Q1) If there are 100 radioactive atoms with a half-life of 30 seconds, how many radioactive atoms will remain after one half-life (30 seconds)? How many will remain after two half-lives (60 seconds)? Three half-lives?
So basically, after 30 seconds it will be 50.After 60 sec. 25 and after 90 sec. it will be 12.5...?
am I right?
Q2) Suppose you analyzed a sample of isotope A. It contained 25 radioactive isotope A atoms, and 103 stable daughter atoms. Approximately how old is the sample?
A1) Oh, half-lives, the radioactive equivalent of "two steps forward, one step back." After one half-life (30 seconds), half of the radioactive atoms will remain, so that's 50 atoms. After two half-lives (60 seconds), we'll take another step backward, so you'd have 25 atoms. And after three half-lives (90 seconds), it's another backward step, leaving you with 12.5 atoms. So, you got the right idea! Half the fun is watching those atoms go on their dance of decay.
A2) To determine the age of the sample, we need to find out how many half-lives have passed. Starting with 25 radioactive atoms, we know that each half-life results in half the atoms remaining. So, if after each half-life we have 50% remaining, we'd need to reach 100% (i.e., 0 radioactive atoms remaining) for the sample to be considered "old." Assuming there was initially 100% (25 radioactive atoms) and the rest (103 stable daughter atoms) is the result of decay, we can calculate by repeatedly dividing 25 by 2 until we get to zero. This gives us approximately 7 half-lives. Since each half-life is 30 seconds, the age of the sample would be around 210 seconds. So, it's an ancient little thing in radioisotope years!
A1) Yes, you are correct for the first question. After one half-life (30 seconds), half of the radioactive atoms will remain, which means there will be 50 radioactive atoms left. After two half-lives (60 seconds), half of the remaining atoms will decay, so there will be 25 radioactive atoms left. After three half-lives (90 seconds), half of the remaining atoms will decay again, leaving you with 12.5 radioactive atoms.
A2) To determine the age of the sample, we can use the ratio of the number of unstable (radioactive) atoms to stable daughter atoms. In this case, the ratio is 25 radioactive isotope A atoms to 103 stable daughter atoms.
To find the age, we need to use the concept of half-life. Since during each half-life half of the radioactive atoms decay, the ratio of unstable to stable atoms will change accordingly.
In this case, the ratio is 25:103, which means that we have completed less than one half-life because the number of stable atoms is much higher than the unstable atoms.
To find the exact age, we can use the formula:
age = half-life * (number of half-lives completed + fraction of half-life remaining)
Since we are given the ratio of unstable to stable atoms, which is 25:103, we can determine that less than one half-life has occurred. Therefore, the age of the sample cannot be determined precisely with the given information.
Q1) You are correct in your understanding of radioactive decay. After each half-life, the number of radioactive atoms remaining is reduced by half.
To find the number of radioactive atoms that will remain after one half-life (30 seconds), you would divide the initial number of atoms (100) by 2. So, after 30 seconds, there will be 100/2 = 50 radioactive atoms remaining.
After two half-lives (60 seconds), you would divide the number of radioactive atoms remaining after one half-life (50) by 2 again. So, after 60 seconds, there will be 50/2 = 25 radioactive atoms remaining.
Similarly, for three half-lives (90 seconds), you would divide the number of radioactive atoms remaining after two half-lives (25) by 2 once more. Therefore, after 90 seconds, there will be 25/2 = 12.5 radioactive atoms remaining.
However, it is important to note that radioactive decay occurs on an individual atom basis, so we cannot have a fractional number of atoms remaining. In practical terms, after 90 seconds, we would expect that either 12 or 13 radioactive atoms would remain, depending on the specific scenario.
Q2) To estimate the age of the sample, you can use the concept of half-life. In this case, the stable daughter atoms indicate that radioactive decay has occurred.
From the problem statement, you have 25 radioactive isotope A atoms and 103 stable daughter atoms. This means that 25 isotope A atoms have decayed into 103 stable daughter atoms.
Since radioactive decay follows an exponential decay model, we expect half of the original isotope A atoms to decay in one half-life. So, if 25 isotope A atoms have decayed, that means that originally there were 2 * 25 = 50 isotope A atoms.
From this information, you can calculate the number of half-lives that have occurred by solving the equation 50 / 2^x = 103, where x represents the number of half-lives.
Using logarithms or trial and error, you can find that x is approximately 4.32 (rounded to two decimal places). This means that approximately 4.32 half-lives have occurred.
Given that the half-life of isotope A is known, you can multiply the number of half-lives by the length of one half-life to estimate the age of the sample. If the half-life is 30 seconds, then the estimated age would be around 4.32 * 30 seconds = 129.6 seconds (or approximately 2 minutes and 9.6 seconds).