The nuclear activity (rate of radioactive decay) of a given radioisotope sample changes with which of the following external conditions, if any?

Introduction / Context:
This question is about radioactivity and nuclear decay. It asks whether the activity of a radioisotope, meaning the rate at which nuclei disintegrate per unit time, depends on external factors such as temperature, pressure or chemical environment. Understanding this independence is important in nuclear chemistry, dating methods and practical use of radioisotopes in medicine and industry.

Given Data / Assumptions:

• We are considering a particular radioisotope sample.
• Possible external influences listed are temperature, pressure and chemical environment.
• Activity is defined as the number of disintegrations per unit time and is related to half life and decay constant.
• We assume normal ranges of temperature and pressure encountered in laboratory and practical conditions, not extreme astrophysical environments.

Concept / Approach:
Nuclear decay is a random process governed by the internal structure of the atomic nucleus. The decay constant and half life of a radioisotope are characteristic nuclear properties and do not depend significantly on ordinary physical or chemical conditions. While chemical reactions involve electrons in outer shells and are sensitive to temperature, pressure and chemical environment, nuclear reactions involve protons and neutrons deep in the nucleus and are largely unaffected by such external factors. Therefore, for practical purposes, the activity of a radioisotope does not change when you heat, cool, compress or chemically react the substance, provided the nuclei remain the same.

Step-by-Step Solution:
1) Recall that activity A of a radioactive sample is given by A = λN, where λ is the decay constant and N is the number of undecayed nuclei.2) The decay constant λ is a fundamental nuclear property related to the structure and energy levels of the nucleus.3) Unlike chemical reaction rates, which vary strongly with temperature and sometimes with pressure, the nuclear decay constant does not change significantly with these external conditions.4) Changing the chemical form of a radioisotope, for example by forming different compounds, alters the electron cloud but leaves the nucleus unchanged, so λ and thus the activity remain essentially the same.5) Normal laboratory changes in temperature, pressure or chemical environment do not appreciably affect nuclear decay rates.6) Therefore, none of the listed external conditions cause a meaningful change in the activity of a given radioisotope sample.

Verification / Alternative check:
Practical evidence from radiometric dating, medical imaging and industrial tracers shows that half lives of radioisotopes are extremely stable over wide ranges of physical conditions. For example, carbon 14 dating relies on the assumption that the half life of carbon 14 has been constant over thousands of years, regardless of temperature or chemical environment. Experiments with radioactive samples heated, cooled or dissolved in different chemicals show negligible changes in activity. Only in very rare cases involving electron capture and extreme conditions is there a tiny effect, which is not relevant to standard general knowledge questions. Thus, the best general statement is that the activity does not change with these external conditions.

Why Other Options Are Wrong:
Temperature of the sample: Chemical reaction rates depend on temperature, but nuclear decay rates remain essentially constant over normal temperature ranges.
Pressure applied to the sample: Reasonable pressure changes compress matter slightly but do not significantly alter nuclear structure or decay constants.
Chemical environment of the atoms: Changing compounds and bonding affects electrons but not the nucleus, so activity is not meaningfully changed.
Both temperature and pressure only: This option implies some dependence, which contradicts the nuclear nature of radioactivity.

Common Pitfalls:
A common mistake is to transfer intuition from chemical kinetics to nuclear decay and assume that increasing temperature will speed up decay. Another confusion arises from hearing about extreme conditions in stars or reactors and imagining that everyday temperature or pressure changes have similar effects. For exam purposes, remember that radioactive decay is a nuclear phenomenon governed by internal nuclear forces and is essentially independent of normal physical and chemical conditions. Therefore, when such options are presented, selecting that none of these conditions change the activity is usually correct.

Final Answer:
The activity of a radioisotope sample does not change with ordinary temperature, pressure or chemical environment, so the correct choice is none of these external conditions.