Question

A Geiger counter is used to measure the decay of a radioactive isotope produced in a nuclear reactor. Initially, when the sample is first removed from the reactor, the Geiger counter registers 15,000 decays/s. 15 h later the count is down to 5500 decays/s. At what time after the sample's removal from the reactor is the count 1200 decays/s?

Accepted Answer

Step 1: Recognize that this problem involves radioactive decay, which follows an exponential decay model. The decay rate can be expressed as $$ N(t) = N_0 e^{-\lambda t} $$, where $$ N(t)  is $$the decay rate at time $$ t $$, $$ N_0  is $$the initial decay rate, $$ \lambda  is $$the decay constant, and $$ t  is $$the time elapsed. Step 2: Use the given data to calculate the decay constant $$ \lambda . $$You are given $$ N_0 = 15000 $$ decays/s at $$ t = 0 $$ and $$ N(t) = 5500 $$ decays/s at $$ t = 15 $$ hours. Substitute these values into the decay equation $$ N(t) = N_0 e^{-\lambda t} $$ and solve for $$ \lambda $$: $$ \lambda = \frac{1}{t} \ln\left(\frac{N_0}{N(t)}\right) . $$Step 3: Once $$ \lambda  is $$determined, use it to find the time $$ t $$ when the decay rate is $$ N(t) = 1200 $$ decays/s. Substitute $$ N_0 = 15000 $$, $$ N(t) = 1200 $$, and the calculated $$ \lambda $$ into the decay equation $$ N(t) = N_0 e^{-\lambda t} . $$Rearrange the equation to solve for $$ t $$: $$ t = \frac{1}{\lambda} \ln\left(\frac{N_0}{N(t)}\right) . $$Step 4: Convert the time $$ t $$ from hours to the desired unit if necessary (e.g., seconds or minutes). Ensure the units are consistent throughout the calculation. Step 5: Verify the solution by substituting $$ t $$ back into the decay equation $$ N(t) = N_0 e^{-\lambda t}  to $$confirm that $$ N(t) $$ equals 1200 decays/s. This step ensures the calculation is accurate and consistent with the problem's conditions.

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주요 개념

Radioactive Decay

Exponential Decay

Geiger Counter