### Video Transcript

Two samples, one gram and two grams, of the radioactive element radon-222 are compared to each other. Which of the following properties of the two samples are the same? (A) The number of radon-222 atoms, (B) the half-life, (C) the volume, (D) the amount of radiation emitted every 30 seconds, or (E) the number of unstable nuclei.

So we have two samples of radon-222 with masses of one gram and two grams. For any element or compound, the mass is directly related to the amount of matter present. In other words, the mass is related to the number of atoms. So, as one sample has double the mass of the other, it will have double the number of radon-222 atoms. Thus, option (A), the number of radon-222 atoms, cannot be a shared property of the two samples, as two samples of the same substance with different masses will not have the same number of atoms. So option (A) is not the answer to this question.

The question tells us that radon-222 is radioactive. This means that radon-222 can undergo radioactive decay. In radioactive processes, the amount of matter does not affect the rate at which a fraction of a substance decays, as determined by the half-life. The half-life is the amount of time it takes for half the substance to decay away, regardless of how much sample there is. This means that both the one-gram and the two-gram sample of radon-222 will have the same half-life. Half-life is called an intrinsic property, a property that’s the same no matter what the mass is. Extrinsic properties, on the other hand, change with mass. So it seems as though option (B), the half-life, is the same for each of the samples, which would mean that option (B), the half-life, is the correct answer to this question. But to confirm, let’s look at the other options.

Radon is a noble gas and almost always exists in a gaseous state. The volume of a gas depends on the size of its container. Gases usually fill a container and have the same volume as their container. We’ve only been given the masses of the radon-222 samples, not the volume of the container in which they are held. So, based on the information in the problem, we do not know if the two samples will have the same volume or not. So option (C), the volume, is not the answer to this question.

Option (D) is the amount of radiation emitted every 30 seconds. We mentioned that the half-life is the same regardless of the amount of sample. But the amount of decay that occurs does change. A sample emits twice as much radiation per time frame if it has twice as many radioactive particles. So the one-gram and two-gram samples of radon-222 will emit different amounts of radiation every 30 seconds. So option (D), the amount of radiation emitted every 30 seconds, is incorrect.

Option (E) is the number of unstable nuclei. As we mentioned before, mass is directly related to the number of atoms. The number of atoms of an isotope is directly related to the number of unstable nuclei. So two samples with different masses will have different numbers of unstable nuclei. Thus, option (E), the number of unstable nuclei, is not a common property of the two samples and is thus incorrect.

So the answer to the question “Which of the following properties of the two samples are the same?” is (B), the half-life.