### Video Transcript

Taking the half-life of a sample of nobelium-253 to be 1.62 minutes, how much time would pass until only one gram of nobelium-253 remained from an initial sample of 32 grams? Give your answer to two decimal places.

Half-life, given the symbol 𝑇 one-half, is the amount of time required for one-half of the radioactive nuclei in a sample to decay. Radioactive nuclei are unstable because the repulsive forces between the particles in the nucleus outweigh the attractive forces. These unstable nuclei can become stable by undergoing radioactive decay, the spontaneous emission of radiation.

So, if we had 12 atoms of radioactive iodine-131 and half of the sample decayed to form stable xenon-131, the half-life would be the amount of time it takes for this process to occur. Interestingly enough, half-life is an intrinsic property. This means that the half-life will be the same regardless of the amount of material present. In our example, six atoms decayed during the first half-life. But if we wait for another half-life, six more atoms won’t undergo decay. Instead, half of the radioactive atoms still present in the sample will decay. So, after two half-lives, three radioactive atoms of iodine-131 still remain.

We can see from this example that after a half-life has passed, one-half of the radioactive sample will still remain. With this information in mind, let’s take a look at the question. In this question, nobelium-253 is the radioactive isotope. We are told that initially the sample of nobelium-253 had a mass of 32 grams. We are also told that the half-life of nobelium-253 is 1.62 minutes. We know that after a half-life occurs, one-half of the sample will have decayed and the other half will still be radioactive nobelium-253. So the mass of nobelium-253 which remains after one half-life will be equal to 32 grams divided by two, or 16 grams.

We ultimately want to know how much time would pass until only one gram of nobelium-253 remains. So let’s continue to go through half-lives, dividing the mass of the sample by two until only one gram remains. After two half-lives, eight grams remain. After three half-lives, four grams remain. After four half-lives, two grams remain. And finally, after the fifth half-life, only one gram remains.

We now know that the sample must go through five half-lives in order to decay from 32 grams to one gram. If one half-life is 1.62 minutes, then five half-lives will be 8.1 minutes, which to two decimal places is 8.10 minutes. So the amount of time that would pass until only one gram of nobelium-253 remains is 8.10 minutes.