Radioactivity concepts: identify the single wrong statement among the following about alpha and beta emissions and radioactive decay laws.

Introduction / Context:
Conceptual questions on radioactivity often mix statements about nuclear emissions (alpha, beta) and the decay law. The task is to spot the statement that contradicts basic nuclear physics. Understanding atomic charge right after emission and the probabilistic nature of decay is key.

Given Data / Assumptions:

• Alpha (α) emission ejects a helium-4 nucleus carrying +2 charge.
• Beta-minus (β−) emission ejects an electron carrying −1 charge.
• Decay constant λ and half-life t_1/2 describe exponential decay, largely unaffected by macroscopic thermodynamic variables.

Concept / Approach:
Immediately after α emission, the electron cloud still corresponds to the parent atomic number Z, while the nucleus has Z−2; the atom is left with two excess electrons and is momentarily negative until charge re-equilibration—making option (a) acceptable in the “instant after emission” sense. After β− emission, the nucleus becomes Z+1 while the electron cloud still has Z electrons, leaving a net +1 charge until neutralization—so (b) is acceptable. Radioactive decay is exponential; even after many half-lives, complete disintegration is asymptotic, never literally 100% in finite time. Hence, (c) is the wrong statement. The decay constant is a nuclear property, insensitive to ordinary pressure, temperature, or concentration, validating (d). The exponential law description in (e) is correct.

Step-by-Step Solution:

Verification / Alternative check:
Plugging numbers into N(t) verifies residual fractions at multiple half-lives (e.g., at 2 half-lives, N/N0 = 0.25).

Why Other Options Are Wrong (i.e., why they are not the incorrect statements):

• (a) Correct in the immediate, pre-neutralization sense.
• (b) Correct in the immediate, pre-neutralization sense.
• (d) Correct: λ is insensitive to normal environmental conditions.
• (e) Correct: exponential law governs decay.

Common Pitfalls:
Interpreting (a) and (b) after electron rearrangement (ultimately atoms neutralize); forgetting that “complete” decay never occurs in finite time under the exponential model.

Final Answer:
A radioactive element with a half-life of 20 years will be completely disintegrated in 40 years.