Answer :
Approximately 0.046% of the original sample of Cesium-124 remains after 6 minutes, calculated by using the half-life of 31 seconds and applying the formula for exponential decay.
Calculating the Remaining Fraction of Cesium-124 After 6 Minutes
To determine what fraction of a sample of Cesium-124 remains after 6 minutes, we need to use the concept of half-life. The half-life of a radioactive isotope is the time it takes for half of the sample to decay. Since Cesium-124 has a half-life of 31 seconds, we can calculate how many half-lives occur in 6 minutes (360 seconds).
First, determine the number of half-lives that have passed:
360 seconds ÷ 31 seconds/half-life = approximately 11.61 half-lives.
Next, we use the formula for exponential decay:
Remaining fraction = ½^number of half-lives
Plug in the number of half-lives:
Remaining fraction = ½^11.61 = approximately 0.00046
Therefore, approximately 0.046% of the original sample of Cesium-124 remains after 6 minutes.
The correct answer is After 6 minutes, \ (\frac {1} {2^ {12}} \) or approximately \ (2.44 \times 10^ {-4} \) small fraction of a sample of Cs-124 will remain.
Cesium-124 has a half-life of 31 seconds, which means that after every 31 seconds, half of the remaining Cs-124 nuclei will decay. To find the fraction remaining after a certain time, we can use the formula \(N(t) = N_0 \times 2^{-\frac{t}{T_{\text{half}}}}\), where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the elapsed time, and \(T_{\text{half}}\) is the half-life.
Given that 6 minutes is equal to \ (6 \times 60 = 360\) seconds, plugging in the values gives \ (N (360) = 1 \times 2^ {-\frac {360}{31}} \approx 2.44 \times 10^ {-4} \), which is the fraction remaining after 6 minutes.
The half-life of a radioactive substance is the time it takes for half of the initial quantity of that substance to decay. In this case, Cesium-124 has a half-life of 31 seconds, which means that every 31 seconds, half of the remaining Cs-124 nuclei will decay into other elements. This property can be mathematically described using an exponential decay equation, \(N(t) = N_0 \times 2^{-\frac{t}{T_{\text{half}}}}\), where \(N(t)\) is the quantity remaining at time \(t\), \(N_0\) is the initial quantity, \(t\) is the elapsed time, and \(T_{\text{half}}\) is the half-life.
Given that we want to find the fraction of a sample remaining after 6 minutes (360 seconds), we can plug in the values into the equation. With \ (N_0 = 1\) (representing the initial quantity), \ (t = 360\), and \(T_{\text{half}} = 31\), we calculate \ (N (360) = 1 \times 2^ {-\frac {360}{31}} \approx 2.44 \times 10^ {-4} \). This means that only a very small fraction of the original sample will remain after 6 minutes, highlighting the rapid decay of Cs-124 due to its short half-life. The formula \( \frac{1}{2^{12}} \) gives us a simple numerical expression for the fraction remaining, which is approximately \(2.44 \times 10^{-4}\).
Learn more about the topic of small fraction.
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