T) = log(0.80)

Using logarithm properties:

(4.00/T) * log(1/2) = log(0.80)

Dividing both sides by log(1/2):

4.00/T = log(0.80) / log(1/2)

Simplifying:

T = 4.00 / (log(0.80) / log(1/2))

Calculating this expression:

T ≈ 6.64 hours

So, the half-life of the radioactive isotope is approximately 6.64 hours.

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