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![### Radioactive Decay Problem
A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCi. After 3.70 hours, the activity is 7.60 mCi.
#### (a) Find the decay constant and half-life of the isotope.
The given values can be used to calculate the decay constant (λ) and the half-life (T1/2) of the isotope.
- **Initial activity (A₀):** 10.0 mCi
- **Activity after time t (A):** 7.60 mCi
- **Time elapsed (t):** 3.70 hours
To find the decay constant (λ), we use the formula:
\[ \lambda = \frac{1}{t} \ln{\frac{A₀}{A}} \]
After finding λ, the half-life (T1/2) can be calculated using the relationship:
\[ T_{1/2} = \frac{\ln{2}}{\lambda} \]
Here are the forms to fill with the values:
#### λ =
\[ 1 \boxed{} \] h\(^-1\)
#### \( T_{1/2} \) =
\[ 2 \boxed{} \] h
These calculations determine the rate at which the isotope decays over time and how long it takes for half of the sample to decay.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2F065a7e92-1feb-4707-8b70-c6275f99ca0e%2Fm79lwez_processed.png&w=3840&q=75)
Transcribed Image Text:### Radioactive Decay Problem
A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCi. After 3.70 hours, the activity is 7.60 mCi.
#### (a) Find the decay constant and half-life of the isotope.
The given values can be used to calculate the decay constant (λ) and the half-life (T1/2) of the isotope.
- **Initial activity (A₀):** 10.0 mCi
- **Activity after time t (A):** 7.60 mCi
- **Time elapsed (t):** 3.70 hours
To find the decay constant (λ), we use the formula:
\[ \lambda = \frac{1}{t} \ln{\frac{A₀}{A}} \]
After finding λ, the half-life (T1/2) can be calculated using the relationship:
\[ T_{1/2} = \frac{\ln{2}}{\lambda} \]
Here are the forms to fill with the values:
#### λ =
\[ 1 \boxed{} \] h\(^-1\)
#### \( T_{1/2} \) =
\[ 2 \boxed{} \] h
These calculations determine the rate at which the isotope decays over time and how long it takes for half of the sample to decay.
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